I
ITERATIVE METHODS FOR THE SOLUTION OF EQUATIONS
J. F. TRAUB
BELL TELEPHONE LABORATORIES, INCORPORATED
MURRAY HILL, NEW JERSEY
i
- ii -
TO SUSANNE
/
- Ill -
PREFACE
This book presents a general theory of iteration
algorithms for the numerical solution of equations and systems of equations.
The relationship between the quantity
and quality of information used by an algorithm and the
efficiency of the algorithm are investigated.
Iteration
functions are divided into four classes depending on whether
they use new information at one or at several points and
whether or not they reuse old information.
Known iteration
functions are systematized and new classes of computationally
effective iteration functions are introduced.
Our interest
in the efficient use of information is influenced by the widespread use of computing machines.
The mathematical foundations of our subject are
treated with rigor but rigor in itself is not the main object.
Some of the material is of wider application than to our
theory.
Belonging to this category are Chapter 3 , "The
Mathematics of Difference Relations"; Appendix A, "Interpolation
and Appendix D "Acceleration of Convergence".
The inclusion
of Chapter 1 2 , "A Compilation of Iteration Functions" permits
the use of this book as a handbook of iteration functions.
Extensive numerical experimentation was performed on a computer;
a selection of the results are reported in Appendix E.
1
The solution of equations is a venerable subject.
Among the mathematicians who have made their contribution
are Cauchy, Chebyshev, Euler, Fourier, Gauss, Lagrange,
Laguerre, and Newton.
E, Schroder wrote a classic paper
on the subject in 1 8 7 0 . A glance at the bibliography indicates the level of contemporary interest.
Perhaps the most
important recent contribution is the book by Ostrowski;
papers by Bodewig and Zajta are also outstanding.
Most of the material is new and unpublished.
Every attempt has been made to keep the subject in proper
historical perspective.
Some of the material has been orally
presented at meetings of the Association for Computing
Machinery in 1 9 6 l , 1 9 6 2 , 1 9 6 3 , the American Mathematical
Society in 1 9 6 2 and 1 9 6 3 * and the International Congress of
Mathematicians in 1 9 6 2 .
I wish to acknowledge with sincere appreciation the
assistance I have received from my friends and colleagues at
Bell Telephone Laboratories, Incorporated.
indebted to M. D. McIlroy,
I am particularly
J. Morrison, and H. 0 . Pollak for
numerous important suggestions.
I want to thank A. J. Goldstein,
R. W. Hamming, and E. N. Gilbert for stimulating conversations
and valuable comments.
My thanks to Professor G, E. Forsythe
of Stanford University for his encouragement and comments during
the preparation of the manuscript and for reading the final manuscript.
My appreciation to S. P. Morgan and Professor A. Ralston
T
- v -
of Stevens Institute of Technology who also read the manuscript.
I am particularly grateful to J. Riordan for suggesting numer/—
ous improvements in style.
I want to thank Miss Nancy Morris for always digging up just one more reference and Mrs. Helen Carlson for
editing the final manuscript.
^
I am grateful to
Mrs. Elizabeth Jenkins for her splendid supervision of the
preparation of the difficult manuscript and to Miss Joy Catanzaro
for her speedy and accurate typing.
To my wife, for her never-failing support and encouragement as well as her assistance in editing and proofreading,
I owe the principal acknowledgment.
J. P. TRAUB
- vi -
TABLE 0 F CONTENTS
1
PREFACE
TERMINOLOGY
GLOSSARY OF SYMBOLS
1.
2.
GENERAL PRELIMINARIES
1.1
Introduction
1.2
Basic Concepts and Notations
1.21
Some concepts and notations
1.22
Classification of iteration functions
1.23
Order
1.24
Concepts related to order
GENERAL THEOREMS ON ITERATION FUNCTIONS
2.1
The Solution of a Fixed Point Problem
2.2
Linear and Superlinear Convergence
2.21
Linear convergence
2.22
Superlinear convergence
2.23
The advantages of higher order iteration
functions
2.3
The Iteration Calculus
2.31
Preparation
2.32
The theorems of the iteration calculus
- vii -
3.
THE MATHEMATICS OF DIFFERENCE RELATIONS
3.1
Convergence of Difference Inequalities
3.2
A Theorem on the Solutions of Certain
Inhomogeneous Difference Equations
3 .3
3.4
On the Roots of Certain Indicial Equations
3.31
The properties of the roots
3.32
An important special case
The Asymptotic Behavior of the Solutions of
Certain Equations
4,
3.41
Introduction
3.42
Difference equations of type 1
3.43
Difference equations of type 2
INTERPOLATOR! ITERATION FUNCTIONS
4.1
Interpolation and the Solution of Equations
4.11
Statement and solution of an interpolation
problem
4.12
Relation of interpolation to the calculation
of roots
4.2
The Order of Interpolatory Iteration Functions
4.21
The order of iteration functions generated
by Inverse interpolation
4.22
The equal information case
4.23
The order of Iteration functions generated
by direct interpolation
4.3
Examples
1
- vili -
ONE-POINT ITERATION FUNCTIONS
5.1
The Basic Sequence E
5.11
5.12
The formula for E
s
An example
0
5#13
5.2
g
The structure of E
s
R a t i o n a l Approximations to E
s
o
o
5.21
Iteration functions generated by
r a t i o n a l approximation to E
5.22
5 3
#
a
The formulas of Halley and Lambert
A Basic Sequence of Iteration Functions
Generated by Direct Interpolation
5.31
5.4
5.32
The basic sequence $
ojs
The iteration function $
5.33
Reduction of degree
q
The Fundamental Theorem of One-Point Iteration
Functions
5.5
The Coefficients of the Error Series of E
s
Q
5.51
A recursion formula for the coefficients
5.52
A theorem cohcerning the coefficients
ONE-POINT ITERATION FUNCTIONS WITH MEMORY
6.1
Interpolatory Iteration Functions
6.11
Comments
6.12
Examples
- ix -
6.2
Derivative Estimated One-Point Iteration
Functions with Memory
6.21
The secant iteration function and its
generalization
6.3
6.22
Estimation of
6.23
Estimation of g ^ " ^
6.24
Examples
1
f^" )
1
Discussion of One-Point Iteration Functions
with Memory
6.31
A conjecture
6.32
Practical considerations
6.33
Iteration functions which do not use
all available information
6.34
An additional term in the error equation
MULTIPLE ROOTS
7.1
Introduction
7.2
The Order of E
7.3
The Basic Sequence &
s
7.31
Introduction
7 . 3 2 The structure of g
g
g
7.33
7.4
Formulas for g
s
The Coefficients of the Error Series of g
R
7.5
Iteration Functions Generated by Direct
Interpolation
7.51
The error equation
7.52
On the roots of an indicial equation
7.53
The order
7.54
Discussion and examples
7.6
One-Point Iteration Functions with Memory
7.7
Some General Results
7.8
An Iteration Function of Incommensurate Order
MULTIPOINT ITERATION FUNCTIONS
8.1
The Advantages of Multipoint Iteration Functions
8.2
A New Iterpolation Problem
8.21
A new interpolation formula
8.22
Application to the construction of
multipoint iteration functions
8.3
Recursively Formed Iteration Functions
8.31
Another theorem of the iteration calculus
8.32
The generalization of the previous theorem
8.33
Examples
8.34
The construction of recursively formed
iteration functions
8.4
Multipoint Iteration Functions Generated by
Derivative Estimation
8.5
Multipoint Iteration Functions Generated by
Composition
8.6
Multipoint Iteration Functions with Memory
i
- xi -
9.
MULTIPOINT ITERATION FUNCTIONS:
CONTINUATION
9.1
Introduction
9.2
Multipoint Iteration Functions of Type 1
9.3
9.4
9.21
The third order case
9.22
The fourth order case
Multipoint Iteration Functions of Type 2
9.31
The third order case
9.32
The fourth orjder case
Discussion of Criteria for the Selection of
an Iteration Function
10.
ITERATION FUNCTIONS WHICH REQUIRE NO EVALUATION
OF DERIVATIVES
10.1
Introduction
10.2
Interpolatory Iteration Functions
10.3
11.
10.21
Direct Interpolation
10.22
Inverse Interpolation
Some Additional Iteration Functions
SYSTEMS OF EQUATIONS
11.1
Introduction
11.2
The Generation of Vector-Valued Iteration
Functions by Inverse Interpolation
11; 3
#
Error Estimates for Some Vector-Valued
Iteration Functions
1
- xii -
11.31
The generalized Newton iteration
function
11.32
A third order iteration function
11.33
Some other vector-valued iteration
functions
11.34
11.4
A test function
Vector-Valued Iteration Functions which
Require No Derivative Evaluations
12.
A COMPILATION OF ITERATION FUNCTIONS
12.1
Introduction
12.2
One-Point Iteration Functions
12.3
One-Point Iteration Functions with Memory
12.4
Multiple Roots
12.41
Multiplicity known
12.42
Multiplicity unknown
12.5
Multipoint Iteration Functions
12.6
Multipoint Iteration Functions with Memory
12.7
Systems of Equations
APPENDICES
A.
INTERPOLATION
A.l
Introduction
A.2
An Interpolation Problem and Its Solution
A.21
Statement of the problem
A.22
Divided differences
A.23
The Newtonian formulation
A.24
The Lagrange-Hermite formulation
A.25
The interpolation error
!
i
- xiii -
A.3
A.4
The Equal Information Case
A.31
The Newtonian formulation
A.32
The Lagrange-Hermite formulation
A.33
The interpolation error
The Approximation of Derivatives
A.41
Statement of the problem
A.42
Derivative estimation from the
Newtonian formulation
A.43
Derivative estimation from the
Lagrange-Hermite formulation
A.5
The Error in the Approximation of Derivatives
A.51
Discussion
A.52
First proof of the error formula
A.53
Second proof of the error formula
ON THE jTH DERIVATIVE OF THE INVERSE FUNCTION
SIGNIFICANT FIGURES AND COMPUTATIONAL EFFICIENCY
ACCELERATION OF CONVERGENCE
D.l
Introduction
2
D.2
Aitken s 5
D.3
The Steffensen-Householder-Ostrowski Iteration
f
Function
Transformation
- xiv -
E«
NUMERICAL EXAMPLES
E.l
Introduction
E.2
Growth of the Number of Significant Figures
E.3
One-Point and One-Point with Memory Iteration
Functions
F>
E,4
Multiple Roots
E.5
Multipoint Iteration Functions
E.6
Systems of Equations
AREAS FOR FUTURE RESEARCH
BIBLIOGRAPHY
r
I
-
XV
-
TERMINOLOGY
Page
asymptotic error constant
1.2-12
basic sequence
1.2-18
derivative estimated iteration function
6.2-3
informational efficiency
1.2-16
informational usage
1.2-16
interpolatory iteration function
4.1-5
iteration function
1.2-4
multipoint iteration function
1.2-11
multipoint Iteration function with memory
1.2-11
one-point iteration function
1.2-10
one-point iteration function with memory
1.2-10
optimal iteration function
1.2-18
optimal basic sequence
1.2-18
order
1.2-12
order is multiplicity-dependent
1.2-13
order is multiplicity-independent
1.2-13
I
I
- xvi -
GLOSSARY OP SYMBOLS
The following list, which is intended only for
reference, contains the symbols which occur most frequently
in this book.
- xvii -
a
j
A
j
(
x
f
)
( j )
(x)/j!
aj(x)/a (x)
( x )
1
Lagrange-Hermite coefficient
a zero of f
a
a
B,
(x)
,Hm-l| )
x
mam(x)
w v
B s ,n
1,3
P
k,a
7
( a )
A - ; » ( t )
t=x.
dominant zero of g, ( t )
Q
C
asymptotic error constant
C[x,l]
binomial coefficient
7
(t)
'1,3
Newton coefficient
inf orma11ona1 us age
d
(s)
1,3
(t)
'1,3
e
x - a
e
x^ - a
i
EPF
t=X.
informational efficiency
ir
- xviii -
Page
E
a certain family of iteration
g
5*1-2
functions
*E o
n»o
"4s
a family of iteration functions
generated by derivative estimation
a family of iteration functions
Q
n, o
6.2-5
6.2-14
generated by derivative estimation
S fx,f.m) = E ( x , f
l/m
o
f
,l)
function whose zero is sought
S
S
*f£ )
an estimate of f^ ^
f[x ,7 ;
±
0
x
±
-
1
,7 j
7.3-3
1.2-1
6.2-5
x _ ,7 ]
1
1
n
n
confluent divided difference
A.2-4
f
a certain kind of interpolatory
function
8.2-2
3
the inverse function to f
1.2-2
1
a£ )
S
an estimate of
6.2-14
k-1
g
k j a
(t)
k
t - a £
tJ
3.3-1
the inverse of the Jacobian matrix
11.2-1,
LP.
iteration function
1.2-4
I
class
class oi
of iteration function of
order p
1.2-13
- xix -
class of Iteration functions with
informational usage d and order p
Jacobian matrix
g (x,f,m) =
s
Z-\ ^(m)(x-a)
1
t
the multiplicity of a
the number of points at which old
information is reused
0
u(x) = Sv^x-a)
order (in sense of order of
magnitude)
mu(x) = 2cD^(m)(x-a)
order
hyperosculatory polynomial for f
names of Iteration functions
family of iteration functions
generated by Inverse interpolation
family of Iteration functions
generated by direct Interpolation
family of iteration functions
generated by rational approximation to E
s
hyperosculatory polynomial for 3
s(n+l)
-
XX
-
Page
R
6.2-4
S(n+1)
s
p
s,j
( m )
P,3
T
S
s+1
(x,f,m) = x - £
p ^ ( m ) Z (x,f,l)
j=l
s - 1 derivatives are used in many
functions
1.2-17
s - 1
6.2-4
Stirling numbers of the first kind
7.3-7
Wjx,f,m) = ^
7.3-5.
a ^(m)Zj(x,f,l)
t
Stirling numbers of the second kind
1
E (x) =
Zt U-a)
g
f(x)/f'(x)
V(x)
cp(x) - a
(x-a)
1.2-6
2.3-3
P
cp(x) - a
u (x)
2.3-5
P
y (x,t,m)
=
3
l/m
Z (x,t ,l)
3
approximant to a
Yj(x)
( • l )
J
" V
j
)
f y )
jU*'(y)]
Z j
(x)
ij • • «
J
Y (x)u (x)
J
7.3-7
5.5-4
tta
u(x)
W ( x )
7.3-4
s
7.3-3
1.2-4
1.2-7
J
y=f(x)
5.1-13
11.3-9
- xxi -
Page
approximate equality between
numbers
1.2-8
same order of magnitude
1.2-8
the set of x for which the
proposition p(x) is true
1.2-9
1.0-1
CHAPTER 1
GENERAL PRELIMINARIES
The basic concepts and notations to be used
throughout this book will be introduced in Section 1.2.
1.1
Introduction
The general area into which this book falls may be
labeled algorithmics.
By algorithmics we mean the study of
algorithms in general and the study of the convergence and
efficiency of numerical algorithms in particular.
More specifically, we shall study algorithms for
the solution of equations.
Our approach will be to examine
the relationship between the quantity and quality of information used by an algorithm and the efficiency of that algorithm
We shall investigate the effect of reusing old information and
of gathering new information at certain felicitous points.
Our interest in the efficient use of information is
influenced by the widespread use of high-speed computing
machines.
The introduction of computers has meant that many
algorithms which were formerly of only academic interest
become feasible for calculation.
They are, in fact, used many
times in many establishments on a wide variety of problems.
The efficiency of these algorithms is therefore most important
Furthermore, there are situations where the acquisition of
more than a certain amount of data is prohibitively expensive.
It is then imperative that as much information as possible be
squeezed from the available data.
Here again the question of
efficiency is of paramount importance.
I
1.1-2
Iteration algorithms for the solution of equations
will be studied in a systematic fashion.
In the course of
this study, new families of computationally effective iteration algorithms will be introduced and certain well-known
iteration algorithms will be identified as special cases.
It is hoped that this comprehensive approach will moderate,
if not prevent, the rediscovery of special cases.
This
uniform approach will lead to certain natural classification
schemes and will permit the uniform rigorous error analysis
and the uniform establishment of convergence criteria for
families of iteration algorithms.
The final verdict on the
usefulness of the new methods will not be available until
the new methods have been tried on a variety of problems arising in practice.
At present, however, extensive numerical
experimentation on test problems support the theoretical error
analysis.
Although we shall confine ourselves to the solution
of real equations and systems of real equations, the field of
potential application of this work is of much broader scope.
Thus, analogous techniques may be applied to such problems as
the solution of differential and integral equations and the
calculation of eigenvalues.
The generalization of our results
to abstract spaces is of interest.
The reader is referred to
Appendix P for some additional discussion of this point.
\ i
I
1.2-1
1.2
Basic Concepts and Notations
1.21
Some concepts and notations.
The symbols to
be introduced below are global; they will preserve their
meaning for the remainder of this book.
A few of these
symbols will be used with other than their global meanings
(for example, as dummy indices in summation) but context will
always make this clear.
Our investigation concerns the approximation of a
zero of f(x), or equivalently, the approximation of a root
of the equation f(x) = 0 .
The zero and root formulations
will be used interchangeably.
There does not seem to be a
good phrase for labeling this problem.
If one says that one
is solving an equation, one may be concerned with the solution of a differential equation.
The adjectives algebraic
and transcendental are usually used to distinguish between
the cases where f is or is not a polynomial.
Perhaps the
best generic term is root-finding.
We will restrict ourselves to f(x) which are real
single-valued functions of a real variable, possessing a
certain number of continuous derivatives in the neighborhood
of a real zero a.
The number of continuous derivatives
assumed will vary upwards from zero.
zeros is not essential.
The restriction to real
With the exception of Chapter 11,
f(x) will be a scalar function of a scalar variable; in
Chapter 11, f(x) will be a vector function of a vector
variable.
r
1.2-2
The "independent variable" will sometimes appear
explicitly and will sometimes not appear explicitly.
This
will depend on whether the function or the function evaluated
at a point in its domain is what is meant.
depend on the readability of formulas.
f and f(x).
See Boas F l . 2 - 1 ,
It will also
Thus we will use both
pp. 67-681.
Derivatives of f will be denoted either by f
(l)
' with
v
f(°) = f, or by f' f» ...
• If f does not vanish in a
(l)
neighborhood of a and if f ' is continuous in this neighbor(l)
9
9
v
hood, then f has an inverse 3, and !r ' is continuous in a
neighborhood of zero.
A zero a is of multiplicity m if
m
f(x) = (x-a) g(x),
where g(x) is bounded at a and g(a) is nonzero.
always take m as a positive integer.
We shall
Ostrowski [ 1 . 2 - 2 ,
deals with the case that m is not a positive integer.
Chap. 5 ]
If
m = 1 , a is said to be simple; if m > 1 , a is said to be
nonsimple.
If a is nonsimple, it is called a multiple zero.
Perhaps the most primitive procedure for approxi-
mating a real zero is the following bisection algorithm.
a and b be two points such that f(a)f(b) < 0 .
Let
Then f has at
least one real zero of odd multiplicity on (a,b).
For the
purpose of explaining the algorithm it is sufficient to take
the interval as (0,l) and to assume that f ( 0 ) < 0 , f(l) > 0 .
Calculate f(J>).
If f Q ) = 0 , a zero has been found.
If
f(l) < 0 , the zero lies on (|,l) and one next calculates f ( | ) .
If f(^) > 0 , the zero lies on ( 0 , - | ) and one next calculates
f(-j^).
The zero will either be found by this procedure or the
zero will be known to lie on an interval of length 2 ~
the bisection operation has been applied q times.
q
after
By then
estimating the value of the zero to be the midpoint of this
last interval, the value of the zero will be known to within
a maximum error of 2 ~ ^ " " .
1
Once the zero has been bracketed
it takes just q evaluations of f to achieve this accuracy.
Observe that the accuracy to which the zero may be found is
limited only by the accuracy with which f may be evaluated.
Indeed, it is only necessary to be able to decide on the
sign of f.
The bisection algorithm just described is an example
of a one-point iteration function with memory, as defined
below.
Since no use is made of the structure of f, such as
the values of its derivatives, the rate of convergence is not
very high.
On the other hand, the method is guaranteed to
converge.
Gross and Johnson [ 1 . 2 - 3 ] use a property of f to
achieve faster convergence.
and Lehmer [ 1 . 2 - 5 ] .
See also Hooke and Jeeves
[1.2-4]
Kiefer [ 1 . 2 - 6 ] gives an optimal search
strategy for the case of a maximum of a unimodal function.
By using additional information we can do far better
than the bisection algorithm.
The natural information avail-
able are the values of f and the values of its derivatives.
We shall use the words information, samples, and data quite
interchangeably.
Let
Let
x
x
x
•••> i-n
x
i> i-i9
n
+
1
a
PP
r o x l m a n t s
t o
a
-
be uniquely determined by information obtained at
x
x
t l i e
i* j__]_* • • • * i - *
n
into
b e
x
i
+
1
be called cp.
x
X
function that maps -J/ JL_I> • • •
Thus
We call cp an iteration function.
The abbreviation I.F. will
henceforth be used for iteration function and its plural.
We shall also write y _j for f(x _^)
±
and
±
Since the information used at
x
i
_j
a r e
for
Let
^^^..j)-
the values of f and
its derivatives, we may write
(l )
Q
cp = cp
i'
i'
'
l
7
i-l'
x
> ±-n
i-l'
'
1-1
'
'
(t )"
n
l-rr
l-rr
l-n
(1-2)
We shall not permit I.F. as general as thisj the types of
I.F. to be studied will be specified in Section 1.22.
I
1.2-5
Rather than writing c p ( x , x _ , .. . , x _ ) , it is more
i
convenient to write cp or cp(x) .
i
1
i
n
Observe further that cp is a
functional depending on f and should perhaps be written cp(x,f) ,
This notation is not necessary and will therefore not be used
except in Chapter 7 .
Even the simplest iteration algorithm must consist
of initial approximation(s), an I.F., and numerical criteria
for deciding when "convergence" is attained.
be concerned with the I.F.
We shall only
Thus, for us, iteration algorithm
will be the same as I.F.
Two I.F. are almost universally known.
Newton-Raphson I.F. and the secant I.F.
cp(x.)
-
x .
-
They are the
For the former,
(1-3)
— 7 ,
while for the latter,
x
cp(x ,x _ ) = x
1
1
1
±
- f
±
x
l l-l
f -f
I l-1
I
I
f
f
l * i-l«
We shall henceforth call the former Newton's I.F.
I.F. is closely related to regula
falsi.
(1-4)
The secant
This last method,
as it is usually defined, keeps two approximants which
bracket the root; the secant I.F. always uses the latest two
approximants.
V
1.2-6
In much of our work we shall deal not with f, but
with a normalized f defined by
u =
I
9
f' ^ 0 .
If f' = 0 , f ^ 0 , the u is undefined.
(1-5)
If f' = 0 , f = 0 ,
which is the case at a multiple zero, we define u = 0 .
The
importance of u in our future work cannot be overestimated.
One reason for its importance is that
11m
x
a x-a
m
For simple zeros,
11m
x
1
a
u(x)
x-a
=
1,
Now, x - a is the error in x but is not known till a is known.
On the other hand, u is known at each step of the iteration.
In the new notation, Newton's I.F. becomes
cp = x - u,
(1-6)
For later convenience we introduce some additional
notation.
Let
e
i
=
x
i ~ a,
e = x - a.
(1-7)
1.2-7
Thus e
i
is the error of the ith approximant.
a x)
.
j (
A
V
(
x
Let
»
)
3 It'{x)>
;
•HfflB
J , m
a
v
(
x
)
.
J
(
x
ma(x) '
)
(y)
=
g
(
J
)
(1-8)
( y )
( - p ^ ^ ' f y )
J ! [ s
,
( y ) ]
J
y-f(x)
The first of these is the Taylor series coefficient
for f, while the second is a "normalized Taylor series coefficient" for f.
The third is a "generalized normalized Taylor
series coefficient" which will be used when the multiplicity
m is greater than unity.
Note that B. -,(x) = A.(x).
The
fourth is a "normalized Taylor series coefficient" for 3.
The last Is a Taylor series coefficient for 3 with a different
normalization and with f(x) substituted after differentiation.
We shall use the symbols
the following conventions:
0 , ~ and ^ according to
1.2-8
If
lim g(x.) = C,
l — oo
1
we shall write
g(x ) - C
1
or
If
lim g(x) = C,
x -* a
we shall write
g(x)
C
or
How the limit is taken will be clear from context.
where C is a nonzero constant, we shall write
f = 0(g)
or
f * Cg.
For approximate equality between numbers, the symbol
be used.
Thus
i(i+v^) ~ 1 . 6 1 8 ^ 1 . 6 2 .
The use of these symbols is illustrated in
1.2-9
EXAMPLE 1 - 1 .
Let
M ef
+ L e\,
±
i+1
±
where
e, ^ 0 ,
and where e
±
M.
K / 0,
N
±
- L ^ 0,
Is nonzero for all finite i.
e
+
i+i - *A
Then
e
^( i>
since
L
e
i i
L.
Also,
e
±
+
*
1
M e
1
,
+ L e
1
since
= M
±
±
-+ M.
We shall use the notation
= {x|p(x)|
to denote the set of x for which the proposition p(x) is true
Thus
J = [x||x| < lj
denotes the interior of the unit interval.
1-2-10
1.22
Classification of iteration functions.
We
shall classify L P . by the information which they require.
Let
be determined only by new information at x^.
information is reused.
No old
Thus
x
i+l
=
1
'
Then cp will be called a one-point I.F.
9
t " )
Most I.F. which have
been used for root-finding are one-point I.F.
The most commonly
known example is Newton's I.F.
Next let
be determined by new information at x.^
and reused information at x.
i-1*
x
i+l
=
X
x
x. .
i-n
#
Thus
x
<P(I'* i - l " • ^ i-n^ •
Then cp will be called a one-point I.F. with memory.
U"
1 0
)
The
semicolon in (1-10) separates the point at which new data are
used from the points at which old data are reused.
This type
of I.F. is now of special interest since the bid information
is easily saved in the memory (or store or storage) of a
computer.
The case of practical interest is when the same
information, the values of f and f' for example, is used at
all points.
The best-known example of a one-point I.F. with
memory is the secant I.F.
1.2-11
Now let
x ,x
1
i - 1 >
be determined by new information at
.. . , x _ , k ^
1
1.
k
No old information is reused.
Then cp will be called a multipoint I.F.
known examples of multipoint I.F.
Thus
There are no well-
They are being introduced
because of certain characteristic limitations on one-point
I.F. and one-point I.F. with memory.
Finally let
at
x
• • • ^i-k
x
i> i-i>
a
be determined by new information
n
d
#
, x
reused information x ^ ^ - l ' * * i - n
#
Thus
x
i-l-l
=
x
x
9
x
x
x
^ i ' i - 1 * * * * i*-k* i-k-l* * * * * i - n ^
9
ft > k.
(1-12)
Then cp will be called a multipoint I.F. with memory.
The
semicolon in (1-12) separates the points at which new data
are used from the points at which old data are reused.
There
are no well-known examples of multipoint I.F. with memory.
1.2-12
1.23
Order.
the order of an I.F.
converging to a.
We turn to the important concept of
Let x ,x ,...,x ,... be a sequence
Let e
Q
1
1
i
= x^ - a.
If there exists a real
number p and a nonzero constant C such that
6
1
+
ei
1
(1-13)
"C,
p
e
then p is called the order of the sequence and C is called the
asymptotic error estimate.
The remainder of this section will consist of
commentary on this definition.
We wish to associate the
concept of order with the I.F. which generates the x ^
emphasize this point, we can write ( 1 - 1 3 )
I cp(x) - a|
lv±-L—^-^
|x-a|
c#
To
as
/
,v
(1-14)
p
We will associate an order to an I.F. whether or not the
sequence generated by cp converges.
The order assigned to
an I.F. is the order of the sequence it generates when the
sequence converges.
Recall that cp is a functional which depends on f.
Hence the order of cp may be different for different classes
of f.
For the type of f and cp that we shall study, we shall
insist that cp be of a certain order for all f whose zeros are
1.2-13
of a certain multiplicity.
We shall always assume that we
are in the neighborhood of an isolated zero of f and that the
order possibly depends on the multiplicity of the zero but is
(With the exception of Chapter 7 ,
otherwise independent of f.
the zero will always be simple unless the contrary is explicitly
stated.)
To indicate that cp belongs to the class of I,P. of
order p, we shall write
cp e I .
p
(1-15)
The following definitions relate order to multiplicity.
If the order is the same for zeros of all multi-
plicities, we say that the order is multiplicity-independent.
If the order depends on the multiplicity we say that the
order is multiplicity-dependent.
If in particular the order
is greater than one for simple zeros and one for all nonsimple
zeros, we say that the order is linear for all nonsimple zeros.
(The adjectives linear and quadratic will sometimes be used
instead of first and second.)
Observe that if the order exists then it is unique.
Assume that a convergent sequence has two orders, p^ and Pg.
Let p
Q
= p
n
+ 6 , 6 > 0.
Then
1,2-14
and
lim:.
i -+ oo
=
9 ,
which contradicts the assumption that the last limit is nonzero.
If the order is integral, then the absolute values
may be dropped in the definition of order.
We shall find that
I.F. with memory never have integral order.
If cp(p+D
is
continuous and if
cp(x)
-
a
=
C ( x - a )
then cp is of order p and C - c p
VP;
p
[ l
+
(1-16)
O ( x - a ) ] ,
Equation ( l - l 6 ) may
(a)/p!.
be written as
cp(x)
-
a
~
C ( x - a )
p
.
In his classic paper of 1 8 7 0 , E. Schroder
defined order as follows:
cp(a)
=
a;
c p ^ ( a )
= 0 ,
[1.2-7]
c p ( x ) is of order p if
J
= l,2,...,p-l;
c p ^ ( a )
f
Q.
(1-17)
This definition is only valid for I.F. of one variable with p
continuous derivatives.
Although many authors have followed
Schro'der's lead, we prefer to state ( 1 - 1 7 )
of Theorem 2 - 2 .
in the conclusion
However, a generalization of ( 1 - 1 7 ) will
serve as the definition of order for the case of systems of
equations in Chapter 1 1 .
1.2-15
The advantage of high-order I.F. will be discussed
in Section 2 . 2 3 .
EXAMPLE 1 - 2 .
For Newton's method,
f i + i - A* \a),
(a)
i
2
2
A
h
2
=
-.
2 f /
e
For the secant method,
|
e
±
, ^p
|
-
A (a),
p = i(l+v/5)
-
1.62.
In Section 7 - 8 we will give an example of an I.F.
whose order is incommensurate with the scale defined by ( 1 - 1 3 ) .
1.2-16
1.24
Concepts related to order.
A measure of the
Information used by an I.P. and a measure of the efficiency
of the I.P. are required.
A natural measure of the former
is the informational usage, d, of an I.F. which we define as
the number of new pieces of information required per iteration.
Since the information to be used are the values of f and its
derivatives, the informational usage is the total number of
new derivative evaluations per iteration.
We use here the
convention that the function is the zeroth derivative.
Ostrowski [ 1 . 2 - 8 , p. 1 9 ] has suggested the "Horner" as the
unit of informational usage.
To indicate that cp belongs to
the class of I.F. of order p and informational usage d we
write
(1-18)
To obtain a measure of the efficiency, we make the
following definition:
The informational efficiency. EFP, is
the order divided by the informational usage.
EPF = d
Thus
#
(1-19)
Another measure of efficiency called the computational
efficiency, which takes into account the "cost" of calculating
different derivatives, will be discussed in Appendix C.
An
alternative definition of informational efficiency is
*EFF = p
(1-20)
1.2-17
Ostrowski [ 1 . 2 - 9 , p. 2 0 ] calls *EFF an efficiency index; see
Appendix C for a discussion of *EFF.
Our definition of
informational efficiency was chosen because it permits the
simple statement of certain important results.
EXAMPLE 1 - 3 .
p = 2,
For Newton's I.F.,
d = 2,
EFP = 1 ,
*EFF = ^ 2 .
For the secant I.F.,
P = 4(1+^5)
-
1-62,
d = 1,
EFP = *EFF = i ( l + v / 5 ) .
For one-point I.F. and one-point I.F. with memory
there can be only one new evaluation of each derivative.
If
the first s - 1 derivatives are used, the informational
usage will be s and
EPF
(1-21)
= ^.
Let n be the number of points at which old information is reused in a one-point I.F. with memory.
r = s(n+l).
Let
(1-22)
1.2-18
Hence r is the product of the number of pieces of new information with the total number of points at which information is
used.
The quantities s, n, and r will characterize certain
families of I.F.
These symbols will occasionally be used
with other meanings as long as there is no danger of confusion
We shall prove in Section 5 ~ 4 that for one-point I.F
EFF
1.
We anticipate this result to define a one-point I.F.
as optimal if EFF = 1 .
For optimal one-point
I.F.,p=d=s.
A basic sequence of I.F. is an infinite sequence of I.F. such
that the pth member of the sequence is of order p.
cept is defined only for I.F. of integral order.
This con-
An optimal
basic sequence is a basic sequence all of whose members are
optimal.
In Section 5 - 1 the properties of a particular basic
sequence, which will be used extensively throughout this book,
will be developed.
2.0-1
CHAPTER 2
GENERAL THEOREMS ON ITERATION FUNCTIONS
In this chapter I.F. will be discussed without
regard to their structure.
In Section 2.1 the existence and
uniqueness of the iterative solution of a fixed point problem
will be proven under the assumption that the I.F. satisfies
a Lipschitz condition.
In Section 2.2 the difference between
linear and superlinear convergence will be studied in some
detail while in Section 2.3 an "iteration calculus" will be
developed for I.F. which possess a certain number of continuous derivatives.
2.1
The Solution of a Fixed Point Problem
We propose to study the solution of
cp(x) = x
(2-1)
by the iteration
x
(2-2)
x
1 +
i = q>( i) •
If a satisfies ( 2 - l ) , then a is called a fixed point of cp.
The problem of finding the fixed points of a function occurs
in many branches of mathematics.
To see its relation to the
solution of f(x) = 0 we proceed as follows.
function
Let g(x) be any
such that g(a) is finite and nonzero.
cp(x) =
x
Let
(2-3)
- f (x)g(x) .
Then a is a solution of f(x) = 0 if and only if a is a solution of
(2-1).
We shall show that under certain hypotheses the
fixed point problem has a unique s o l u t i o n a n d that the iteration defined by ( 2 - 2 ) converges to this solution.
Because of
the restrictive nature of our hypotheses* the proof will be
very simple .
Many other proofs have been given both for real
functions and in abstract spaces and under a variety of
hypotheses.
[2.1-1],
Collatz [ 2 . 1 - 2 ,
Ford [ 2 . 1 - 4 ] ,
Chap. 4 ,
See, for example Antosiewicz and Rheinboldt
Chap. II], Coppel
Householder [ 2 . 1 - 5 ,
[2.1-3],
Sect. 3 - 3 ] , John
Sect. 6 ] , and Ostrowski [ 2 . 1 - 7 ,
Chap. 4 ] .
[2.1-6,
»
2.1-2
We assume that cp is defined on some closed and
bounded interval J = [a,b] and that its values are in the same
interval.
Then if x
is in J, all the x. are in J.
o
i
To
9
guarantee that ( 2 - 1 ) has a solution we must assume the continuity of cp. We have
LEMMA 2 - 1 .
J = [a,b] to J.
Let cp be a continuous function from
Then there exists an a, a < a < b, such that
cp(a) = a.
PROOF.
Since the function is from J to J,
cp(a) ^ a,
Let h(x) = cp(x) - x.
cp(b) £ b.
Then
h(a) ^ 0,
h(b) £ 0 ,
and the intermediate value theorem guarantees the existence
of a such that h(a) = 0 .
The result follows immediately.
In order to draw additional conclusions we must
impose an additional condition on cp. Let
Ms)
- cp(t)| < L|s-t|,
for arbitrary points s and t in J.
0 < L < 1,
(2-4)
Observe that this
Lipschitz condition implies continuity.
that the solution of cp(x) = x is unique.
We can now show
i
li
2.1-3
LEMMA 2 - 2 .
Let cp be a function from J to J which
satisfies the Lipschitz condition ( 2 - 4 ) . Then cp(x) = x has
at most one solution.
PROOF.
tions,
and
Assume that there are two distinct solu.
Then
a
1
=
cp(a ),
1
a
2
=
cp(a )
2
and
a
l " 2
a
=
a
" <p( ) •
2
An application of ( 2 - 4 ) yields
l ^ - a g l
=
|cp(a )
x
-
cp(a )|
2
^ L l o ^ - a i g l
<
l a ^ - a g l ,
which is a contradiction.
The existence and uniqueness of a solution a of the
equation cp(x) = x has now been verified.
It is a simple
matter to show that the sequence defined by x^ ^ = cp(x^)
converges to this solution.
THEOREM 2 - 1 .
Let J be a closed, bounded interval
and let cp be a function from J to J which satisfies the
Lipschitz condition
2.1-4
|cp(s) - cp(t)| << L|s-t|,
for arbitrary points s and t in J.
= cp(x^) .
point of J and let
0 1 L <
Let X
q
1,
be an arbitrary
Then the sequence {x^}
converges to the unique solution of cp(x) = x in J.
PROOF.
Lemmas 2 - 1 and 2 - 2 assure us of the exist-
ence and uniqueness of a solution a of the equation cp(x) = x,
This solution is the candidate for the limit of the sequence
defined by x^ ^ = cp(x ) .
Thus
i
x
1 + 1
- a = cp(x ) - a = cp(x ) - cp(a) .
±
±
From the Lipschitz condition we conclude that
L < 1.
l i+l" ' ^ Ljx^al ,
x
a
Thus
| x
and x^
±
- a |
£
L
1
| x
Q
- a |
a.
If one did not know that c p ( x ) = x had a unique solution, one would have to show that the x
criterion.
i
satisfy the Cauchy
A proof based on the Cauchy criterion may be found
in Henrici [ 2 . 1 - 8 ,
Chap. 4 ] .
1
B
2.1-5
An estimate of the error of the pth approximant
which depends only on the first two approximants and the
Lipschitz constant may be derived as follows.
Prom the
Lipschitz condition we can conclude that for all i,
K + i ~
x
i l
^
l 1
x
I I"
x
0
Let p and q be arbitrary positive integers.
X
X
p+q " p
=
( p+q- 4<i-l>
x
x
+
X
X
( p+q-l- p+q-2
P
(
I •
)
2
~
5
)
Then
+ ^pH-l'V
+
and
X
X
l p+q- pl * l
X
P +
q-
X
P +
q-ll
' p + q - r p - K l - 2 • + ••• + l p l- pl
+
X
X
X
X
+
An application of ( 2 - 5 ) shows that
|x
p
p + q
q
1
- x | £ L (l+L+...+L " )|x -x |,
p
1
o
or
Let q —• oo.
Then
' V
a
|
x
x
l l- ol«
This gives us a bound on the error of the pth approximant.
Observe that if L is close to unity, convergence may be very
slow.
A good part of our effort from here on in will be
devoted to the construction of I.P. which converge rapidly.
To do this we shall have to impose additional conditions on cp
2.2-1
2.2
Linear and Superlinear Convergence
In Section 2.1 the analytical problem, cp(x) = x,
suggested the iterative procedure
= cpCx^) . This is not
typical of the problems on which we will work.
In general we
will be given the equation f = 0 and we will want to construct
a functional cp which will be used as an I.F.
In Section 2.1
we showed that if cp satisfies a Lipschitz condition, then the
iteration will converge to the unique solution of the fixed
point problem.
From now on, rather than seeking weak suffi-
cient conditions under which the iterative sequence converges,
we will be interested in constructing I.F. such that the
sequence converges rapidly.
rather strong conditions.
We will be quite ready to impose
In particular, we shall assume that
f has a zero in the interval in which we work.
2.2-2
2.21
Linear convergence.
Let us assume that cp'
is continuous in a neighborhood of a.
We will certainly
insist that
(2-7)
cp(a) = a.
For if x.
a and cp is continuous, then
a =
lim
i
—•
x
± + 1
=
oo
lim cp(x ) = cp( lim x ) = cp(a)
i —• oo
i —• oo
±
±
Now
x
i+l
=
x
°P( i)
=
a
<P( )
+
c
x
a
P'(^i)( i" )^
where £^ lies in the interval determined by x^ and a.
An
application of ( 2 - 7 ) yields
x
i+l ~
a
" c p ^ ^ ) (x -a)
±
(2-8)
We shall require that |cp'(£^)| <i K < 1 in the neighborhood
of a.
Since cp' is continuous it will be sufficient to require
that | cp' (ct) | £ K < 1 .
Then there is a neighborhood of a such
that
0 £ L < 1
I cp' (x) | £ L,
and
x
a
±+l~ \
^
L
x
i~ l•
a
Then
l ^ - a l
and x
±
—• a.
£
L
| x
0
- a
«
2.2-3
If a is such that for every starting point x
Q
in a
sufficiently small neighborhood of a the sequence {x^} converges to a, then a is called a point of attraction.
terminology is due to J. P. Ritt.
See Ostrowski
This
[2.2-1,
p. 2 6 ] for a discussion of this and the definition of point
of repulsion.
follows.
In this terminology we can state our result as
If cp' is continuous in a neighborhood of a and if
cp(a) = a and |cp'(a)| <C L < 1 , then a is a point of attraction.
We can conclude something more.
ent from zero.
Let e
i
Let
cp'(a)
be differ
Then cp' does not vanish in a neighborhood of a
= x^ - a.
Then ( 2 - 8 ) may be written as
e
It is clear that if e
o
e
i+l " $'(£>±) ±-
is not zero and cp' does not vanish,
^
then e.^ does not vanish for any finite i.
That is, the itera-
tion cannot converge in a finite number of steps as long as
the iterants lie in the neighborhood of a where cp' does not
vanish.
Hence
e
i+l
,/> \
and
e
-
i+1 _
e
•
•/ \
cp'(a)
i
This is the case of linear or first-order convergence.
2.2-4
In the case of superlinear convergence It is not
necessary to require that
|cp'(a)|
<
converges in some neighborhood of a.
1; the iteration always
ir
i
2.2-5
2.22
Superlinear convergence.
P ^ 11*
is continuous.
, is
cp(a) = a.
We now assume that
As before we require that
Then
P
q/ ^U
x
i + l
<P( i.)
=
x
=
a
+
cp' ( a ) ( x
±
- a )
+
. . .
+
where £^ lies in the interval determined by x
should be clear that
cp
)
^
is of order p only if
j = l,2,...,p-l, and if c p ^ (a) is nonzero.
( x
i
±
- a )
and a.
cp^^(a)
p
,
It
0,
=
Hence we impose
the following conditions on cp:
cp(a)
=
a;
=0,
<p^ ^(a)
J
= l,2,...,p-l;
j
? 0.
cp^ ^(a)
P
(2-9)
Then
<p
p
s-
= x
1
- a.
p )
,
Uj)
p
±—
pi
i+1
where
(
p^
i'
P
p
Since q/ ^(a) is nonzero, q/ ^ does not
vanish in some neighborhood of a.
Then the algorithm cannot
converge in a finite number of steps provided that e
Q
is
nonzero and that the iterants lie in the neighborhood of a
p
where q/ ^ does not vanish.
Section 2 . 2 3 . )
(See also the beginning of
Furthermore,
fj±i-.<p
e
i
(p)
(«)
( 2
-
1 0
)
Hartree [2.2-2] defines the order of an I.F. by
the conditions (2-9) . This definition of order cannot be
used for one-point I.F. with memory.
Hence we prefer to
state these conditions as part of a theorem.
We summarize
the results in
THEOREM 2-2.
Let cp be an I.F. such that c p ^ is
continuous in a neighborhood of a.
e
x
a
Let j_ = j_ ~ *
Then cp
is of order p if and only if
cp(a) = a;
cp^(a) = 0 ,
J = l,2,...,p-l;
c p ^ ( a ) ^ 0.
Furthermore,
e
l+l
P
e
(p)
, q> (a)
Pi- '
Recall that we assign an order to cp whether or
not the sequence generated by cp converges.
Sufficient condi-
tions for a sequence to converge are derived below.
2.2-7
EXAMPLE 2-1.
Let f"'be continuous.
Let cp = x - f / f ,
(Newton's I.P.).
A direct calculation reveals that
cp(a) = a, cp'(a) = 0, cp"(a) = f"(a)/f'(a) ^ 0.
Hence Newton's
I.F. is second order and
A (a),
2
A
2
f"
= 2f' •
We require that f"' be continuous in order to satisfy the
hypothesis of Theorem (2-2) that cp" be continuous.
It will be
shown in Chapter 4 that we only need f" continuous.
We observed in Section 2.21 that a sequence formed
by a linear I.F. need not converge in any neighborhood of a.
We shall show that a sequence formed by a superlinear I.F.
always converges in some neighborhood of a.
In the following
analysis the cases of linear and superlinear convergence are
handled jointly.
We start our analysis from
(2-11)
Let
i
2.2-8
(the notation is specified at the end of Section 1 . 2 1 ) and
let c p ^ ^ be continuous on J.
Let x
P
p
J Ux)
e J and let
Q
I
for all x e J.
Since x
Q
e J,
£
M I
0
e |
M,
Q
r .
Hence
Let
MT
= L < 1.
P - 1
(2-12)
Then
|
Hence
e J.
e
|
i
1 LT <
r .
We now proceed to prove by induction that if
( 2 - 1 2 ) holds, then for all i,
x
±
Assume that ( 2 - 1 3 ) holds.
|e
i + 1
l
|e | £ L^T.
e J,
(2-13)
±
Then
= i M j I e j P ^ M l e j P ^ l e J
£ m|r|
p _ 1
|e | i l l t
j
±
= L
1
+
1
r < r .
r
2.2-9
Hence ( 2 - 1 3 ) holds for 1 + 1 which completes the proof by
induction.
Since
L < 1,
1
| e | 1 L ]?,
±
we conclude that x
±
-* a.
THEOREM 2 - 3 .
We have proven
p
Let q/ ^ be continuous on the
interval J,
J
Let x
Q
= |x |x-a| <
r^.
e J and let
|q>
for all x e J.
( p )
(x)l
£
Let
MT
Then for all 1 , x^ e J and x
i
P - 1
< 1.
-»• a.
M
2.2-10
2.23
functions.
The advantages of higher order iteration
If cp' Is continuous in a neighborhood of a and if
cp is first order, then the sequence generated by cp will converge to a if and only if |cp'(a)| < 1 unless one of the x
becomes equal to a.
i
That the last restriction is necessary
is shown by the example
cp(x)
=
i
2x
for
|x| £ 1
for
|x| > 1
Here cp' is certainly continuous in the neighborhood of zero
and
cp'(O)
2 but the iteration converges for all X
=
unit interval.
Baring such cases,
than unity for convergence.
cp'(a)
q
in the
will have to be less
If, on the other hand, p > 1 and
c p ^ ^ is continuous in the neighborhood of a, then there is
always a neighborhood of a where the iteration is guaranteed
to converge.
This is one of a number of advantages of superlinear
convergence over linear convergence.
Perhaps the most impor-
tant advantage is roughly summarized by the statement that
x
a
i+l S
x^.
r
e
e
s
with a to p times as many significant figures as
See Appendix C.
The higher order I.F. will often require fewer
total samples of f and its derivatives.
Recall that the
informational efficiency of an I.F. is given by the ratio of
i
ir
2.2-11
the order to the number of pieces of new information required
per iteration.
As will be shown in Chapter 5 , there exist
I.F. of all orders whose informational efficiency is unity.
We defined such I.F. as optimal.
To simplify matters, let cp
and ^ be optimal I.F. of orders 2 and 3 respectively.
x^ be generated from x
Q
by the application of cp three times
and let y^ be generated from x
twice.
tion,
Let
Q
by the application of ^
Although each process requires six pieces of informao
q
- a = 0[(x -a) ] whereas y^ - a = 0[(x -a)^].
Q
Q
Ehrmann [ 2 . 2 - 3 ] considers a certain family of I.F. whose
members have arbitrary order.
He discusses which order is
"best" under certain assumptions.
The main drawbacks to the use of high order onepoint I.F. are the increasing complexity of the formulas and
the need for evaluating higher derivatives of f.
In
Chapters 8 and 9 we shall see how these disadvantages may be
overcome by using multipoint I.F.
Observe that if f satisfies a differential equation
it may be cheaper to calculate the derivatives of f from the
differential equation than from f itself.
In particular, let
f satisfy a second order differential equation.
and f
After f
have been calculated, f" is available from the differ-
ential equation.
By differentiating the differential equation
one may calculate f"'.
as it is feasible.
This process may be continued as long
1
i
2.2-12
It might appear that it is difficult to generate
I.F. of higher order for all f.
Such is not the case and
algorithms for constructing I.F. of arbitrary order have been
given by many authors.
Ehrmann [ 2 . 2 - 5 ] ,
See, for example, Bodewig
Ludwig [ 2 . 2 - 6 ] , E. Schroder
Schwerdtfeger [ 2 . 2 - 8 ] ,
and Zajta [ 2 . 2 - 9 ] .
[2.2-4],
[2.2-7],
Many of these
algorithms depend on the construction of I.F. such that
cp(a) = a,
q/ )(a) = 0 ,
J
(2-l4)
j = l,2,...,p-l.
We shall propose numerous techniques for generating I.F. of
arbitrary order which also satisfy certain other criteria.
It will turn out that it is not necessary to use ( 2 - l 4 )
to
construct these I.F.
From two I.F. of first order, it is possible to
construct an I.F. of second order by the technique of
Steffensen-Householder-Ostrowski.
The reader is referred to
Appendix D.
In many branches of numerical analysis one constructs approximations by insisting that the approximation
be exact for a certain number of powers of x.
There is one
class of I.F., those generated by direct interpolation
(Section 4 . 2 3 ) ,
for which the I.F. yield the solution a in
one step for all polynomials of less than a certain degree.
This is not a general property of I.F.
In fact, powers of x
could not be used if for no other reason than that x
root of multiplicity m.
m
has a
2.2-13
EXAMPLE 2 - 2 .
Let
m
cp
- x -
X
jr?'
Then cp is second order and yields a in one step if f is any
linear polynomial.
Let
if
Then if/ is second order.
f
2
= x - YT + f .
Let f = x.
Then ^ will not yield the
answer in one step and will not converge if |x | > 1 .
o
2.3
The Iteration Calculus
We will develop a calculus of I.F.
The results to
be proven in this section will play a key role in much of the
later work.
They will not apply to I.F. with memory.
The
nature of our hypotheses will be such that the order will be
integral.
the I.F.
No assumptions will be made as to the structure of
Theorems 8 - 1 and 8 - 2 also belong to the iteration
calculus but are deferred to Chapter 8 because their applications are in that Chapter.
Some of these results, for the
case of simple roots, may be found in the papers by Zajta
[2.3-1]
and Ehrmann [ 2 . 3 - 2 ] .
2.3-2
2.31
Preparation.
In this section we do not
restrict ourselves to simple zeros.
The multiplicity of the
zero is denoted by m and we indicate that cp is a member of
the class of I.F. whose order is p by writing cp e I . We
always insist that cp is of order p for all functions f whose
zeros have a certain multiplicity.
However cp may be of
order p for simple zeros but of a different order for multiple
zeros.
cp e I
Hence our hypotheses contain phrases such as "Let
for some set of values of m."
EXAMPLE 2 - 3 -
Let u = f/f
and let
u
cp = x - u,
1
Then cp-^ e I
zeros.
2
f°
r
cp = x - mu,
2
simple zeros whereas cp-^ e 1^ for multiple
On the other hand, cp e I
2
and cp^ e I
2
cp^ = * -
2
for zeros of multiplicity m
for zeros of arbitrary multiplicity.
See
Chapter 7 for details.
P
Let ^ / ^ be continuous in a neighborhood of a.
If cp e I , then from Theorem 2 - 2 ,
p
cp(a) = a;
cp^(a) = 0
for
j = l,2,...,p-lj
cp^(a) ^ 0
(2-15)
2.3-3
The expansion of cp(x) in a Taylor series about a yields
cp(x)
-
=
a
^ p V
0
U - a )
p
(2-16)
,
where i lies in the interval determined by a and x.
Now,
P
i is not a continuous function of x but q/ ^(£) may be
defined as a continuous function of x as follows.
X
V(x) = <P< ) (x-a)
q
for
x ^ a,
V ( a )
-
p
^
Let
,
p
{
a
)
.
*
Then V(x) is continuous wherever cp(x) is continuous and x ^ a.
A p-fold application of L'Hospital's rule reveals that
l
x
l
m
v
a
(
x
)
.
a ^ l a l .
p
•
Hence
P
cp(x) - a = V(x)(x-a) ,
(2-17)
where V(x) is a continuous function of x and V(a) is nonzero.
Thus one may characterize I.F. of order p by the condition
that cp(x) - a has a zero of multiplicity p at a.
By comparing ( 2 - 1 7 ) with ( l - l 4 ) we observe that
C = V(a)
where C is the asymptotic error constant.
2.3-4
Equation ( 2 - 1 7 ) may be put into a form which is
more convenient for many applications.
Let f' be continuous
in a neighborhood of a and let a be a simple root.
Then f'(a)
is nonzero and
f (x) = f '(T])(x-a) .
Let
M x ) = Jkp-
for
x ^ a,
* ( a )
= f'(a) .
Then
f(x) -
(2-18)
X ( x ) ( x - a ) ,
where A(x) is continuous if f is continuous and 7\(a) / 0,
Prom ( 2 - 1 7 ) and ( 2 - 1 8 ) we conclude that
cp(x) - a = T(x)f (x),
(2-19)
p
p
where T(x) = V(x)/A (x).
Then T(x) is continuous wherever
cp^ ^(x) and f'(x) are continuous and f'(x) does not vanish.
p
Furthermore,
T(a) -
(P)
a
<P ( ) ^ 0 .
pi[f'(a)]
n
p
If the multiplicity m is greater than unity, then
f is no longer proportional to x - a and so we cannot obtain
an expression like ( 2 - 1 9 ) which involves f.
However,
r
2.3-5
u = f/f' has only simple zeros and hence ( 2 - 1 9 )
generalized.
is easily
Let f ( ) be continuous and let a have multim
plicity m with m j> 1 .
Proceeding as before we conclude the
x
s
existence of continuous functions A-^(x) and ^g( )
f(x)
=
A ( x ) ( x - a )
1
m
,
f ' ( x )
=
^ ( x M x - a )
1
u
1
c
1
h
-
that
1
,
with
Hence
u(x)
= -fj^y
=
p(x)(x-o),
(2-20)
with
P(a)
Finally, from ( 2 - 1 7 )
and
- i .
(2-20),
(2-21)
p
cp(x) - a = W(x)u (x),
and W(x) continuous and with
W ( x ) = ^ L ,
w
(
a
)
=
p
The importance of ( 2 - 1 7 ) ,
future work cannot be overestimated.
have occasion to use ( 2 - 2 1 )
a
^
p
p (x)
i
a
I
^
0
.
'
(2-19),
and ( 2 - 2 1 )
in our
In particular, we will
in the form
= cp(x) - W ( x ) u ( x ) .
p
(2-22)
I
2.3-6
2.32
The theorems of the Iteration calculus.
The
hypotheses of the theorems to be proven below will not be
weighed down by the statement of continuity conditions on the
derivatives of cp and f; they should be clear from the theorems.
The notation is the same as in Section 2 . 3 1 .
THEOREM 2 - 4 .
Let
cp (x)
x
I
e
some set of values of m and let
cp^(x)
these values of m,
.
PROOF.
cp (x)
e
Q
I
J
cp ( )
3
p
e
x
2
p
=
cp fcp (x)].
=
a
2
f
o
Then for
1
1 2
(2-17),
From
p
P-i
cp (x)
1
=
a
•+
V
1
r
2
( x ) ( x - a )
,
cp [cp (x)]
=
cp (x)
2
+
V
( x ) ( x - a )
2
2
.
Then
cp (x)
3
=
2
1
cp (x)
-
3
a
=
V
a
2
+
[ c p
1
V
2
[
C
p
( x )
1
( x ) ] V ^
2
][<p (x)
-
1
( x ) ( x - a )
P
l
P
2
af ,
2
,
Let
P
V (x)
V
=
3
2
[ c p
1
( x ) ] V
2,
1
d
( x )
The fact that
C
3
=
V
3
( a )
completes the proof.
=
V
2
( a ) V
P
2
( a )
=
C
^
2
± 0
(2-23)
2.3-7
NOTE.
the
of
Observe
a s y m p t o t i c
t h e
e r r o r
a s y m p t o t i c
c o n s t a n t
e r r o r
COROLLARY.
some
where
Then
s e t
of
t h e
f o r
t h e s e
g i v e s
o f
c o m p o s i t e
t h e
p .
i
=
p
e
o f
cp(x)
cpj
=
a n y
I
,
p
I
=
f o r
I . F .
terms
i n
t h e
c o n s t i t u e n t
=
1,2,
. .
m.
I
o f
1,2,
. , k
Cqpj
(...(cpj
m,
of
q>(x)
e
. . . , k ,
then
( x ) ) . . . ) )
t h e
numbers
l
.
l
p
<p(x)
2 * *
,
p
l , 2 , . . . , k .
I
n
p a r t i c u l a r ,
k
e l , , .
K
p
EXAMPLE
qp (x)
1
2-4.
=
L e t
cp (x)
=
2
x
-
u ( x ) ,
(Newton's
I . F . ) .
Then
$1 e ^2'
^2
1
6
9
C
2
l
=
2
C
=
A
2^ ^
a
A
2
83
7
2f *
Hence
cp (x)
3
=
I . F .
L e t
p e r m u t a t i o n
v a l u e s
f o r
formula
o f
p
i f
t h e
L e t
v a l u e s
a r e
(2-23)
c o n s t a n t s
cp^(x)
f o r
t h a t
cp [cp (x)]
2
1
6
l
4
,
C
=
[ A
2
( a ) ]
3
.
I
2.3-8
THEOREM 2 - 5 set of values of m.
Let <p(x) e I , with p > 1 , for some
Then for these values of m there exists
a function H(x) such that
cp(x) - x = - u(x)H(x),
H(a) ± 0 .
Since cp(a) = a, there exists a function G(x)
PROOF.
with G(a) = 0 such that cp(x) =• x - G(x) .
cp'(a)
=0
=
1 -
G'(a)
Then for p > 1 ,
.
Therefore G(x) has a simple zero at a and G(x) = u(x)H(x) for
some H(x) with H(a) V 0 .
Theorem 2 - 5 may be rephrased as follows.
If a is a
p-fold zero, p > 1 , of the function c p ( x ) - a, then it is a
simple zero of the function
and
cpgCx)
=
x
- f
cp(x)
x .
Since
ep-^(x)
-
x
-
f(x)
are both of order one, the theorem is
( x )
false when p = 1 .
EXAMPLE 2 - 5 if the order of c p ( x )
Since < p ( x ) - x has only simple zeros
is greater than unity, any I.F. which is
of order p for functions f ( x ) with simple zeros will be of
order p if f ( x ) is replaced by c p ( x )
replaced by c p ( x )
- x .
- x in Newton's I.F.,
Thus if f ( x ) is
II
I
2.3-9
and f(x)
e Ig.
Let cp(x) be Newton's I.P.
Then
u(x)
x
This generalization of Newton's I.P. is of order two for roots
of arbitrary multiplicity; it will be discussed in Chapter 7.
THEOREM 2 - 6 .
Let <p, (x) e I
some set of values of m.
and cp (x)
n
0
e
I_
for
Then for these values of m there
exists a function U(x), with U(x) bounded at a and U(a) ^ 0 ,
such that
p
<P (x) = cp (x) + U(x)u (x),
2
x
with
p = minfp-^Pg],
if
p
2
? p ;
2
(P-,)
p = p ,
P
>
P
x
if
p
lf
if
P
2
x
= p
=
P
2
2
and
and
x
(PT)
(a) £ cp
(a);
(a) = cp
(a).
2
^
2
1.
2.3-10
PROOF.
t h a t
p
>
1
p
Case
1.
p
p
/
1
Assume
in
p a r t i c u l a r
Then
P-i
cp-^x)
=
a
cp (x)
-
cp^x)
2
+
W
( x ) u
x
=
-
u
p
L
( x ) ,
cp (x)
2
=
a
+
W
( x ) u
P
^(x)
W
2
( x )
-
W
1
x
2
- P
( x ) ,
2
( x ) [ u ( x ) ]
D e f i n e
The
f a c t
t h a t
U(x)
=
W
2
( x )
-
U(a)
=
W
2
( a )
^
2.
p
Case
=
1
p
W
0
( x ) [ u ( x ) ]
p
c o m p l e t e s
( P
cp
and
2
1
n
l - P
2
t h e
)
p r o o f
of
Case
( P J
x
x
(a)
/
( x )
-
cp
( a ) .
2
P-,
q> (x)
2
D e f i n e
U(x)
U(a)
=
= W
c o m p l e t e s
2
W
2
( a )
t h e
"
cp^x)
( x )
-
-
W
p r o o f
=
-
L
W - ^ x ) .
{ a )
l
u
of
-
^
Case
[
( x ) [ W
2
The
f a c t
"
(
^
a
W
^ ( x )
=
cp[ )(a)
q
( x ) ,
3.
c p
2
^
cp^ ^(a)
q
P
x
t h e r e
and
=
P
)
-
^
e x i s t s
t h e
(
a
)
(a)
.
?
0
2.
and
2
( x ) ] .
t h a t
( P i ) ,
Case
1
an
p r o o f
(a)
i n t e g e r
may
be
(Pn)
=
q,
q>
q
2
>
p
c o m p l e t e d
U n l e s s
such
as
t h a t
a b o v e .
1
!
2 . 3 - H
The case of most interest for later applications is
( P
p
=
l
p
1>,
,
<Pl>, X
^ ^2
*
2 ' ^1
THEOREM 2 - 7 .
W
e
P
r o v e
a
converse to Theorem 2-6,
Let q>-(x) e I
, and let
l
cp (x) = cp (x) + U(x)u (x) for some set of values of m.
p
p
2
Then
1
for these values of m, cp (x) e l . , where
p
P
P
P
= min[p,p ],
2
if
1
=
2
if
p ,
p
x
p ^
= p
2
P l
-
and
U(a) ^ -
4
P )
(a)m
p l
j
p)
cp{ (a)m
P
>
2
if
p ,
p
Case 1 .
PROOF.
p
<p (x) = a + W (x)u
x
x
= p
1
and
Q
p ^ p .
1
Assume p > P .
Then
1
l
p
(x),
cp (x) = <p (x) + U(x)u (x),
2
(x)
1
W ( x ) + U(x)[u(x)]
1
P
P P l
.
l
The fact that
W ( a ) = W-^a) ^ 0 completes the proof of Case 1 .
2
P
1
Define W ( x ) = W ( x ) + U(x)[u(x)]
2
p
U(a) = -
Hence
cp (x) - a = u
p
r
2.3-12
Case 2.
p = P
P
1
p
and U(a) f - <pj ^ (a)m /p J.
Then
cp (x) - a = uP(x)[W (x) + U(x)].
2
1
Define Wg(x) = W ^ x ) + U(x) .
The fact that
4 (a)mP
W ( a ) = W (a) + U(a) =
,
+ U(a) ^ 0
p )
1
2
1
p
completes the proof of Case 2.
Case 3 .
P = P
P
1
p
and U(a) = - cpj ^ (a)m /p J .
This
case may be completed in a similar manner.
The "comparison theorem" just proved will permit
us to deduce the order of a given I.F. if it differs by terms
of order u
p
from an I.F. whose order is known.
'*
The following theorem permits the calculation of
the asymptotic error constant of an I.F. of order p if the
asymptotic error constant of another I.F. of order p is known.
Recall that the asymptotic error constant C is defined by
C =
l
l
m
x -» a
- «
(x-a)
(2-24)
p
Absolute values are not required in (2-24) since p is an
integer.
r
2.3-13
THEOREM 2 - 8 .
Let cp-^x) e I
some set of values of m.
o
(
»
I
e
p
for
Let
)
-
-
;
/
x
(x-a)
Let C.^ and
and cpg(x)
,
)
p
be the asymptotic error constants of cp^ and cpg
Then for these values of m,
C
= C
1
11m
+
x
PROOF.
G(x).
a
A p-fold application of L'Hospital's rule
yields
4
p
llm G(x) = -s
x — a
EXAMPLE 2 - 6 .
)
( « )
p
Then cp-^Cx) e I
2
±
p )
(°)
= C
d
'
- C
1
Let
cp (x) = x - mu(x),
1
q>i
cp (x) = x 2
•
and cp (x) e Ig, for m arbitrary.
As will be
2
shown in Section 7 * ^ 1
mu(x) = x - a - ^ i " !
(x-a)
m^ '
2
3
+ 0[(x-a) ],
r
2.3-14
where a (x) = f^ '(x)/ml.
Then It is easy to verify that
m
1
~
raa
C
m
m
2
-
Since
=
^[cp (x)
+
1
and C
lim
m
x
as predicted by Theorem 2 - 8 .
cp(x)
2a
m+1
m+1
_
n
G(x) = -
"m+1
ma
a
9
m
Let
cp (x)]
=
2
x
-
| u ( x )
m + u'(x)
are equal in magnitude but opposite in sign,
2
it is clear that cp(x) e
for all m.
A more useful result than the one obtained in
Theorem 2 - 8 is given by
THEOREM 2 - 9 .
Let ^ ( x ) e I
for some set of values of m.
cp (x)
2
H(x)
cp
1
ir
2
(x)
^
a,
(x)
u
=
1
cp (x)
. Then for the values of m,
2
= C
I
•
be the asymptotic error constants of
C
e
f
x
s
1
2
and let cp (x)
Let
Let C
2
and C
-
=
p
+
l l m
H
(x) .
cp-j^x)
and
2!. 3 - 1 5
PROOF.
It Is easy to show that
lim u(x)
x -+ a
1
=
From the previous theorem,
lim
x
H(x) =
a
lim G(x) x-a
x -*• a
and the result follows
EXAMPLE 2-7.
Let m = 1.
cp (x) = x - 2
2
cp (x) = x - u(x) - A ( x ) u ( x ) ,
1
Then
2
2
-
A ^ f x ) '
u
( x )
are of third order and
(i)
A_ = j7j7
f
C
2
1
= 2A (a) -
A
3
( a ) ,
j
Since
cp (x) = x - u(x) 1 + A (x)u(x) + A (x)u (x)
2
2
2
2
+
0 [ v T ( x ) ] ,
r
2.3-16
2
lim H(x) = -A (a)
x —• a
Hence C
2
= Ag(a) - ^(a) .
THEOREM 2 - 1 0 .
Let m = 1 and let U(x) be an arbi-
trary function which may depend on f(x) and its derivatives
and which is bounded at a.
Let q>-,(x) e I
and let
Jtr
U(a)
t
-
1
p
P![f'(a)] '
Then
cp(x) = ^ ( x )
(2-25)
p
+ U(x)f (x)
is the most general I.P. of order exactly p.
PROOF.
From ( 2 - 1 9 ) ,
(
p
q,,(x) = a + T (x)f (x),
1
1
( a )
T^a) = *
.
Pl[f'(a)]
p
p
1
By hypothesis,
p
«p(x) = cp-^x) + U(x)f (x) .
Then
p
<p(x) = a + f (x)[T (x) + U(x)].
1
( 2 - g 6 )
2.3-17
Define
T(x) = T (x) + U(x).
x
Since T(a) is nonzero, ep(x) e I .
p
The observation that
p
U(x)f (x) is the most general addend to cp-^(x) which le^ds to
(2-26)
completes the proof.
Note that we do not assume that U(a) ^ 0 .
It is
clear that
q
cp(x) = p (x) + *U(x)u (x),
C
is also of order p.
(2-27)
Setting U(x) = *U(x)u " (x) puts ( 2 - 2 7 )
q
into the form of ( 2 - 2 5 ) •
not hold if m > 1 .
q > p
1
p
Note also that Theorem 2 - 1 0 need
For example,
cp(x) = cp (x) + U(x)
1
is also of order p if m > 1 .
The following two theorems lead to another characterization of I . . F . of order p.
2.3-18
THEOREM 2 - 1 1 .
of m.
Let cp(x)
6
I
p
for some set of values
Then for these values of m, there exists a function Q ( x )
such that Q(a) ± 0 and f[cp(x)] = Q(x)f (x) .
p
PROOF,
Let a be a zero of multiplicity m.
As was
shown in Section 2 . 3 1 * there exists a continuous function M x )
such that
= *(x)(x-a)
f(x)
m
and
Then
f[<p(x)]
-
m
=
[<p(x)
a] Mcp(x)]
=
[V(x)(x-a) ]
=
V
p
r a
A[cp(x)]
m
P
(x)Mcp(x)][(x-a) ] .
Since M a ) ^ 0 , A(x) does not vanish in some neighborhood
of a and we may write
f[cp(x)]
m
p
p
= V (x)X[cp(x)]A- (x)f (x).
Define
Q(x)
m
p
= V (x)7v[cp(x)]7v- (x).
r
!
2 . 3 - 1 9
The fact that
m
Q ( a )
"f
=
p!
(m)
(a)
ml
?
0
completes the proof.
As a converse to Theorem 2 - 1 1 ,
THEOREM 2 - 1 2 -
we have
p
Let f[cp(x)] = Q(x)f (x) for some
set of values of m with Q(a) ^ 0 .
Then for these values of m,
x
cp( ) e I .
p
PROOF.
theorem.
Let A(x) be defined as in the previous
Then
m
f [cp(x)j = [cp(x) - a] A[cp(x)],
and from the hypothesis,
p
p
mp
f[cp(x)] = Q(x)f (x) = Q(x)A (x)(x-a) .
Define V(x) by
1
p
V ^ x ) = X- [cp(x)]A (x)Q(x).
Then
cp(x) - a = V(x)(x-a)
p
and the fact that V(a) ^ 0 completes the proof.
r
!
2.3-20
Theorems 2 - 1 1 and 2 - 1 2 show that cp(x) e I
If and
P
only If f [q>(x)] = Q(x)f (x) with Q(a) ? 0.
EXAMPLE 2 - 8 .
Let m = 1 and let
cp(x) = x - u(x)H(x),
H(x) = x ,
1
( ) (x)*
A
x
A
u
2^ )
x
=
JT"
2F
7
This is Halley's I.P. which will be studied in Section 5 . 2 .
Then
f[cp(x)]
= f(x)
H(x)
- u ( x ) H ( x ) f (x)
+
= 1
A (x)u(x)
2
+ |u (x)H (x)f"(x) +
2
2
+ A|(X)U (X) +
2
0[u (x)],
3
0[u (x)].
3
Thus
f[cp(x)]
= f(x)
-
u(x)f"(x)
= 0[u (x)]
3
=
- A (x)u (x)f'(x)
2
2
+ |u (x)f"(x)
2
+ 0[u (x)]
3
0[f (x)].
3
Therefore Halley's I.P. is third order for m = 1 .
THEOREM 2 - 1 3 .
P
d f
Let m = 1 and cp(x) e I .
Tcpfx)!
dx*
(p)
= f'(a)cp (a)
x=a
Then
2.3-21
PROOF.
Let i/(x) = x - f (x) .
Clearly ^(x) e I .
2
Let
¥(x) = T^tcp(x)] = q>(x) - f[<p(x)].
By Theorem 2 - 4 , f(x) e I .
Hence
P
cp(x) - f [<p(x)] = a + V ( x ) ( x - a ) .
1
(2-28)
Since cp(x) e I ,
ir
P
<p(x) = a + V(x)(x-a) .
Therefore
P
f[cp(x)] = [V(x) - V (x)](x-a) .
1
(2-29)
A second expression for f[cp(x)] may be derived as follows.
Define A(x) by
f(x) = A(x)(x-a),
A(a) = f'(a).
Then
p
f[cp(x)] = A[<p(x) ][cp(x) - a] = Mcp(x)]V(x)(x-a) .
(2-30)
From ( 2 - 2 9 ) and ( 2 - 3 0 ) ,
A[cp(x)]V(x) = V(x) - V ( x ) .
1
(2-31)
r
2.3-22
From
(2-28),
(P)(„, .
9
p
v(a) - A . a f[g<x)]
" M f W l
x=a
Taking x = a in ( 2 - 3 1 ) yields the desired result.
We turn to a generalization of the previous theorem,
THEOREM 2-14.
Let m = 1 and <p(x) e I .
Then
J
d f [CD(X)1
dx'
0 < j < 2p
(j)
f'(a)cp (a),
x=a
PROOF.
Let
= d^/dx^.
Theorem 2-11 that D^f[cp(x)]| _
It is clear from
= 0 for 0 < j < p.
Since
q/^(a) = 0 for 0 < j < p, the theorem is proved for these
values of j .
Let p £ j < 2 p ,
Now
2
f(x) = f'(a)(x-a) + x(x)(x-a) ,
p
cp(x) = a + V(x)(x-a) .
r
2.3-23
It is clear that if f and cp possess a sufficient number of
continuous derivatives, then T and V may be defined so as to
possess as many continuous derivatives as required.
f[cp(x)] = f (a)[cp(x) - a] + [cp(x)][cp(x) - af
T
We have
p
= f'(a)v(x)(x-a) + S(x).
Then
J
D f [cp(x)j = f'(a)
£
C[j,k]D
k=0
J_k
k
P
where C[j,k] denotes a binomial coefficient.
J
D f [cp(x)]|
x=a
= f'(a)p!C[j,p]D
J_p
Hence
V(x)|
It is easy to show that
D
J
-
p
V ( x ) |
which completes the proof.
x
=
a
J
V(x)D (x-a) + D S(x),
= - 4 f H < p ( J ) (
a
) ,
x=a
.
CHAPTER 3
THE MATHEMATICS OF DIFFERENCE RELATIONS
In this chapter we shall lay the mathematical
foundations prerequisite to a careful analysis of the convergence and order of one-point I.F. and one-point I.F. with
memory.
The two theorems of Section 3 ^ will be basic for
that analysis.
It is assumed that the reader is familiar with the
elementary aspects of difference equation theory.
not the case, Hildebrand [ 3 . 0 - 1 , Chap. 3 ] * Jordan
If this is
[3.0-2,
Chap. 1 1 ] , Milne-Thomson [ 3 . 0 - 3 L a n d Norlund [ 3 . 0 - 4 ] may be
used as references.
3.1-1
3.1
Convergence of Difference
LEMMA 3 - 1 .
and let q -
2
j = o
Let
^y
L
7
e
Q
t
Inequalities
* 7 ^ *
• • • >7
M
a
b
e
P
o
b
e
t
i
nonnegative
n
s
i
v
integers
constant and let the
e
6^ be a sequence of nonnegative numbers such that
n
5
I I^
n
m
+
J
and such that
5,
1
T,
i
*
0,1,...,n.
(3-1)
Then
Mr
I m p l i e s
t h a t
6^-*0.
PROOF.
q _ 1
< i
'
Let
L = Mr
L e t
t
be
t h e
f i r s t
s u b s c r i p t
5
n + l *
M
5
f o r
q _ 1
(3-2)
.
which
n - t
II
i s
y^
(
5
n - j )
n o n z e r o .
7
Then
j
•J-t+1
* Mr " 5 _
q
1
n
We
now
p r o v e
b y
i n d u c t i o n
5
f o r
a l l
i .
Note
t h a t
i + l
(3~l)
t
= L6 .
n
t
t h a t
^
L
5
and
i - t '
(3~3)
L
<
(3-3)
1,
i m p l y
6
±
+
1
£
r.
!
3.1-2
Let ( 3 - 3 ) hold for 1 = 0,1,.. .,k-l.
5
M 5
k+i^
£t
n
j=t+i
* ^^^k-t
x
L 5
J
k-t
Prom ( 3 - 3 ) we conclude that
This completes the induction.
§
-
( v /
Then
— 0.
LEMMA 3 - 2 .
Let y
Q
be a positive integer and let
7 , 7 , ...,y be nonnegative integers.
1
2
Let q = Z^ 7y
n
Q
Let
M and N be positive constants and let 6^ be a sequence of
nonnegative numbers such that
5
I+I^
M 5
I°
5
n
+ 5
( i ±-j)
7 j +
N 5
I
o + 1
*
and such that
6
< r,
1 = 0,1,...,n.
Then
q
2
Implies that 6^ -+ 0.
r
~ o a-1
°Mr
+ NT
q
y
o
< 1
(3-4)
3.1-3
PROOF.
L e t
~
q
L
-
2
o
7
MT
a - 1
7
+
q
NT
o
(3-5)
.
Then
5
n l *
M5
+
n° II ( n n - P
5
+5
3
+
N5
n
j = l
n
5
n
+
l
*
5
=
We
now
p r o v e
by
8
M T
n
2 "
n
i n d u c t i o n
5
f o r
a l l
i .
Note
Let
'
t h a t
(3-6)
0
u
M r
^
(3-4)
L
k + l
*
5
k
q
_
1
(
r )
2
+ N r
6
i '
and
f o r
7 . - 1
5
II
1
7
=
0
+
j
L 6
N T
n
7
°
.
t h a t
i + l
h o l d
"
7
i
L
(3-6)
=
<
(3-6)
1 ,
i m p l y
0 , l , . . . , k - l .
6
i
+
£
1
r .
Then
n
Mr
=
L6,
j = l
T h i s
6
4
—
c o m p l e t e s
0.
t h e
i n d u c t i o n .
From
(3-6)
we
c o n c l u d e
t h a t
!
3.2-1
3-2
A Theorem on the Solutions of Certain Inhomogeneous
Difference Equations
Let
n
X
a
*j i+j - °*
be a homogeneous linear difference
coefficients.
%
-
1
(3-7)
equation with constant
The indlcial equation corresponding to ( 3 - 7 )
is the algebraic equation
N
I
"
,
t
J
(3-8)
= 0.
J=0
Now, if all the roots of ( 3 - 8 ) are simple, the general solution
of ( 3 - 7 ) is given by
N
4=1
where the
are the roots of ( 3 - 8 ) and where the c^ are con-
stants determined by the initial conditions.
that if all the
It is obvious
have moduli less than unity, then all
solutions of ( 3 - 7 ) converge to zero.
We wish to extend this result to the case of a
linear difference equation whose homogeneous part has constant
coefficients and whose inhomogeneous part is a sequence converging to zero.
Hence we shall consider
3.2-2
n
I
K
j
i + J
a
"
N
K
=
0-9)
1
.nowhere
3^
-*•
0 .
Let
f i r s t - o r d e r
Observe
b y
and
t
»
t h a t
f o l l o w s .
t h e
=
q
P
i
f i r s t
c o n s i d e r
d i f f e r e n c e
t h a t
-K
us
p ^ .
-*
L e t
0 .
I
i + l
s o l u t i o n
of
s h a l l
The
and
n
be
V
or
and
a d d i n g
a l l
° n
t h e
t h e
"
i
a
n
+
K
=
c a s e
of
t h e
i s
e q u a t i o n
w i t h
l e s s
may
be
-
;>
n
1
i s
t h a n
g i v e n
u n i t y
"summed"
t .
as
Then
o°l " H'
o°l+l
K
+
K
o
a
n - 2
+
K
o
a
n - l
=
=
p
-
+
n - 2 '
P
l e a d i n g
e q u a t i o n s
0l'\
e q u a t i o n
t h a t
+
whose
(3-10)
i *
p
i n d i c i a l
assume
l+l
l
a
e q u a t i o n
o
K
a r b i t r a r y
°t+2
t h e
+
d i f f e r e n c e
a
M u l t i p l y i n g
s p e c i a l
e q u a t i o n
0
We
t h e
n
- 1 -
term
i s
a
n
_ j
b y
(-K )^
Q
y i e l d s
I
i = t
P l P ? "
1
"
1
-
( 3 - 1 1 )
3.2-3
Hence
n-1
\o \
*
n
IPiT^laJ
+
1 1
£
1
1
IPJIPJ " " .
(3-12)
0 we can fix I so large that the sum has magnitude
Since
less than e / 2 .
Since p.^ is less than unity we can then take
n so large that the first term on the right side of ( 3 - 1 2 )
has magnitude less than e / 2 .
We conclude that
—> 0.
This
is the desired result for the case of a first-order difference
equation.
We shall now derive a generalization of (3~ll) for
the case of an Nth order difference equation.
Thus we start
with
_N
P
K
i+j " i '
1
N " '
Let py
j .= 1 , 2 , . . . , N be the roots of the indlcial equation,
Let d
be constants whose values we shall choose at our
n i
later convenience.
that n - N ^ I.
Let I and n be arbitrary integers such
Then
n^N
I
_N
d
1=1
ni Z V i + J
j=0
n-N
=
Z
d
p
ni i'
1=1
3.2-4
or
n^N
X
i+N
d
n^N
Z s-i s s=l
K
ni
1=1
Z
a
d
p
ni i*
1=1
Changing the order of summation yields
n^N
s
t+N-1
ni
n-N
*
I
s*t+N
s-i
s
°=
n
I
i«s-N
n l V i
d
n-N
I
°s
s=n-N+l
+
X
i=s-N
d
.K
.
ni s - i
-1+2+3.
Consider 2 ,
This is zero if we choose d
solution of the homogeneous difference equation.
convenient to define d
where the constants
Consider 3 .
as a
It is
by
n i
d
n i
m
-
i
v
r
1
-
1
(
will be chosen later.
Using ( 3 - 1 3 ) we can show that
n
n^N
N-l
N
_N
s=n-N+l
i=a-N
A=0
r=N-7\
j=l
3
-
i
3
)
3.2-5
We have the N coefficients D-^Dg, ... , D at our disposal.
N
choose them so that the coefficient of
is unity and the
coefficients of the a _^ are zero for 7\ > 0.
n
that k
We
Thus, recalling
- 1,
n
N
(3-14)
N
I
r-N-y
U s i n g
t h e
f a c t
N
I
* r
V
j
4
*
"
1
° '
=
*
1,2,...,N-1.
=
j = l
t h a t
N
1\
*r.Pi
I
= 0,
j
-
1,2, . . .,N,
r=0
i t
i s
l e n t
n o t
t o
d i f f i c u l t
t h e
t o
p r o v e
t h a t
t h e
s y s t e m
(3-14)
i s
e q u i v a -
s y s t e m
N
D
where 6^
p
j j
=
5
r,N-l'
r
=
0 l ... N-l
9
is the Kronecker symbol.
9
(3-15)
The determinant of
this system is a Vandermonde determinant.
(3-15)i
9
Hence Dj satisfying
and consequently ( 3 - 1 4 ) , exist and are indeed the
ratios of certain Vandermonde determinants.
of the Dy
3 reduces simply to o^.
With this choice
We conclude that
3.2-6
n-N
a
n -
t+N-1
<W*i
I
"
1=1
s
a
I
s
I
s=l
d
K
ni s-i
1=1
We summarize our results in
LEMMA 3 - 3 .
Let
N
IN
u
p
\j i+J " i '
K
N
J=0
be a linear difference equation whose homogeneous part has
constant coefficients.
be simple.
Let the roots of the indicial equation
Let
N
d
3=1
where the
are the roots of the indicial equation and where
the Dj are determined by.
N
I
Vj
5
" r,N-l'
r
" 0,1,...,N-l;
3.2-7
5
r
N - l
i
s
t
h
e
K
r
o
n
e
^
i
k
e
s y m b o l .
r
Then
n-N
a
n
I
-
t + N - i
n
d
A
Z
-
t
and
We
n
a r e
are
a r b i t r a r y
now
THEOREM
°
s
s=l
1 = 1
where
s
r e a d y
3-1.
d
n i
<3-17)
s - i '
K
1 = 1
i n t e g e r s
t o
l
such
t h a t
n
-
N
j>
t .
p r o v e
L e t
N
I
be
a
l i n e a r
c o n s t a n t
t i o n
be
Then
o
±
d i f f e r e n c e
0
and
f o r
PROOF.
L e t
have
a l l
i
+
e q u a t i o n
c o e f f i c i e n t s .
s i m p l e
V
t h e
m o d u l i
s e t s
of
j
-
P i
whose
homogeneous
r o o t s
of
l e s s
t h a n
i n i t i a l
t h e
p a r t
i n d i c i a l
u n i t y .
L e t
has
e q u a -
£^
-+0.
c o n d i t i o n s .
Let
s
t+N-1
G
3 l
-
D
J
Z
3 =
a
1
s
Z
1 = 1
- s - i P j
1
"
1
'
(3-18)
3.2-8
Using ( 3 - 1 6 ) ,
(3-17),
and ( 3 - 1 8 ) we can write
n-N
N
°n = E n A " I
1=1
j=l
ilPy
d
G
The notational adjustments which would be necessary if one
of the roots of the indicial equation were equal to zero are
obvious.
Since
N
d
ni
-
1
v r
-
1
1
-
J=l
and since the
having moduli less than unity, it is clear
that there exists a constant A such that
I
n-N
d
nil
<
A
-
i=l
Let e > 0 be preassigned and arbitrary.
Since ^
-* 0, there
exists a number L such that
< 2A
Fix £ such that I ;> L.
f o r
a 1 1
Observe that
1
^ L
is independent of n.
With I fixed there exists a constant B such that
N
3.2-9
Let
p
= •max[|.p |,|p
1
2
|,...,|p
N
|
].
Let T| be so large that
P
^
2B"
Then for all n > T|, we have
n-N
l *
n
| . *
I
N
| d
n
l
| |
P
l
|
+
i=t
a
nl
<
2T
A
Z
I ^ J I P j l ^
J=l
+
^
B
=
e
which completes the proof.
For the applications that we have in mind, the
inhomogeneous part of the difference equation will converge
to a nonzero constant.
We wish to show that all solutions
of the inhomogeneous equation will converge to a constant.
This result follows easily from the above theorem.
We have
3.2-10
COROLLARY.
Let
N
j=0
be a linear difference equation whose homogeneous part has
constant coefficients.
Let the roots of the indicial equa-
tion be simple and have moduli less than unity.
Let co^
cd.
Then
CD
'i
N
for all sets of initial conditions.
PROOF.
Let
CD
a
i
85
*i
+
~N
I
Then
N
N
CD
X * .
"i+J
r
*±+j
N
J=0
Let
= cd^ - cd.
+
*
=
° V
J
Then the conditions of the theorem apply
and we conclude that
0 and hence that
0)
i
N
0=0
3.3-1
3.3
On the Roots of Certain Indicial Equations
The properties of the roots of the polynomial
equation
k-1
g
kja
(t) = t
k
- a
£
t
J
= 0
(3-19)
j=0
will be derived.
This will turn out to be the indicial
equation for certain families of difference equations which
will be encountered in our later work.
The case a = 1 has
been treated by Ostrowski [ 3 . 3 - 1 , pp. 8 7 - 9 0 ] .
3.3-2
3.31
If k = 1 , the
The properties of the roots.
only root of ( 3 - 1 9 )
is t = a.
For the remainder of this
section we shall assume k ^> 2 .
We shall permit a to be any
real number such that
ka > 1 .
(3-20)
Since
g
k,a
( l )
=
1
t = 1 is not a solution of ( 3 - 1 9 ) •
^ a ^
=
( t _ l ) g
k,a
( t )
k
+
'
It is convenient to define
=
t
k a
"
"
1
(
a
+
1
H
k
+
a
(3-21)
-
Hence G.k. ,(at ) has a root at t = 1 and roots at the roots of
0
1
By Descartes rule, g, ( t ) has exactly one real
jk , a
positive simple root. This unique root will be labeled p. .
ic j a
Since
Q
k-1
g
k ,, (a) = - £ aJ,
j=l
a
S
k
,
a
( D
=
1
- ka,
we have
LEMMA 3 - 4 .
The equation
k-1
g
k,a
( t )
=
t
k
" a £
J=0
t
J
= 0
g
k
a
(a+l) = 1
3.3-3
has a unique real positive simple root,
max[l,a] < £ ^
k
Thus p
k
, and
< a + 1.
a
has modulus greater than one.
a
We shall
show later that all other roots have moduli less than one.
the next lemma we shall prove that p.
In
is a strictly increasK. a
o
$
ing function of k.
LEMMA 3-5.
PROOF.
P .
k
l j f i
< P
k # a
.
This follows from the observation that the
recurrence relation
t&krl.aM
implies that gjj ( P _ i )
a
k
a
l s
-
a
=
*k,a^)
negative.
The following inequality will be needed below.
LEMMA 3 - 6 .
Let ka > 1 .
^
>H
Then
(i • £) k
PROOF.
Let k be fixed and define
k
2
Then J'(a) = (a+l) (ka-l)/a and therefore J(a) is a strictly
increasing function of a for ka > 1.
The observation that
completes the proof.
Descartes
1
rule shows that (3,
.k
Is a simple root.
fa
Furthermore
LEMMA 3-7.
PROOF.
G
Then G
k
a
k , a
(
All the roots of g
k
&
( t ) = 0 are simple
Define
t
)
=
^ - ^ S i c a ^ )
-
t
k
+
1
"
(
a
+ D t
( t ) has only one nonzero root,
v
(a+1).
The fact that v is positive completes the proof.
k
+
a .
r
I
3.3-5
In the following two lemmas we derive better bounds
LEMMA 3 - 8 .
k+t
<
a + 1
> < Pk,a < a + 1.
Therefore, for a fixed
lim
NOTE.
P
k
= a + 1,
a
Since ka > 1 implies that (a+1)k/(k+1) > 1,
we: need not write
max[l, ^
PROOF.
Let v = (a+1)k/(k+1).
n
k,a
G
(a+1)] <
( v )
( v )
_
=
a
a
(a+l)
"
k+1
k + 1
Then
/_ , l^"
V
k,
1
k
+
An application of Lemma 3 - 6 shows that G, ( v ) < 0 , which
0
ic
ja
together with an application of Lemma 3 - 4 completes the proof,
LEMMA 3 - 9 .
(a+l)
k
k
'
a
(a+l)
where e denotes the base of common logarithms.
k
r
3.3-6
PROOF.
Let v = (a+1)k/(k+1).
The only positive
root of the equation G," ( t ) = 0 is t = v - (a+l)/(k+l).
Q
Therefore G, ( t ) does not change sign in the interval
K. , a
0
v £ t
a + 1.
Since
G^ (a+1) = 2k(a+l)
k _ 1
> 0,
a
G, _(t) is convex in the interval.
K. f a
Since the secant line
through the points [v,G, ( v ) ] , [a+l,G, (a+l)] intersects the
Kj a
ic f a
t axis at
0
a
t = a + 1 -
(a+1)
whereas the tangent line at [a+l,G, (a+l)] intersects the
ic j a
t axis at
t = a + 1
(a+l) '
k
we conclude that
a + l - - ^ ( l ^ )
(a+l) V
^
K
k
O
,
< a
--K,ak
P
+
l -
a
( a + l )
k
k«
An application of the well-known inequality (l + l / k ) < e
completes the proof.
Il
3-3-7
We will next study the polynomial generated by
dividing g, ( t ) by the factor t - p.
.
Q
Define c,.
j
Ka
5
0-£
J£ k -
1,
by
-£§77"
K
'
I
a
W
J
*
( 3
'
2 2 )
j=0
We first prove
LEMMA 3-10.
k-1
V
c
=
°J
k a
K
j=0
PROOF.
ka-1
P -i*
'
a
The result is obtained by setting t = 1 in
(3-22) and observing that g. ( l ) = 1 - ka.
iv , a
Q
In the following three lemmas we shall prove that
all the roots of the quotient polynomial have moduli less
than unity.
It will be convenient to sometimes abbreviate
3.3-8
LEMMA
PROOF.
c
I t
1,
=
Q
C j
3-11.
i s
Cj
>
f o r
0
e a s y
=
t o
-
p"
£
0
show
a
£
j
£
k
-
1.
t h a t
p
6
,
t=0
(3-23)
A
second
f o r m u l a
f o r
t h e
c^
may
be
d e r i v e d
b y
n o t i n g
t h a t
k-1
p
k
- a £
3 * - = 0.
(3-24)
t=0
B r i n g i n g
t h e
t h e
e q u a t i o n
l a s t
-
k
and
j
terms
d i v i d i n g
c^*»
£
a
(3-24)
of
b y
k - 1
p
t = o
L e t
0
=
p '
1
.
f a c t
r i g h t
s i d e
1
J
-
of
p"" ,
6
£
£
k
1.
t = i
Then
°3
The
t h e
y i e l d s
0
- a y.
- p *
t o
t h a t
6^1
j - l
a
0
c o m p l e t e s
LEMMA
c
=
(
l
" i - e
t h e
J
)
(3-25)
»
p r o o f .
3-12.
c
j + l
<
°y
k
>
2
>
1
1
H
k
"
2.
li.
3.3-9
Case 1 .
PROOF.
°o°2
<
c
l
o r
f
r
o
m
3
j = 1.
We must show that
2 3
( ~ )'
a
< a
- ^k,a "
a
a
2
< ^k,a- > '
This simplifies to 0 < 1 + a - 0. _ which is true from
xV , CI
Lemma 3 - 4 .
Case 2 .
J > 1.
Then from ( 3 - 2 5 ) , the result is
equivalent to
(i-e , k - J - l )(i-e
or 0 < 1 - 29 + 6* which holds since
LEMMA 3 - 1 3 .
All the roots of the equation g
other than the root 0,
k
a
(t) = 0 ,
. have moduli less than one.
K,
PROOF.
6^1.
a
The proof depends on the following theorem
which Ostrowski [ 3 . 3 - 2 , p. 9 0 ] attributes to independent
discoveries by Kakeya [ 3 . 3 ~ 3 ] and Enestrom.
If in the equation g(x) = 2 j ^
Q
t> _jX
n
all
coefficients bj are positive, then we have for each
root i that
141
<: max
l ^ K n
b /b
J
3.3-10
Applying this theorem to (3-22) we find, by virtue
of Lemmas 3-11 and 3-12, that
c
c
C
l^Kk-l J-1
c
o
Furthermore c,/c
jl
o = k. ^ a - a < 1 from (3-23) and Lemma 3~4.
This completes the proof for the case k > 2. For k = 2,
= t + p
and
Ul
= P
Q
- a
- a < 1.
2 > a
The major results of this section are summarized in
THEOREM 3-2.
Let
k-1
If k
1, this equation has the real root p
k ^ 2 and ka > 1.
simple root p,
0
a
_ = a. Assume
j.,a
Then the equation has one real positive
n
and
max[l,a] < p
v
Q
< a +.1.
3.3-H
Furthermore,
r~
o +4.1
a
i
~ —
(
s s
a
§a
riz
+
l
)
k
< P„
a
^k,a
< a + 1 -
—
—
( a + l )
where e denotes the base of common logarithms.
k*
Hence
lim 6, „ = a + 1. All other roots are also simple and
v
k,a
have moduli less than one.
r
3.3-12
3.32
An important special case,
A case of special
interest occurs when a is a positive integer.
(We will not
be interested in nonintegral values of a until we investigate
multiple roots in Chapter 7 . )
Lemma 3-4- niay be simplified to
read
a < P ^
k
a
< a + 1,
k > 1.
Values of p,
for low values of k and a may be found in
it j a
Table 3 - 1 . Observe that p.
has almost attained its limit,
ic j a
a •+ 1 , by the time k has attained the value 3 or 4 .
particularly true if a is large.
consequences later.
This is
This will have important
3.3-13
TABLE 3 - 1 .
"a
1
1
VALUES OF p
2
3
4
k
1
1.000
2.000
3.000
4.000
2
1.618
2.732
3.791
4.828
3
1.839
2.920
3.951
4.967
4
•1.928
, 2.974
3.988
4.994
"5
1.966
2.992
3.997
4.999
6
1.984
2.997
3.999
5.000
7
1.992
2.999
4.000
5.000
3.4-1
3.4
The Asymptotic Behavior of the Solutions of Certain
Difference Equations
3 .41
Introduction.
Consider the difference equa-
tion
n
e
i+i -
II
K
e
where K is a constant and s is a positive integer.
logarithms in ( 3 - 2 6 )
(3-
ty
2 6
)
Taking
leads to a linear difference equation
with constant coefficients whose indicial equation is
n
n + 1
t
- s
t
j
= 0.
(3-27)
This is a special case of the equation studied in Section 3 . 3
with k = n 4- 1 and a = s.
(3-27),
The properties of the roots of
derived in the previous section, permit us to analyze
the asymptotic behavior of the sequence {e^}•
Now consider the difference equation
n
e
where
K.
i+i
=
M
i
II
e
l-y
(3-28)
Is the asymptotic behavior of the sequence
generated by this equation the same as that generated by
(3-26)?
The answer turns out to be in the affirmative.
It
will turn out that the errors of certain important families
of I.P. are governed by difference equations of the form specified by ( 3 T 2 8 ) .
3-4-2
In Section 3 * 4 3 we shall show that the asymptotic
behavior of the sequence
n
e
M
e
i+i = i i
n
( v
e
i
)
S
+
N
i
e
i
+
1
^
(3-29)
J=I
where M.^
K, N
±
—• L, is the same as the asymptotic behavior
of the sequence generated by ( 3 - 2 6 )
or ( 3 - 2 8 ) .
Difference
equations of the type defined by ( 3 - 2 9 ) will be encountered
in Chapter 6 .
We shall designate difference equations of the
type defined by ( 3 - 2 8 ) and ( 3 - 2 9 ) as of type 1 and type 2,
respectively.
3.4-3
3.42
s t u d y
t h e
ence
D i f f e r e n c e
a s y m p t o t i c
e q u a t i o n s
b e h a v i o r
of
of
t h e
t y p e
1,
s o l u t i o n s
We
of
s h a l l
t h e
d i f f e r -
e q u a t i o n
n
e
where
s
i s
a
p o s i t i v e
i + i
=
M
i n t e g e r .
We
M
and
i f
t h e
m a g n i t u d e s
t h e n
t h e
none
of
t h e
t h e n
e^
i s
t h e r e
| e
1
+
1
sequence
M
4
l
not
e x i s t s
| / | e . j j * \
a
z e r o
S e c t i o n
i t
n e c e s s a r i l y
0
f o r
any
f i n i t e
1.23)
a
i f
s u f f i c i e n t l y
s m a l l .
z e r o .
Observe
i f
of
t h a n
i f
t h a t
° )
are
n
e
i .
n o n z e r o
t h a t
e
none
g r e a t e r
show
3
(3-31)
t o
i f
t o
s h a l l
>$^,> * • • >
and
p
^
K,
z e r o
c o n v e r g e s
( s e e
e
-
±
c o n v e r g e s
number
p r o v e
i s
of
of
a r e
n
I
o
- e - , . . . , e ^
n
We
u n i t y
s h a l l
a
number
a r e
prove
such
c o n s t a n t .
such
*
l*
t h a t
z e r o ,
t h a t
t h a t
I t
p
i s
e a s y
e x i s t s ,
t o
t h e n
u n i q u e .
L e t
5
S i n c e
e x i s t s
c o n v e r g e n t
a
c o n s t a n t
±
=
s e q u e n c e s
M
such
[ e
±
| ,
are
t h a t
r
=
s ( n + l ) >
n e c e s s a r i l y
bounded,
(3-32)
t h e r e
3.4-4
for all 1 .
Then
n
B
1
+
*
1
II
M
B j .
4
J = 0
Let
5.
^
r ,
An application of Lemma 3
1
= 0 , 1 , . .
.,n.
1 with all the y^ equal to s and
_
with r replacing q shows that if
M r
r
-
<
1
1 ,
then 6 -»• 0
i
Assume that
and K are nonzero.
We will investi-
gate how fast the sequence {e^} converges to zero.
o
±
= In 5
±
= ln|e |,
J
±
±
= lnJMj.
Let
(3-33)
Then from ( 3 ~ 3 0 ) , we have that
_n
a
J
i+l " i
+
Z
s
°±-y
(3-34)
J=0
Let t be an arbitrary parameter whose value will be chosen
later.
Let
c_O = 1,
cj(t)
3
= t
- s
^
l
t,
j *
l,2,.o.,n+l,
1=0
(3-35)
r
3.4-5
and let
D (t)
±
= a
(3-36)
- tff _ .
1
±
1
Then it is not difficult to see that ( 3 - 3 4 ) is identical
with
X
°^\
+
{
l-3
t
)
+
c
n
+
l
(
t
K - n
J
= i
^-37)
for all values of t.
Consider the equation
n
c
n + 1
(t)
= t
n + 1
y
t V « 0.
t=0
-s
(3-38)
Observe that this is precisely the indicial equation of the
difference equation derived from ( 3 - 2 6 ) by taking logarithms.
It is a special case of the equation
k-1
S , (t) - t
k
a
k
- a
£
t
J
= 0
(3-39)
J-0
studied in Section 3 . 3 with k = n + 1 , a = s.
The analysis
of ( 3 - 3 9 ) was based on the assumption that ka > 1 .
Since ka = (n-fl)s = r
s
Let r > 1
the conclusions of Section 3 . 3 apply.
Hence all the roots of ( 3 - 3 8 ) are simple and there is one
f
3.4-6
real positive root greater than unity; all other roots have
moduli less than unity.
The real positive root is labeled
o • We shall abbreviate 6
n+JL y
s
11+1 by
• sp and choose the
1
parameter t equal to p.
o
Hence
c
n + 1
( p ) - 0.
(3-40)
Let
C
j = Cj(p),
= Dj(p).
Then ( 3 - 3 7 ) becomes
n
The c. are just the coefficients of the polynomial gotten by
J
dividing c
[See ( 3 - 2 3 ) J
^ t ) by t - p.
Hence the indicial
equation corresponding to the difference equation ( 3 - 4 l )
only simple roots whose moduli are less than unity.
—• K, we conclude that
-* ln|K|.
has
Since
All the conditions of
the Corollary to Theorem 3 - 1 now apply and we conclude that
D
±
- ^ L .
c
j=0
j
(3-42)
3.4-7
An application of Lemma 3 - 1 0 , with ka = (n+l)s = r and with
^k,a
*
Pn+1,8
=
p
'
s
h
0
W
S
t
h
a
t
n
I
C
J ' P i -
From (3-33), (3-36), (3-40), ( 3 - 4 2 ) ,
(3.43)
and (3-43), we conclude
that
6
1+1*
_^ | |(p-l)/(r-l)
K
We summarize these results in
THEOREM 3-3.
Let
n
e
M
i+i - I
n
e
i-j
with
| e | ^ T,
1 = 0,1,...,n.
±
Let s be a positive integer and let r = s(n+l) > 1 .
M
±
-»• K and let
| M
±
|
£
M
for all 1 .
M r
r
_
1
<
Let
1.
Let
3.4-8
Then e
A
-•0.
Let p be the unique real positive root of
n
t
Let M
i
n + 1
and K be nonzero.
- 8
-
i
K
| ^ - ^ ^
r
- ^ .
(3-44)
i
Table 3 - 1 gives values of p = P
of n and s.
0 .
Then
p
e
£
n + 1
s
for low values
Note that no a priori assumption has been made
of the existence of a number p such that
* o .
e
p
if
If the existence of such a number is assumed a priori, it is
not difficult to prove that it must satisfy the indicial
equation ( 3 - 3 8 ) .
3.4-9
3.43
Difference equations of type 2,
We shall
study the asymptotic behavior of the solutions of the difference equation
n
e
i+i
=
M
n ( i-j- i)
J=I
e
e
i i
e
s +
N
e
i i
+ 1
'
( 3
~
4 5 )
where
M ^ K / O ,
N
-» L,
±
and where n and s are positive integers.
if the magnitudes of
e^
E
® * ^S
*•* >
0
a
E
N
^
e
(3-46)
We shall show that
sufficiently small, then
0 and there exists a number p greater than unity such
that I
1 /| e JL |
p
converges to a nonzero constant.
InJPaet,.
we shall demonstrate that the asymptotic behavior of the
sequence generated by (3-45) is identical with the asymptotic
behavior of the sequence generated by
n
e
M
i l = i
+
II
e|.j,
M
±
- K
which was studied in the previous section.
Let
Q
±
= |e |,
±
r = s(n+l)
and let
|M | £ M,
±
|N j ^ N
±
(3-4?)
I
3.4-10
f o r
a l l
i .
Then
n
s+1
5
i + i
^
M
5
n
I
(
5
i - j
+
i )
5
s
N
+
5
I
• J - l
L e t
5
An
a p p l i c a t i o n
w i t h
q
e q u a l
of
t o
i
Lemma
r ,
t h e n
5
i
-»•
t h a t
M
r
0 , 1 ,
. .
a l l
. , n .
t h e
y^
e q u a l
t o
f i n i t e
1.
s
and
i f
r - l
+
<
x
^
0 .
Assume
w r i t e
r - s
-
w i t h
3-2,
sjiows
2
1
<> R ,
(3-4-5)
t h a t
e^
i s
n o n z e r o
f o r
a l l
We
may
as
n
e
i
+
l
H 4-y
II
-
(3-48)
w i t h
T
i
=
M
i
A
i
+
N
i
9
^3-49)
i '
where
»i
"
5
J
We
s h a l l
=
( l - A ) " .
- ~n~~^
n
1
d e m o n s t r a t e
t h a t
-*• 1
and
6^
•—• 0 .
•
(3-50)
3.4-11
To show that \^ -* 1 we note from (3-45) that
e
1 + 1
/e -*0.
Therefore
1
e
0
~*
j _ A j _ « j
f
o
r
a
1
1
finite j and the
result follows.
To show that 6^
(3~45) and
0 we proceed as follows.
Prom
(3-50),
n
e
M
x
± - i-i ±-i
n
e
i-i-j
+
N
i-i l-ie
Hence
n
n
"
6 1
M
i - l V l
e
L l - n
+
N
i-1
n ^ "
'i.
n
3
1
•
< 3 - 5 D
4-i
Repeat this process for the second term on the right side of
(3-51)*
Carry out this reduction a total of n - 1 times.
The problem is reduced to proving th^at
e
e
i + 1
/ ^ ~* 0 and this
is clearly true from (3-45) •
Since 7^
T
i
K.
1 , 6^
0, and M
i
K, we conclude that
Theorem 3-3 niay now be applied to (3-48) and we
arrive at
THEOREM 3-4.
Let
n
s+1
j-i
I
3.4-12
with
| e
±
|
i
r ,
1
=
0 , 1 , . . , , n .
Let s be a positive integer and let r = s(n+l) > 1 .
M
i
•* K / 0 , N
±
Let
L, and let J M j ^ M, |N | £ N for all 1
±
Let
2
Then e^
0.
r - s
M
r
r - l
+
jjpS
^
Let p be the unique real positive root of
t
n + 1
n
- s £
^
= 0.
J-0
Assume that e
i
^ 0 for all finite i.
f U l "
,
K
Then
| < P - l ) / < r - l ) .
4.0-1
CHAPTER 4
INTERP0LAT0RY ITERATION FUNCTIONS
In this chapter we shall study I.F. which are
generated by direct or inverse hyperosculatory interpolation;
such I.F. will be called interpolatory I.F.
The major results
concerning the convergence and order of interpolatory I.F.
will be given in Theorems 4-1 and 4-3.
A sweeping generaliza-
tion of Fourier's conditions for monotone convergence to a
solution will be given in Theorem 4-2.
4.1-1
4.1
Interpolation and the Solution of Equations
4 . 1 1 statement and solution of an interpolation
problem.
The reader is referred to Appendix A for a discussion
of hyperosculatory interpolation theory.
Certain salient
features of that discussion which are needed for the development of the theory of this chapter will be repeated here.
Consider the following rather general interpolation
problem.
We seek a polynomial P such that
(k«)
P
J
x
( i_j) =
f
(k.)
( i-j)
x
f
o
r
J =0,l,...,n;
(4-1)
^ 1;
k^ = 0,1, ... ,7^-1,
x_
±
k
± x_
±
t
if
k ^ I.
That is, the first 7^-1 derivatives of P are to agree with
the first 7j-l derivatives of f at the n + 1 points
x
i ' i _2* • •
x
l^t
n
q
=
I
7
r
Then there exists a unique polynomial P
n,7 ,7 »
Q
degree q - 1 which satisfies ( 4 - 1 ) .
1
of
• • - ,7
n
For the sake of brevity
we write
P
n,7
= p
n , 7 , 7 , .. . , 7 '
Q
1
n
(h-g)
'
v
i
4.1-2
where 7 signifies the vector y y^
...,y .
9
7^ = s for all yy
If, in particular,
then we write the interpolatory polynomial
as P
.
n, s
n
q
Let f( )(t) be continuous in the interval determined
by x , x
1
1 - 1
,..-jX
1 - n
, t . Then
n
f(q)[| (t)]
f(t)
- P
B f y
7
II (t-xj.j) , (4-3)
J
(t) .
x
x
where ^ ( t ) lies in the interval determined by ±> i-i>
Let f
interval J.
q
zj( ^
q
be nonzero and let f^ ^ be continuous on an
Let f map J into K.
is continuous on K.
problem for 3 as follows.
(kj
Q
(
y
x
• • • > j_-
i-p
=
We can state the interpolation
We seek a polynomial Q such that
(k.)
i-j^
S
Then f has an inverse S and
( y
f
o
r
J
=
0,1, ...,n;
(4-4)
kj = o,i,...,7^-i, 7j ^ i; y _
±
k
Then there exists a unique polynomial
degree q - 1 which satisfies (4-4) .
/ y _
±
if
t
7 ,y^ . . . , 7
Q
5
+ i.
n
of
For the sake of brevity
we write
Q
If,
n,7
^n,y s7 s
Q
1
..-,7 *
n
in particular, y^ = s for all 7^, then we write the
interpolatory polynomial as
^(t).
(4-5)
r
4.1-3
The error of the interpolation is given by
«
s(t) = Q ^ ( t ) + —
7
( a )
[Mt)]
^
"
II
where 6^(t) lies in the interval determined by
^i'^i-l* • • • '^i-n'^ *
8
(t-y,^) ,
(4-6)
f
4.1-4
4.12
of roots.
Relation of interpolation to the calculation
x
x
x
Let ±> i-i9
• • • > i-
zero a of the function f.
b e
n
+
1
a
n
PP
r o x l r n a n t s
t
o
a
To calculate a new approximant
it is reasonable to calculate the zero of the polynomial
which interpolates f at the points x^,x^_^,...,x .
The
1-n
x
x
x
process is then repeated for the set i+i' i'•••> i-n+l•
One drawback of this procedure is that a polynomial equation
must be solved at each step of the iteration.
A second draw-
back is that the polynomial will have a number of zeros, some
of which may be complex, and criteria are required to select
one of these zeros as
the point
x
i + 1
-
Once such criteria are established
is uniquely determined by the points
X • • X • -i • . . . . X . . We define $
I 1-1
I-n
n,7
maps i> i„i> • • • > i i+i
as the function which
J
x
x
x
i r r t o
x
#
n
The difficulties of having to solve a polynomial
equation may be avoided by interpolating
#
y
at the points y ^ y ^ i * • • * i - n '
polynomial at zero.
a
n
d
The point
x
the inverse to f,
evaluating the interpolatory
is uniquely determined;
x
x
let the function which maps i> ±^i > • • • * i L
i n t
n
x
o j_
+ 1
be
labeled cp nj/
For either of the two processes described above,
x ,x ,...,x
Q
1
n
must be available.
One method for obtaining
these starting values is described in Section 6 . 3 2 .
If
4.1-5
I.F. which are generated through direct or inverse
hyperosculatory interpolation will be called interpolatory I.F.
Hyperosculatory interpolation is not the only means by which
I.F. may be generated.
Other techniques will be studied in
various parts of this book.
It is a very useful technique
however and gives a uniform method for deriving I.F. and of
studying their properties and, in particular, their order.
The most widely known I.F. are all examples of interpolatory
I.F.
A useful by-product of generating I.F. by hyperosculatory
interpolation is the introduction of a natural classification
scheme.
II
4 . 2 - 1
4.2
The Order of Interpolatory Iteration Functions
4.21
The order of iteration functions generated by
x
inverse interpolation.
X
x
Let J_* J__]_j • • • ' i -
mations to a zero a of f.
b
e
n
+
1
a
n
PP
r o x l
~
Let 0^ ^ be the polynomial which
interpolates S at the points y ^ y ^ i * • • • >y±-
i
n
t J r i e
s
e
n
s
e
°f
n
Define a new approximation to a by
(4-4).
x
±+i -
0
^J
)
'
.Then Repeat this procedure using the points i + i j ^ • • • > i- +l
x
x
x
n
We shall investigate how the error of
depends on the
errors at the previous n + 1 points.
We observed ( 4 - 6 )
3
- ^,7
( t )
(q)
+
that
[e.(t)]
"
qT
7
n
(Wi-J>
J
-
where 0^(t) lies in the interval determined by
y , y _ , . . . , y _ , t , and where q =
1
1
1
1
x
i+l
-
a
where 0 ^ = 9^(0) .
- - -^f
( q )
f
i +
x
t
=
±
±
j = i-j
i • ( i+i)
t
=
°*
[I (y -j) >
(e )
_
a
«
in terms of either the y^_j or the
y
e
T
n
e
n
7,
S
x
Let
S
jy
n
f
' ( W
W
e
^.
e
c
a
n
( 4 - 7 )
write ( 4 - 7 )
Since
i + i '
J
4.2-2
where T\
±+1
lies in the interval determined by x
± + 1
and a, we
conclude that
n
M
y±+i = * i n
q
*l-y
(4-8)
{(i)
(-i) z (e )
±
M
* i
where p
± + 1
= f (Tl ) .
=
~
q!«;(p i> '
1+
Since
1+1
we can also conclude that
(4-9)
n
e
i + i - (i- l )II^ ( )t-y
(e )
M
qe
1
M,
=
-
qi
II
[3'(pi_j)]
J
where p _ j = f (T| _^) .
±
±
It is clear that if none of the set e ,e^,...,e is
Q
zero and if
n
does not vanish on the interval of iteration,
then e^ is not equal to zero for any finite 1.
4.2-3
We shall show, however, that If the initial approximations x ,x ,...,x , are sufficiently close to a, then e
Q
1
n
We will use (4-9) for the proof.
±
-+ 0
Let
J = -jx |x-a| 1 T\.
q
Let f ( ^ be continuous on J and let f' be nonzero on J.
f map the interval J into the interval K.
tinuous on K and 0' is nonzero there.
l
8
(
q
^ '
^ V
|.'(y)li»
A
Q
1
n
e J.
is con-
Let
for all y e K; that is, for all x e J.
Let x , x , . . . , x
Then 3 ^
Let
a
Let
2
Then
max[|e |,|e |,...,|e |] £ r.
o
1
n
We shall show that if
M p
then all the
e J.
q-1 ^
1 }
(4-10)
!
4.2-4
Since x , x , . . . , x
Q
1
e
M
l n ll = l n«
+
n
e J, |M | £ M.
Hence
n
II
lei.jl^iHT^r,
where the last inequality is due to (4-10).
induction.
Let x
±
We proceed by
e J, for i = 0,1,...,k.
Hence |M | £ M
k
and
J=o
which completes the induction.
Since all the x.^ e J, | j
^ M
for all i.
A modification of this proof could be used to show
that e^ -»• 0.
Instead we note that
n
5
i l ^
M
+
7
II
(^-j)
=
and use Lemma 3-1 to arrive at
LEMMA 4-1.
Let q =
7y
Let
J = -jx | x-a | £ r |
4.2-5
and let f ^
x ,x ,...,x
Q
1
be continuous and f
n
e J.
M
i
Let
Let
for all x e J and let M = ^ A ^ -
Then e
nonzero on J.
r
q - 1
Suppose that
<
1
>
0.
Since
-*• 0, x
±
-*• a, and we can conclude that
4.2-6
4.22
The equal Information case.
We will now
specialize to the case of Interest for our future work.
7j - s,
j = 0,1,...,n
Let
f
Then the same amount of information will be used at each point.
Thus the first s - 1 derivatives of the interpolatory polynomial are to agree with the first s - 1 derivatives of 3 at
y
y
#
y
i' i-l'•• ' i-n
#
L
e
t
r = s(n+l).
The results of the previous section hold for this case if we
replace q by r and 7j by s.
In particular,
n
-
*
M
i
II
y±-y
3=o
(4-11)
r
(-i) ^^(e )
±
*
M
i
-
"
r ! « ' ( p
1
+
1
)
'
and
n
e
M
i +
i - i
3=0
e
II
l-y
(4-12)
r
r)
i-i)
^
(e
)
n
±
M. = '1
r !
II
r * ' ( P i - j ) ] '
3=0
1
4.2-7
Let the conditions of Lemma 4 - 1 hold with q replaced by r
and 7 , replaced by s.
Then |M | <; M for all i, e
±
±
0 , and
J
M - * Y ( a ) , where
1
r
v
r
( x ) was defined ( 1 - 8 ) as
r
( r )
_ _ (-D 3 (y)
r'[*'(y)]
y=f(x)
r
All the conditions of Theorem 3-3 are now satisfied and we
conclude that
K ± l l
| j {p-Mr-\
_
Y
a )
(4.
1
where p is the unique real positive root with magnitude
greater than unity of the equation
t
n+l _
s
n
\ J
t
=
o.
J=0
Furthermore, * M — - ( - l ) a ( 0 ) , where CZ (y) is defined ( 1 - 8 )
r
±
as
r
r
3
)
4.2-8
Hence
i l m i - .
|a (o) fp-^r-i).
( 4
.
1 4 )
Iy i
x
It is not difficult to see that one can pass from ( 4 - l 4 )
( 4 - 1 3 ) by observing that y _j =
±
to
f'(\_ )e _y
3
±
Our results concerning the convergence and order
of I.F. generated by inverse interpolation are summarized in
THEOREM 4 - 1 .
Let
J = jx |x-a| £ rj-.
Let r = s(n+l) > 1.
on J.
r
Let f ( ^ be continuous and let f ' S ^
Let x , x , . . . , x
Q
1
defined as follows:
^ 0
e J and let a sequence (x ) be
n
i
Let
g
be an interpolatory polynomial
for 3 such that the first s - 1 derivatives of 0^
g
are equal
to the first s - 1 derivatives of 2f at the points y^fY^-^* • • • ^ i Define
x
i+l
=
Vs
( x
i
; x
x
l-l"-" i-n) -
0)
%3^ '
4.2-9
Let e ^
= x ^ j - a. Let
,r-l
Suppose that MT
< 1,
for all x e J, and let M = ^ A g .
Then, x
±
e J for all 1 , e -* 0 , and
±
(4-15)
where p is the unique real positive root of
n
t
n + 1
(4-16)
- s
«J=o
and where
r
Y (x) =
r
{r)
(-D Z (v)
r! [S'(y)]
]
y=f(x)
Also
i±^^|a
where
(o)|M/M
4.2-10
We close this section with a number of comments.
We have proven that the order of any I.F. generated by
inverse interpolation is given by a certain number p without
making any a priori assumptions about the asymptotic behavior
of the sequence of errors.
If we had assumed a priori the
e
e
existence of a number p such that I ^ ] _ I/I i I
P
+
converges to
a limit, then it would have been much easier to prove that
this number p was determined by the indicial equation ( 4 - l 6 ) .
then cp
n, s
If n = 0 , then m
is a one-point I.F.; if n > 0 ,
n, s
is a one-point I.F. with memory. The order p is an
integer if and only if n = 0,- that is, if the I.F. has no
memory.
Observe that the asymptotic error constant of the
sequence {y^} depends upon OL^ whereas the asymptotic error
constant of the sequence { x ^ depends upon Y^.
We shall see
that whenever we deal with direct interpolation, the asymptotic error constant of the sequence (x ) will depend upon A^.
i
Recall ( 1 - 8 ) that A
r
is the same function of f as <2 is of 3.
r
Thus, in a certain sense, {y^} plays the same role for inverse
interpolation that (x^} plays for direct interpolation.
Values of p for different values of n and s may be
found in Table 3 - 1 with k = n + 1 and a = s.
4.2-11
4.23
The order of iteration functions generated
by direct interpolation.
We now investigate the case where a
new approximation to a is generated by solving the polynomial
which interpolates f.
We immediately turn to the case where
all the 7j are equal to s.
x
x
x
Let j^ j__]_* • • • ' i zero a of f.
b
e
n
+
n
1
a
PP
r o x l m a t i o n s
to
a
Let P
be the polynomial whose first
n, s
s - 1 derivatives are equal to the first s - 1 derivatives
x
X
x
of f at J_j J__I' • • • > j_- '
n
Define a new approximation to a by
x
x
Then repeat this procedure for i i> i>
+
x
• • • ' j_- +l
#
n
Since P
is a polynomial of degree r - 1 , where
n, s
x
r = s(n+l), j_ 2
w
i
l
+
(4-17) •
'
1
n
o
t
generally be uniquely specified by
It is not even clear a priori that P
has a real
*
n, s
zero in the neighborhood of a.
We shall prove that under
suitable conditions P^
does possess a real zero in the
n, s
neighborhood of a.
Under certain hypotheses we shall, in
fact, be able to prove much more.
4.2-12
x
X
x
Let J = jx |x-a| <; r|- and let J_>J__;L> • • • * i Let f
,
be nonzero on J.
J #
If the ^..j bracket a, then it Is
clear that P^ has a real zero in J.
^
nj s
Hence it is sufficient
n
x
to investigate the case where all the -j__j l i
of a.
e
n
e
o
n
o n e
side
We shall first prove
LEMMA 4-2.
Let
J = |x||x-a| £ r|.
Let f ^
of a.
be continuous on J and let f' / O . o n J.
Let
Let
for all x e J.
Suppose that
^ i a L U , ; ! .
v
2
r
Then P
has a real root, x.,-., which lies in J.
n,s
-L+J-
(4-18)
I
]
4.2-13
PROOF.
To prove the lemma It is sufficient to prove
the result for the case that
> a,
and f' > 0.
prove that
Hence P
n
s
j = 0,1, .. .,n,
( x ) * fCXj^) is positive.
1
(a-r) is negative.
n, s
Q
p(t) - f ( t ) . -
f
(
r
Since P
| p ^
)
5
M
We shall
interpolates f,
n} s
Q
8
(t-x^) ,
«J=o
where £(t) lies in the interval determined by
x^ j X j ^ _ - ^ , .,. *x^_^, t.
Then
p(orr) = f(a-r) -
where £ s £(a-r).
f (
*fo>
n
3
(a-r-x^j) ,
Furthermore,
r
p(a-r) = - rf(n) - - ^ - f ^ U )
where T| lies in (a-r,a).
n
J]
Hence P(a-r) < 0 if
J=o
(r+
s
X j L
. a) ,
r
4.2-14
Since
2
1
the proof Is complete.
Hence P
(4-18)
holds.
M
has a real root in J provided that
We shall now show that under certain conditions
we can prove a much stronger result without demanding that
(4-18)
holds.
LEMMA 4 - 3 . Let
J = jx |x-a| £ rj-.
Let f ^
be continuous on J and let f 'f( ) / 0 on J. Let
x >x^,...,x
0
side of a.
r
n
e J and assume that these points all lie on one
Let these points be labeled such that x
n
is the
closest point to a. Let
f(x )f
n
( r )
(x ) > 0,
n
f'(x )f( (x ) < 0 ,
r)
n
n
r even,
(4-19)
r odd.
(4-20)
r
4.2-15
Then P„ „ has a real root x^
such that
n,s
n+i
minta.x^] < x
PROOF.
n + 1
< max[a,x ].
n
There are four possible cases depending on
the signs of f(* ) and f .
n
We shall prove the result for
only two cases which will give the flavor of the proof; the
other cases may be handled analogously.
Case 1.
f(* ) > 0, f' > 0.
that P„ (a) < 0.
n, s
%
Since P„ interpolates f,
ny s
a
p , ( t ) = f(t) - f
n
We need only prove
n
(
r
)
s
i;pn
8
n
( t - x ^ j ) . (4-2i)
Hence
P
II
r)
=-^T^ (i)
n,s^
(
s
X l
. a) ,
r
J=0
x
where £ = £(a) . For this case ^ _ j - a > 0.
r_1
r
is negative if (-l) f^ ^ is negative.
Hence P
n
(a)
Q
The proof of Case 1
may now be easily completed.
Case 2.
that P ^ ( a ) > 0.
n
s
f
(
x
n
) < 0, f
> 0.
We need only prove
From (4-21),
*»,.<«> - - ^ P -
n
3
(c-x^j) .
r
4.2-16
For this case a - x _ j > 0.
Hence P ^ ( a ) Is positive if
±
n
s
r
-f( ) is positivej that is if
r
f(x )f( ^(x ) > 0,
n
f'(x )f
n
n
(r)
( x ) < 0.
n
Let the hypotheses of the preceding lemma hold,
We can conclude that if a
a < x
If x
< x , then
R
< x ,
± + 1
i = n,n+l,... .
±
(4-22)
< a, then
x
Let (4-22) hold.
i ^ i + l ^ CL'
x
=
n
'
n + 1
'»"
•
(4-23)
Then the sequence {x^} is monotone decreas-
ing and bounded from below.
x
Hence it has a limit and
i + l " i - j ~* °'
J-0,l,...,n.
x
Label the limit £.
f
1
Let t = x
-
£
4
± + 1
F
in (4-21).
n
(4-24)
Then
8
("iw-'i.,) .
4.2-17
x
where | = l( j_ )
a
n
+1
P
(x ,) = 0 .
n,s i + 1 '
x
x
~* a.
1
where we have used the fact that
Let i
4j
f(5) = 0 .
d
oo. An application of (4-24) yields
Since f' is assumed to be nonzero. £ = a and hence
The same conclusion would have been obtained if we
had started with (4-23) • We summarize our results in
THEOREM 4 - 2 . Let
J = jx |x-a| £ rj*.
Let r = s(n+l) > 1 .
f
, (r)
f
'
o on J.
r
Let f^ ^ be continuous on J and let
Let x ,x ,...,x_ e J and assume that these
o 1
n
n
points all lie on one side of a.
Suppose that
f(x )f( )(x ) > 0 ,
r
±
±
f'(x )f( )(x ) < 0 ,
r
±
±
where i is any of 0 , 1 , . . . , n .
r even
(4-25)
r odd
(4-26)
Define a sequence { x ^ as
follows:
Let P
be an interpolatory polynomial for f such
n, s
that the first s - 1 derivatives of P_ ^ are equal to the
n, s
first s - 1 derivatives of f at the points i' j__i* • • • > ± x
x
Let jL+^ ^
e a
0
x
x
9
n
31
P ^ ^ (whose existence we have verified in the
preceding discussion) such that x. . is real, P^
i"t"j.
n, s
(x. ...) = 0 ,
i"T"jL
and
min[a,x J <
i
x
i
+
1
< max[a,x ].
i
Then the sequence {x^} converges monotonically to a.
r
4.2-18
This result is well known for the special case
n = 0 , s = 2 which is Newton's I.F.
For this case the condi-
tions of Theorem 4 - 2 are known as Fourier conditions
(Fourier [ 4 . 2 - 1 ] ) .
Theorem 4 - 2 gives a sweeping generalization
of Fourier 6 result.
1
Although the sufficiency of the Fourier
conditions are geometrically self-evident for Newton's method,
they are not self-evident in the general case.
Observe that the hypotheses of Theorem 4 - 2 do not
place any restrictions on the size of the interval where
(r) /
monotone convergence is guaranteed other than that f'f ' f 0 .
Note that the condition that x ,x ,...,x all lie on one side
o 1
n
v
n
of J is automatically satisfied if n = 0 .
(4-26)
If ( 4 - 2 5 ) and
do not hold, then we shall have to place restrictions
on the size of the interval where convergence is guaranteed.
We have already seen in the proof of Lemma 4 - 2 that in the
general case we require a condition on the size of the interval
in order to assure that P_
has a real zero in the interval.
n, s
To prove convergence in the general case we start
again with
f
( r )
te,(t)]
*»..<*>-'<*> — F T —
n
n
j=o
<*-*i-j> s
1
4.2-19
Then
n
p
n
,s
( a )
=
-^
f ( r ) ( C
i
)
e
n ^
J-0
X
A
i-j - I-J - >
Let the conditions of Lemma 4 - 2 hold.
real x
e J such that P
1 + 1
n
s
(x
1 + 1
n
where T l
1+1
8
±+1
^
( A )
-
Then there exists a
) = 0.
P , (a) = (a-x
^
Furthermore,
)P'(Tl
1+1
),
lies in the interval determined by a and
Then
i+r
Assume that p'
does not vanish in the interval determined
n,s
by a and
x
i + 1
*
Then
n
i+1
=
H
"i+1 II "i-ji+i II
l-y
j=0
e
(4-27)
1 + 1
r
!
c
<
w
4.2-20
Observe that if none of the set
zero and if f d o e s
e
e
0
> i>•••*
e
i s
n
not vanish on the interval of Iteration,
then e^^ is not equal to zero for any finite i.
We shall show, however, that if the initial approximations x , x , . . . , x , are sufficiently close to a, then
Q
e
A
0.
1
n
Let
for all x e J.
Let
P
n,s|
^ 2
for all x In the interval determined by a and
H = v /|x .
1
2
Then | H
± + 1
x
i + 1
«
| £ H and
n
5
i+i *
H
II
d
l-y
5
i-j -
An application of Lemma 3-1 shows that if
HT
then e
A
-*• 0.
r _ 1
< 1
K-jl-
l^t
4.2-21
Observe that the argument used In the proof of
Lemma 4 - 1 could not be used because
depends upon
Since
we conclude that
H
i+1 -
r
(-l) A (^).
r
A
r
= ijp-.
All the conditions of Theorem 3-3 are now satisfied and we
conclude that
le
'
i n a U
|A (a)H
/ ( r _ l )
r
(4-28)
where p is the unique real positive root with magnitude
greater than unity of the equation
t
n + 1
- s £
tJ = 0.
Our results concerning the convergence and order
of I.F. generated by 'direct interpolation are summarized in
r
4.2-22
THEOREM 4 - 3 .
Let
J = |x |x-a| £ r j .
r
Let r = s(n+l) > 1.
f
, ( r ) ^ o on J.
Let f^ ^ be continuous and let
Let x ,x ,...,x
f
^T-
for all x e J.
e J.
1
1
1 v,,
Let
|f'| 1 v.
Suppose that
21 r r-l
2
v
r
<
1 #
( 4
_
2 9 )
2
Define a sequence {x^} as follows:
Let P
n
be an interpo-
g
lators polynomial for f such that the first s - 1 derivatives
of P
are equal to the first s - 1 derivatives of f at the
n,s
x
points x ^ x ^ . ^ • . . > ^_ -
Let
n
x
is real, P
n
s
( ^ l)
+
=
Oj
a
n
d
x
j_+i
e
be a point such that x^+i
J
*
T
h
e
e x i s
t e n c e of such
a point is assured by ( 4 - 2 9 ) • Define $
by
n, s
x |+1 ~ n , ( i
$
x
;
x
S
Assume that
P
determined by a and x
1 + 1
1
Hr ^
1
< 1.
;> |i for all x in the interval
n,s
• Let H = v /|x and suppose that
Let e _j = x ^
±
x
i-l' • • " i-n) •
2
1
- a.
2
4.2-23
Then, x^ e J for all 1 , e
IfldiU
-* 0 , and
1
( „ ) ( P - l > ^ ,
u
(4-30)
where p is the unique real positive root of
t
n
+
1
-
8
£
t
=
J
0,
J=o
and where
f
r
NOTE.
The point
( r )
r!f>
#
may not be uniquely defined
by the hypotheses of the theorem.
Additional criteria must
then be imposed in order to make $
a single-valued function.
n, s
The hypotheses of this theorem may seem rather
strong; in the cases of greatest practical interest, however,
some of the conditions are automatically satisfied.
See the
examples of the next section.
The form of ( 4 - 3 0 )
form of ( 4 - 1 5 ) •
is strikingly similar to the
As before, the only parameters that appear
are r and p.
If n = 0 , then <f>
is a one-point I.F,; if n > 0 ,
n, s
then $n, s is a one-point I.F. with memory,
4.3
Examples
In all the examples of this section J will denote
the interval defined by
We shall find that both the Newton I.F. and the secant I.F.
may be derived from both direct and inverse interpolation.
For a given I.F., the conditions sufficient for convergence
may vary depending on the method of generation of the I.F.
This should not be surprising since Theorems 4 - 1 and 4 - 3 were
derived for general families of I.F.
Hence the conditions
sufficient for convergence, for a special case which Is
covered by both theorems, need not agree.
The notation in
this section is the same as in the previous section.
The Yj
are defined by ( 1 - 8 ) .
EXAMPLE 4 - 1 .
We perform inverse interpolation
with n = 0 and s = 2 .
Q
o,2
( t )
=
Hence r = 2 and
*i
+
(^i)*!'
S
i
=
^ i ^
Then
x
l l - Oo^*!*
+
=
Q
o,2
( 0 )
=
X
i "
u
u
II
4.3-2
This is Newton's I.F.
^
*"( i)
1
e
i+l
=
Then
2
9
1
f
"(*i)
e
" i
7 T T i " 5
, 2 2
,
2
^
2
[ f TU, ] e ,
(4-31)
where 0
*
±
= f ( £ ) , p^^ = f (T^) . Let x
±
continuous and f'3" ^ 0 on J.
for all x e J.
Q
e J and let f" be
Let
Let M = 7\ /A| and suppose that Mr < 1.
Then
1
0 and
e
I+l
t t - Y (a).
(4-32)
2
No absolute value signs are required in (4-32) since the
method is of integral order.
EXAMPLE 4 - 2 .
with n = 0 and s = 3•
Q
o,3
We perform inverse interpolation
Hence r = 3 and
2
( t )
=
3
i
+
(t-yi)^! +
"
V
Then
x
i+l - Po,
c
3
( x
i
)
=
*o,3
=
( 0 )
x
i " i " V
u
i K '
x
A
2
-
•&>
and
_
e
Let x
Let £
Q
±
+
1
"
3
6[3'(
P l
n
3
e
±
e J and let f"' be continuous and
13'|
£
^ *
for all x e J.
2
'
/ 0 on J,
Let M =
\/X^
and
p
suppose that Mr
< 1.
Then
e
-+ 0 and
i+i
f*-
The I.F. of the form cp
Y (a).
3
are of such importance
that they are given the special designation E .
s
They will be
studied in considerable detail in Chapter 5 ; their properties
form the basis for the study of all one-point I.F.
EXAMPLE 4 - 3 .
with n = 1 , s = 1 .
We perform inverse Interpolation
Therefore r = 2 and
W
Q
l,l
( t )
- *i
+
l
(t-yi)
where we have used the Newtonian form of the interpolatory
polynomial as given In Appendix A.
x
{X
i + l - *1,1 ±
;
X
i-1
This is the secant I.P.
e
}
=
Q
l,l
( 0 )
Then
=
X
x
l~ i-l
x
f
i" i-l
f
i " i
f
Then
i+l " " 2 S'lp^S'tPi^) i i - l '
e
e
Let x ,x^ e J and suppose that the other conditions of
Q
Example 4 - 1 hold.
Then e
i
-* 0 and
1
l!i±l[-. |Y ( a ) ! ^
|e |P
'
^
'
2 (
±
where p = |(l+v/5) ~ 1 . 6 2 .
4.3-5
EXAMPLE 4 - 4 .
n = 0, s = 2.
We perform direct interpolation with
Therefore r = 2 and
?
f
o,2^
- i
+
(t-^K.
Then
x
(x
This is Newton's I.F. again.
nomial,
=
i + l " *o,2 i>
x
Since P
u
i " i*
Q
is a linear poly-
2
is always uniquely specified.
Since P
Q
2
* f
the hypothesis of Theorem 4 - 3 which is concerned with the
nonvanishing of P
is automatically satisfied by the condin, s
tion on the nonvanishing of f .
Also ( 4 - 2 7 ) becomes
f"U )
2
±
e
Let x
on J.
i+l
=
Q
X
i
)
i-
( 4
^
3 3 )
s J and let f" be continuous and f 'f" ^ 0
Q
Let f(x o }f"(x o ) > 0 .
that if x
2f'(
e
From Theorem 4 - 2 , we conclude
< a, then x^ converges to a monotonically from
below while if x
from above.
Q
> a, then x
±
converges to a monotonically
Furthermore
^ T - A ( a ) .
e
(4-34)
i
Observe that since Yg(a) = Ag(a), ( 4 - 3 2 ) and ( 4 - 3 4 ) do not
contradict each other.
If we perform direct interpolation with n = 1 and
s == 1 we will again derive the secant I.F.
form *
The I.F. of the
will be studied in Section 5.3 with emphasis on
the study of *
Section 10.2.
,.
The case n = 2, s - 1 will be studied in
5.0-1
CHAPTER 5
ONE-POINT ITERATION FUNCTIONS
In this chapter we shall study the theory of onepoint I.F.
These I.F. are of integral order.
One particular
basic sequence, E , will be studied in considerable detail
s
in Section 5 . 1 . By using E
as a comparison sequence, we draw
conclusions about certain properties of arbitrary one-point
I.F.
We consider Theorem 5 - 3 to be the "fundamental theorem
of one-point I.F."
5.1
The Basic Sequence E
2
s
We recall our definition of basic sequence.
A
basic sequence of I.F. is an infinite sequence of I.F. such
that the pth member of the sequence is of order p.
Technique
for generating basic sequences, some of which are equivalent,
are due to Bodewig
E. Schroder [ 5 . 1 - 4 ] ,
[ 5 . 1 - 1 ] , Curry [ 5 . 1 - 2 ] , Ehrmann
Schwerdtfeger
Whittaker [ 5 . 1 - 6 ] , among others.
Householder [ 5 . 1 - 8 ,
[5.1-5L
[5.1-3]*
and
See also Durand
Chap. 3 ] * Korganoff [ 5 . 1 - 9 *
[5.I-7],
Chap. 3 ] *
Ludwig [ 5 . 1 - 1 0 ] , Ostrowski [5.1-11, Appendix J ] , and
Zajta [5.1-12]-
From Theorem 2 - 6 we know that two I.F. of
order p can differ only by terms proportional to u^. Hence
if the properties of one I.F. of order p are known, many of
the properties of arbitrary I.F. of order p may be deduced.
If the properties of a basic sequence are known, then many
of the properties of arbitrary I.F. of any order may be
deduced.
In Section 5 . H we shall study a basic sequence
whose simplicity of structure makes it useful as a comparison
sequence.
5.1-2
5.11
The formula for E .
s_
The I.P. m
were
n, s
T
defined and studied in Section 4 . 2 2 .
are without memory.
If n = 0 , these I.F.
Because of their importance, the cp
O, S
are given the special designation E . The conditions for the
s
convergence of a sequence generated by E were derived in
s
a
a
Section 4 . 2 2 ; we assume that these conditions hold.
In con-
trast with the careful analysis which is required in the
general case, we shall find that the proof that E
o
is of
s
order s is almost trivial.
In order that the material on E be self-contained,
we start anew. Let f' be nonzero in a neighborhood of a and
(s)
let f ' be continuous in this neighborhood. Then f has an
inverse 3f, and 2f v i s continuous in a neighborhood of zero.
g
v
v
o,os be the polynomial whose first s - 1 derivatives agree
with 2f at the point y *» f (x) .
Then
(
)
t
«(*) - ^ . . ( t ) + « ' f f n
8
(t-y) .
and
-
1
,(J>
where e(t) lies in the interval determined by y and t.
Ea => 0
^o,s(0)
''
K
Define
5.1-3
Hence
E
s
"
I
^
^
J
)
f
(3-D
J
,J=0
or
> =*-Z^^.
(-)
S
5
2
Furthermore,
a
»
Eg
s)
(5.3)
s
.+ Lzl^lV (e)f ,
where 6 = 6(0).
In ( 5 - 2 ) , E
o
is expressed as a power series in f ^.
For some applications It is more useful to express E
as a
s
power series in u where u = f / f .
Hence we write
s-1
-1
In practice fj is not known and we must express E
in terms
s
of f and its derivatives.
Y (x)
J
=
With the definition
(-D'W^fy)
JI[S'(y)]
J
y-f(x)
we
may
w r i t e
s^l
s
E
(
x
)
*
=
[
~
Z
Y
J
u
(5-4)
J
and
s
a
=
E
(-D *
+
( s )
(e)
s ! [ S ' ]
8
u
s
t
(5-5)
s
Thus
a
We
may w r i t e
f o r m a l l y
=
E
+
g
0 [ u
s
] .
(5-6)
t h a t
00
a
=
x
"
Z
J
V
(5-7)
'
j = l
The
s t r u c t u r e
of
Assume
a b o u t
z e r o .
Prom
t h e
w i l l
t h a t
s
i n v e s t i g a t e d
does
n o t
v a n i s h
i n
i n
S e c t i o n
an
(5-5),
v
_ (-i) - g( )f ) / _ u _ y
a
( x - a )
S i n c e
3 0( ^( )
be
s
1
s
0
s
s ! [ 3 ' ]
s
'
i n t e r v a l
5.13.
we conclude that
(5-8)
(x-a)s
Let x. = x, x
E
e
= s' l
1+1
85 x
1 1 1 6 1 1
i "
(5-8) may be
written as
Y (a).
Q
Hence E
Is of order s and has Y (a) as its
g
g
asymptotic error constant.
of E
Since the informational usage
is s, its informational efficiency is unity.
is an optimal basic sequence.
(These terms are defined in
Section 1.24.)
It is easy to see that
where
y
A
= f(x ),
±
Hence E
3
1
(-l) " ^(0),
Bodewig [ 5 . 1 - 1 3 ] attributes E
to Euler [5.1-14].
o
s
In the Russian literature, these formulas are credited to
Chebyshev who wrote a student paper entitled "Ca.lcul des
racines d'une equation" for which he was awarded a silver
medal.
This paper which was written in 1 8 3 7 or 1 8 3 8 has not
been available to me.
We show that the formula of E. Schroder
is equivalent to E .
g
Observe that
1
jl
dy
[5.1-15]
evW__\
d
f '(x) dx'
75
K Y
_
1
' " f '(x) •
Then from ( 5 - 2 ) ,
s
L
y.
which is Schroder's formula.
1
(x)
\T^Tx) dxj
TiJFf'
Compare with the formula for a
Burmann series given by Hildebrand [ 5 . 1 - 1 6 , p. 2 5 ] .
5 . 1 2
t h a t
t h e
s e r i e s
E.
An
f o r m a l
i n f i n i t e
s o l u t i o n
S c h r o d e r
of
w h i l e
of
A.
i f
I f
I f
n
a
In
c e r t a i n
s e r i e s
q u a d r a t i c
f o r
a
c a s e s
one
c o n v e r g e s
e q u a t i o n
i s
t o
can
show
a.
s t u d i e d
A
by
[ 5 . 1 - 1 7 ] .
EXAMPLE
i n t e g e r .
e x a m p l e .
=
n
^
- 1
f ( x )
5 - 1 .
2
t h i s
t h i s
=
x
n
C o n s i d e r
f ( x )
l e a d s
a
l e a d s
-
A,
t o
t o
=
x
n
f o r m u l a
a
f o r m u l a
=
( A + y )
f o r
-
A,
f o r
t h e
w i t h
n t h
n
an
r o o t s ,
r e c i p r o c a l
t h e n
»(y)
l
/
n
and
J - l
n
Then
x
In
p a r t i c u l a r ,
+
x
( 5 - 9 )
5.1-8
If n = 2 , this is Heron s method for the approximation of
1
The formula ( 5 - 9 ) was derived by Traub
square roots.
If n = - 1 ,
using the binomial expansion.
E
s-l
= x £ (l-Ax) ,
J
s
[5.1-18]
(5-10)
<J=0
and
00
This geometric series converges to l/A if |l-Ax| < 1 .
Rabinowitz [ 5 . 1 - 1 9 ] points out that ( 5 - 1 0 ) may be used to
carry out multiple-precision division.
See Durand
1
[5.1-20,
1
of In A, sin"' A, and tan"" A.
pp.
for the approximation
5.1-9
5.13
The structure of E . We defined E as
.
s
s
rr
s-i
E
s
y
= x
'
y
J
j-i
J![*'(y)]
J
y=f(x)
It is easy to show that Y^ satisfies the differencedifferential equation,
jY^x) - 2(j-l)A (x)Y _ (x) + Y ^ U ) = 0,
2
J
Y^x) = 1,
1
Ag(x) =
(5-11)
The first few Yj may be calculated directly from this equation,
An explicit formula for the Y^ may be derived from
the formula for the derivative of the inverse function derived
in Appendix B.
We have
'f* I
r
(-l) (j r-l)!
+
II
" ^ j - ,
1=2
V
with the sum taken over all nonnegative integers p
2,
1=2
and where r = 2 £
±
2
p
1 #
(5-12)
such that
(i-DPi = J " 1 ,
For j = l, p
the definition of Y^ and ( 5 - 1 2 )
±
= o for all i.
we have
(5-13)
Prom
5.1-10
THEOREM 5 - 1 . Let
Y,(x) - ( - l ) - y
J
J
j )
(y)
,
«Ji[3'(y)]
where f and
A
( )
X
=
J
y=f(x)
are Inverse functions.
Then
i=2
1
with the sum taken over all nonnegative integers
^
such that
(i-l)p = J - 1 ,
±
i=2
and where r =
0^-
The first few Yj are given in Table . 5 - 1 •
Observe
that Yj is independent of f; it depends only on the derivatives of f.
We conclude that
1=2
j=l
1
(5-14)
where the Inner sum Is taken over all nonnegative Integers p
such that 2 £
±
2
(i-l)^
1
= j - 1 and where r = 2 £ p .
1
2
±
±
5.1-11
TABLE 5-1.
FORMULAS FOR Yj
1
5A| - 5 A A
g
3
+ A
l4Ag - 21AgA
3
4
+ 6AgA^ + 3A^
5.1-12
By replacing the upper limit of the first sum by <»,
we obtain a formal infinite series formula for a.
The first few E
are given byx
s
Eg - x - u,
E3 = E
2
2
- A u ,
2
(5-15)
E
E
4
5
= E
=
E
3
3
4
- (j2A| - A ) u ,
3
- (5A
3
- 5A A
2
3
+
h
k^* .
The following corollaries follow easily from
Theorem 5 - 1 .
9
COROLLARY a.
Y^ is a polynomial in A , A , . .., A^
COROLLARY b.
A^ is the same polynomial in
f * • •9
g
3
that Yj is in Ag • A^ ^••# «Aj«
COROLLARY c.
polynomial is unity.
<
The sum of the coefficients in this
5.1-13
PROOF.
1
f(x) - ( l - x ) " .
The polynomial is an identity in x.
A
At x = 0, j =
to f(x) is Sf(y) = 1 - y
- 1
1
f°
r a l
l J*
Let
The Inverse
and when x = 0, y = 1.
A short
calculation shows that at y = 1, Y^ = 1 for all j .
COROLLARY d,
Y
depends on A
1
1
only as (-l)^A .
1
For some applications it is convenient to work with
Zj - Y
J
j U
.
(5-16)
Then
s-1
E
= x-
s
£
Z
r
(5-17)
We have
COROLLARY e.
where
= 0,
The form of
ip
A
is given by
= - 1 . Thus Zj is a polynomial,
homogeneous of degree zero and isobaric of weight - 1 .
5.1-14
It Is easy to show that Z^ satisfies the differencedifferential equation,
JZj(x) - (j-l)Z _ (x) + u(x)zj_ (x) = 0 ,
J
1
Z (x) = u ( x ) .
1
1
(5-18)
In terms of the forward difference operator A, this equation
may be written as
At-JZjU) ] + Z (x)Zj(x) = 0.
1
In Lemmas 5-1 and 5-2 and in Theorem 5-2, we shall
make use of the fact that E
is of order s.
The following
s
lemma enables us to calculate the asymptotic error constant
of an arbitrary I.P. of order p.
LEMMA 5-1.
Let ep be an I.F. of order p and let C
be its asymptotic error constant.
Let
cp(x) - E (x)
G(x) =
-f
,
(x-a)
x ji a.
p
Then
C
Y (a) +
p
lim G(x) .
x -+ a
1
5.1-15
PROOF.
Take cp = cp and
= E
2
p
in Theorem 2-8 and
observe that the asymptotic error constant of E
is Y (a).
1?
JP
A more useful result is given in
LEMMA 5-2.
Let cp be an I.F. of order p and let C
be its asymptotic error constant.
Let
<p(x) - E (x)
H
(x) =
—
-
,
E
x
a
.
p
u (x)
Then
C - Y (a) +
p
PROOF.
lim
x
lim H(x).
a
From the previous lemma,
H(x) =
a
x
IP
G(x) x-q
= C - Y (a),
lim
x
p
a
since
lim
x-a
Ji(*l
x
"
a
=
i.
r
5.1-16
EXAMPLE
Let
5-2.
<p =
x
-
1-A u*
2
T h i s
i s
S i n c e
l i m
E^
H(x)
H a l l e y ' s
-
=
x
u [ l + A
-
- A ^ ( a ) .
The
of
o r d e r
f o r
a l l
t h e
c a l c u l a t i o n
an
I . F .
t o
be
b y
u ] ,
i t
the
and
be
d e r i v e d
e a s y
i s
may
the
comparing
p a r t i c u l a r l y
be
awkward
s i m p l e s t
of
-
t o
in
show
S e c t i o n
5.21
t h a t
w i t h
u s e f u l
cp
5-2.
and
E p
+
the
i s
of
—•
a
.
u
^ .
T h i s
( x )
p
+
1
p
( x ) '
of
the
I . F .
r e s u l t s
theorem
theorem
p e r m i t s
c o n s t a n t
procedure
i f
an
t h i s
e r r o r
development
E
p
a p p l y
a s y m p t o t i c
l i m
x
t o
u s i n g
f o l l o w i n g
order
-
c o n s t a n t
c a l c u l a t e d
The
i n
'cp(x)
e r r o r
however
I . F .
o r d e r
i t
A|(o) - A ^ a ) .
a s y m p t o t i c
order
I t
THEOREM
i s
A | ( a )
3
2-2.
but
2
w i l l
Then
I n t e g e r - v a l u e d
Theorem
which
= Y (a) -
C
w i t h
I . F .
of
and
t u r n s
Chapter
o n l y
i f
of
out
9.
5.1-17
exists and is nonzero.
C =
Furthermore
lim
x -+ a .
uP(x)
where C is the asymptotic error constant of cp.
PROOF.
From (5-6),
a = E
p + 1
(x) + 0[u
p + 1
(x)].
Therefore,
,(») -a_*W
(x-a)
- V l ^ )
p
+0>
P + 1
(x)]
f x x {
x-
p
u (x)
cp(x) - E
p + 1
(x)
p
/
u ( x )
x
u (x)
N
P
/u(xi}
V x-a
a
V " /
y
Since
we conclude that
lim
x -+ a
<p(*) - E
q>Cx) - a
. (x-a)
p
which completes the proof.
J
p
p + 1
u (x)
(x)
5.1-18
Observe that Lemmas 5-1 and 5-2 each involve two
I.F. of the same order;
Theorem 5-2 involves two I.F. whose
orders differ by unity.
The next lemma is used in Section 5.51.
LEMMA 5-3-
W * >
PROOF.
I
- >.(«) - ^
B,(x)
From (5-18),
JZj(x) - £
( J-l)Z _ (x) + u(x) £
t
j
1
J=2
J-2
Zj-i(x) = 0
J=2
This telescopes to
sZ. (x) - Z (x) + u(x) £
1
z
x
j_i( ) = °
J=2
Since Z-^x) = u(x),
sZ (x) - u(x)
s
1 -
£
J=2
z ^ w
-» u(x)E (x)
s
5.1-19
The fact that
completes the proof.
EXAMPLE 5-3.
E
Let s = 2.
Then
q3 = o
"2 " £UE'
^ 2 = x - u - Ju(l-u'),
E
J
s
and
E^ = x -
u - |u(2A u)
2
= x -
u
- AgU
2
R a t i o n a l
5.2
A p p r o x i m a t i o n s
I t
o f t e n
i s
common
p r e f e r a b l e
t o
The
t i o n
a r r a n g e d
t h e
be
d e g r e e s
of
d e n o m i n a t o r .
Such
t h a t
f o r
f u n c t i o n
I n t o
a
r a t i o n a l
t h e
a p p r o x i m a t i o n
a p p r o x i m a t i o n s
t w o - d i m e n s i o n a l
of
an
c a l l e d
a r r a y
I s
[5-2-2,
Kopal
f u n c t i o n s
t h e
a r r a y
numerator
a
Pade
Chap.
t o
of
a
f u n c -
i n d e x e d
d e p e n d i n g
c e d u r e
and
See
[5.2-3,
Wall
on
and
t o
E
t h e
form
a
E
are
s
i s
a
p o l y n o m i a l
d e r i v a t i v e s
a
Pade
t a b l e
c o n s t r u c t e d
i n
u
we
e x t e n d
of
f,
of
I . F .
so
as
t o
w i t h
The
be
c o e f f i c i e n t s
t h e
u s u a l
r a t i o n a l
p r o -
a p p r o x i m a -
o r d e r - p r e s e r v i n g .
s
For
e a c h
s
we
o b t a i n
s
p a r t i c u l a r
we
o b t a i n
t h e
Most
Traub
by
and
t a b l e .
I X ] ,
are
20].
S i n c e
t i o n s
a
p o l y n o m i a l s
[5-2-1],
K o b e t l i a n t z
Chap.
r a t i o n a l
the
E .
s
knowledge
p o l y n o m i a l s
f u n c t i o n s .
may
t o
of
t h e
m a t e r i a l
[5-2-4],
of
-
1
o p t i m a l
o f t e n
t h i s
I . F .
of
r e d i s c o v e r e d
s e c t i o n
f i r s t
o r d e r
s .
H a l l e y ' s
appeared
In
I . F .
i n
r
5-2-2
I t e r a t i o n
5-21
a p p r o x i m a t i o n
nomial
t o
Y ( u , s - l )
E
s
.
I t
f u n c t i o n s
i s
g e n e r a t e d
c o n v e n i e n t
t o
b y
r a t i o n a l
d e f i n e
a
p o l y -
b y
s-1
Y ( u , s - l )
Then
t o
E
=
Q
x
-
Y ( u , s - l )
Y ( u , s - l ) .
which
Vb
•
x
a r e
-
=
We
£
Y j ( x ) u
w i l l
s t u d y
( x ) .
j
r a t i o n a l
o r d e r - p r e s e r v i n g .
fc:5)-
a p p r o x i m a t i o n s
Define
» + * • • - 1 .
« > °.
where
J
R(u,s,a) = ^
Note
t h a t
p r e s e n t
i n
The
b
a
R
are
+
s
R ^(x)u (x),
s
t h e
Q ( u , s , b ) .
term"
T h i s
i s
J
^
(x)u (x),
a b s e n t
g u a r a n t e e s
i n
t h a t
Q
R ( u , s , a )
^
^ ( a )
a
g q
( x ) . 1.
and
=
a.
p a r a m e t e r s
^ j ( x ) ,
chosen
" c o n s t a n t
Q(u,s,b)> ^
so
J = l , 2 , . . . , a ;
Q
s
^ j ( x ) ,
J
-
1,2,...,b,
t h a t
R ( u , s , a )
-
Y ( u , s - l ) Q ( u , s , b )
=
0 [ u
s
( x ) ] .
(5-19)
r
5.2-3
Then
and by Theorem 2 - 7 , i/ ^ is of order s.
Equivalently,
a
is of order a +
b
•+
satisfied if the R
1.
The conditions imposed by ( 5 - 1 9 )
b
are
.,Q , are chosen so that
s, j s, j
a
r
3
J=0
where cd.v , a
_ » 1, for I £ a, cd,
c, a = 0 , for I > a, and
k * min(t-l,b).
For parameters thus chosen,
Since the j ( x ) , 1 ^ J <. s - 1 , depend only on
Y
1 <L J <1 9 - 1 * and since the R
.(x),Q
these Yj(x), we conclude that the ^
a
fc
o
f^,
.(x) depend only on
are all optimal I.F.
A number of formulas of type $ , , together with their asympa, d
totic error constants, may be found in Table 5 - 2 .
^a,o
Note that
s-1
For s fixed, which of the s - 1 I.F. generated by
this process is to be preferred?
There are indications
(Frame [ 5 . 2 - 5 ] and Kopal [ 5 . 2 - 6 ] )
that the I.F. which lie
near the diagonal of the Pade table are best.
3
)
TABLE 5-2. FORMULAS FOR
v
and C
a,
*a,b
Formulas for f
Asymptotic Error Constants
^a b "
= lim
'
x —•a (x-a;
a
C
7
s = 2: tJj~
r
s = 3: y
2
Newton
c
i,«
^ = x - u[l+Y u]
=
X
= Y (a) - r|(a)
3
2
2
= x - u[l+Y u+Y u ]
d
2
Y
*2,1
=
X
"
(f
2 - Y )u]
2 +
U
- \(a)
3
3
Y -Y u
2
Y
C
3
C
1 , 1 , 2
2
3
H a l l e y
• 1-Y u'
= Y (a)
= Y (a)
2
*L,1
s = k:
= x - u,
l, o
a
b
X
2
1 - Y u + (y , - Y^)u
2
2
2,l =
V
a
l,2 =
V
a
)
( a )
3
" Y (a)
2
)
- 2Y (a)Y (a) + Y^a)
2
3
5.2-5
EXAMPLE 5-4.
If two I.F. have the same order, then
a measure of which I.F. will converge faster is given by the
relative magnitudes of the asymptotic error constants.
f(x) m
n
x
1//n
- A, a = A .
Define C
by
a,b
h
a
^a b "
8
(x-a)
3
a
'
b
Then
s = 3:
_ (n-l)(2n-l)
C
2
n
'°
1 1 ~
1
s
. 4.
6c
,
1
c
3,o
C
2'
2
'
2 ,
2*
12<X
n
-
1
d
(n-l)(2n-l)(3n-l)
3
=
2 4 a
2
1
- (n -l)(2n-l)
"
72a3
'
2
_
n(n -l)
c
and
c
lim
n -+ oo
c
= 4,
1,1
n
p
oo
3
lim
n
lim
eo
c
= •2,l
'
2
3
- ^
2,1
=
9,
Let
5.2-6
The "^o ^
form.
Thus Kiss
a
r
e
n
o
t
t
n
e
o n
l y optimal I.F. of rational
suggests a fourth order formula
[ 5 - 2 - 7 ]
which in our notation may be written as
u(l-A u)
p
cp
=
x
(5-20)
^
l-2A u+A u*
2
Snyder
[ 5 - 2 - 8 ]
replacement,
11
derives ^
1
3
(Halley's I.F.) by a "method of
and a fourth order formula by a "method of
double replacement •
!f
When Snyder's fourth order formula is
translated into our notation, it is seen to be identical
with
(5-20)
Hildebrand
which Kiss derives by entirely different methods.
[ 5 - 2 - 9 ,
Sect.
9 - 1 2 ]
studies I.F. generated by
Thiele's continued-fraction expansions.
The
5.22
the
most
f o r m u l a s
f r e q u e n t l y
has
r e c e n t l y
Ey
I n v e s t i g a t e s
method
of
p o i n t s
out
the
t a n g e n t
~
x
and
H.
352]
p .
^
may
p o i n t s
be
out
i n
Lambert.
the
Perhaps
l i t e r a t u r e
i s
T%ir
b y
[5.2-10],
Frame
i n
1870.
of
t h e
h y p e r b o l a s .
t h a t
and
[5.2-12].
W a l l
c o n v e r g e n c e
[5.2-16]
Bateman
m
I . P .
r e d i s c o v e r e d
[5-2-13,
S c h r o d e r
H a l l e y
b e e n
[5.2-11],
Richmond
H a l l e y
r e d i s c o v e r e d
•1,1
I t
of
I t
t h a t
by
t h e
w h i c h
[5.2"15,
Z a g u s k i n
t h e
d e r i v e d
he
c a l l s
P.
113]
method
method
i s
b y
[5.2-14]
S a l e h o v
method
d e r i v e d
was
of
due
t h e
Domoryad.
t o
[5.2-17] (1694).
I f
f
=s x
1
-
1
9
A,
=
H a l l e y ' s
x [ ( n - l ) x
( n + l ) x
n
I . F .
becomes
+ (n+l)A]
n
f
+
( 5
_
2 1 )
( n - l ) A
and
( x - a )
In
t h e
B a i l e y
c u r r e n t
t h e y
was
l i t e r a t u r e ,
[5.2-18] (l94l).
Uspensky
D a v i e s
and
Peck
R.
r e f e r e n c e .
known
t o
Barlow
i s
Dunkel
c a l l
t h e
method
t o
a s c r i b e d
was
d e r i v e d
[5.2-20]
i t
[5.2-22]
[5.2-23] (l8l4).
a s c r i b e
o f t e n
(5-21)
Newton
[5.2-21] (1876)
no
[5.2-25]
(5-21)
E q u a t i o n
[5.2-19] (1927).
g i v e
M u l l e r
12a*
0
p o i n t s
H u t t o n ' s
n o t e s
K i s s
by
out
method
t h a t
[5.2,-24]
Lambert
t o
t h e
t h a t
but
f o r m u l a
and
[5.2-26] (1770),
5,3
A
B a s i c
Sequence
D i r e c t
of
I t e r a t i o n
F u n c t i o n s
G e n e r a t e d
by
I n t e r p o l a t i o n .
In
S e c t i o n
5 . 1
we
s t u d i e d
t h e
b a s i c
sequence
E
^
o
m
g e n e r a t e d
b y
t h e
sequence
b a g i c
i n v e r s e
I n t e r p o l a t i o n
*
g e n e r a t e d
o
a t
b y
one
p o i n t .
d i r e c t
We
e
Oj s
S
t u r n
t o
i n t e r p o l a t i o n
a t
OyS
one
s
p o i n t .
>•?,
a
we
of
They
d e g r e e
i n v e s t i g a t e
p o l y n o m i a l
I . F .
l a t t e r
p o l y n o m i a l
i t e r a t i o n .
n o m i a l s
These
t o
a
be
g e n e r a t e d .
of
have
I . F .
d e g r e e
t h e
l e s s
s
-
v i r t u e
t h a n
t e c h n i q u e
s o l v e d
have
b u t
or
t h e
1
of
e q u a l
w h i c h
w h i c h
drawback
must
be
b e i n g
t o
r e d u c e s
s
t h a t
s o l v e d
e x a c t
f o r
-
In
t h e
p r e s e r v e s
1 .
d e g r e e
t h e
f o r
a t
e a c h
a l l
p o l y -
5.33*
S e c t i o n
of
o r d e r
t h e
of
t h e
r
5.3-2
5.31
The basic sequence 4>
Oj
defined and studied in Section 4 . 2 3 .
are without memory.
.
s
CT
The I.F. <J>
was
n, s
If n = 0 , these
I.F.
Th$ conditions for the convergence of a
sequence generated by <j>
were derived in Section 4 . 2 3 ;
o, s
assume that these conditions hold.
we
In contrast with the
careful analysis which is required in the general case, we
shall find that the proof that $
trivial.
is of order s is almost
o, s
In order that the material on S>
Q
contained, we start anew.
Let
0
be self-
, S
be the polynomial whose
o, s
first s - 1 derivatives agree with f at the point x
and
f(t) = P , ( t ) +
0
gr
s
4
i f
s
(t-x ) ,
±
Then
( 5 - 2 2 )
where i^(t) lies in the interval determined by x.^ and t.
Define x
i + 1
by
P
x
o,s( i l>
+
Let a real root of
( 5 ~ 2 3 )
" <>•
be chosen by some criteria.
the function that maps x. into x.
be labeled *
.
i
1+1
o,s
x
i+l - *o,
( x
B
i>'
( 5 - 2 3 )
Let
Thus
5.3-3
The order and asymptotic error constant of $
are now
o, s
easily obtained.
Set t = a in (5-22).
° = *o,a<«>
where e
»
i
• * # '
(
a
Then
< e
)
- a and where £^ = £^(a) .
1
) . ; .
Since
where Tl^ ^ lies in the interval determined by x
+
1 + 1
and a,
we conclude that
'
P
t
\
f
o,s^i+l^ i+l ^
e
(
B
)
( ^
s
e
s!
i
#
Le* ^ o e nonzero in the interval determined by x . a n d
P
fc
a
( s)
and let f
v
' be nonzero in the interval of iteration.
Observe that p' - * f ' ( a ) .
o, s
Then
a
s
^-(-l) A (a),
s
e
A
8
- f ^ .
(5-24)
i
Slnoe s is an integer, it is not necessary to take absolute
values in (5-24).
We conclude that $
Q
fl
is
a basic sequence
r
5.3-4
5.32
The Iteration function ^ ^ >
The I.F.
0
is Newton's I.F.
We turn to
studied by Gauchy
[5,3-1].
.
Q
<t>
Q
2
This I.F. was already
See also Hltotumato
[5.3-2].
We
must solve
0 = f(x) + f'(x)(t-x) + | f " ( x ) ( t - x )
2
« 0.
(5-25)
Then
$
a
0,3
will
=
x
be
. £1
a
+ ltd
— grr fvi--_ 4
1
)i
2
'
A
u
fixed point of $
0
U
X
=
,
f '
from E ^ .
2
2f' *
^ if and only if we take
the + sign if f' > 0 and the - sign if f' < 0.
of sign is made, then $
A
If this choice
- differs only by terms of O(u^)
Thus
^0,3
=
~ T»
x
+
JT> ( l - ^ A u ) ^ .
2
(5-26)
For x close to a, severe cancellation is bound to
occur in (5-26).
This may easily be avoided by observing that
2
- b + (b -4ac)^ _
2
a
,
5
-2c
b +
2
(b -4ac)*'
and hence taking
«
=
0
(5_ )
x
2 7
1 +
(l-4A u)t
0
r
5.3-5
This last form is clearly best for computational purposes.
Although generations of schoolboys have been drilled to
rationalize the denominator, it is preferable, in this case,
to irratlonalize the denominator.
The generalized Fourier conditions of Theorem
assume a particularly simple form for $
Q
^.
4-2
Let
f'f'"< 0
over the interval of Iteration.
Then if x
verge to a monotonically from above; if x
verge to a monotonically from below.
Q
Q
> a, the x^ con< a, the
con-
Observe that in the
case of Newton's method, monotone convergence is only possible
from one side.
1
This is because we demand that f(x )f"(x ) > 0
for Newton s I.F. and f must change sign as x goes through a.
5.3-6
5.33
Reduction of decree.
We observed that the
use of $ o f s requires the solution of an (s-l) degree polynomial at each iteration.
We can effect certain degree-
reducing changes which change the asymptotic error constant
but which do not affect the order.
Consider the following change.
the t - x factors in (t-x) ~
is Newton's I.F.)
Replace one of
by Eg - x = -u.
(Eg = x - u
Define
( 5 - 2 8 )
We shall show that although R
Is a polynomial of degree s - 2
s
In t - x, it still leads to an I.F. of order s. Observe that
p
0
, 3 < * >
-
V
*
>
o
-
^
r
^
s
2
and
Since
E
2
2
- a - V(x)(x-a) ,
v ( a ) - Ag(a),
and
f(t) - P , ( t )
0 #
+
f
(
3
)
f
p
2
<*-*> - <t-E )
1
f
)
l
(
t
. ,
)
B
>
5.3-7
we conclude that
s
0 «• f(a) = R ( a ) + (-l) (x-a)
f
s
(
B
)
( 0
sJ
.
(s-l)l
'
where i lies In the interval determined by x and a.
x , „ by
V
W
Define
i+1
R (x
s
i + 1
)
= 0.
Then
R (a) - B
where T )
1 + 1
R (Tl
B
1 + 1
)(x
1 + 1
-a),
lies in the interval determined by x
Assume that R ( T 1
S
1 + 1
'i+1
) does not vanish.
±
Then if e
+
±
1
and a.
0,
s
(-l) [A (a) - A ^ ^ ^ A g C a ) ] .
g
EXAMPLE 5-5.
R ( t ) = 0.
3
Let s = 3.
Then we must solve
Prom (5-28),
0 - f(x) + (t-x)[f'(x) + if"(x)(-u)]
or
*
=
X
" I^AgH-
( -29)
5
5.3-8
This is Halley's I.F. which we have now generated by linearizing a second degree equation.
'i+1
(5-29),
From
2
A ^ a ) - A (a)
2
= Y (a) - Y (a),
3
as we found in Section 5 . 2 1 .
EXAMPLE 5 - 6 .
Let s = 4 .
0 = f(x) + f'(x)(t-x) +
(t-x)
Then
;
\ f"(x) - \ f"{x)u
or
0 = u + (t-x) +
(t-x) [A -A u]
cp = x -
2.u
2
2
3
and
1 + [1 - 4u(A -A u)]i
2
3
Then
'i+1
-
[A (a) - A ( a ) A ( a ) ] .
4
One could also arrive at ( 5 - 3 0 )
2
3
(5-30)
by expanding <p into a power
series in u and applying Lemma 5 - 2 .
r
5.3-9
T h e r e
o f
P
O
o f
^
(t)
c a n
t
x
a r e
b e
n u m e r o u s
l o w e r e d
o t h e r
w i t h o u t
w a y s
b y
c h a n g i n g
w h i c h
t h e
t h e
d e g r e e
o r d e r .
I f
o n e
S
y
t h e
-
t e r m s
t h e
d e g r e e
t h e
a s y m p t o t i c
c o n t e n t
w o u l d
i n
b e
P
r e p l a c e
0
j
( t )
3
( t - x )
b y
2
/
0
=
t w o
f ( x )
b y
o n e
r e p l a c e d
b u t
w o u l d
m o r e
+
( E g - x )
f ( x )
w e r e
1
b y
n e i t h e r
b e
E^
t h e
c h a n g e d .
-
x ,
t h e n
o r d e r
We
n o r
s h a l l
e x a m p l e s .
I n
5-7.
=
" "
c o n s t a n t
w i t h
EXAMPLE
8
l o w e r e d
e r r o r
o u r s e l v e s
( t - x )
( t - x ) f ' ( x )
=
2
+
u
2
.
4-
i ( t - x )
2
f " ( x ) ,
T h e n
( t - x ) f ' ( x )
+
^ f ' ( x )
o r
<p =
w h i c h
i s
j u s t
o,4
(
t
)
=
f
(
-
u
-
p
u A
2
,
E ^ .
EXAMPLE
P
x
x
>
+
5-8.
I n
( t - x ) f ' ( x )
+ | ( t - x )
2
f " ( x )
+
\
( t - x )
3
f ' " ( x ) ,
5.3-10
replace one of the t - x In the quadratic term by E^ - x and
replace (t-x)
in the cubic term by ( E g - x )
2
= u .
Then
0 - f(x) + (t-x) f'(x) - \ f " ( x ) ( u + A u ) + \ f"'(x)u
2
J
2
u
cp = x -
1 + A u +
g
This is y
1
2
[k^
derived in Section 5,21,
- Ag^u'
5.4-1
5.4
The Fundamental Theorem of One-Point Iteration Functions
We review some of the terminology introduced in
Section 1.24 which will be used in this section.
The
informational usage, d, of cp is defined as the number of new
pieces of information required per iteration.
If an I.F.
belongs to the class of I.F. of order p and informational
usage d, we write cp e ^ 3 . *
T
^ e informational efficiency
of cp is defined by EFF = p/d.
If EFF = 1 ,
9
EFF,
cp is an optimal I.F,
In this section we consider both simple and multiple zeros.
If the order of cp is independent of the multiplicity m of the
zero, then we say that its order is multiplicity-independent.
The reader familiar with I.F. has no doubt observed
that one-point I.F. of order p depend explicitly on at least
f and its first p - 1 derivatives.
usage of the I.F. is at least p.
Hence the informational
A theorem which gives a
formal proof of this fact is given below.
It is this theorem
which causes us to label as optimal those I.F. whose informational efficiency is unity.
The theorem is quite simple to
prove and the result is in the "folklore
analysis.
11
of numerical
Its importance is that it makes us look for types
of I.F. whose informational efficiency is greater than unity.
Multipoint I.F. and I.F. with memory are not subject to the
conclusions of this
"fundamental theorem of one-point
I.F."
Recall that E s , which was studied in Section 5 . 1 , is of order
p = s.
This fact will be emphasized in this section by
writing E .
We give two equivalent formulations of
THEOREM 5 - 3 *
point I.F.
Let m = 1 .
Let cp denote any one-
Then there exist cp of all orders such that
EFF(cp) = 1 and for all cp, EFF(cp) <; 1 .
Moreover cp must depend
explicitly on at least the first p - 1
derivatives of f,
Let m = 1 . .
ALTERNATIVE FORMULATION.
any one-point I.F.
if cp e
H^r>*
Then for all p, there exist cp e
then d ^ p .
on the first p - 1
Let cp^ e Ip.
I
and
Moreover cp must depend explicitly
derivatives of f.
PROOF.
of all orders.
Let cp denote
Since E^
I . there exist optimal
p ° p p'
e
I.F.
This disposes of the first part of the proof.
From Theorem 2 - 1 0 , the most general I.F. of
order p is
cp = cp + U f
p
x
where U is any function bounded at a.
I
We take cp = E_.
P
is given by ( 5 - 2 ) ,
any terms in l/f.
to take for E
E
E
p
depends
1
p
=
x
.
P
f i ^ i
L
Hence U cannot contain
The most convenient form
( j )
J!
g
explicitly on f , f , f (
highest power appearing in E
p
be cancelled by U f .
is f
p _ 1
,
f
J „
p
_
1
) .
Since the
none of its terms can
Since cp is a one-point I.P., none of
5 . 4 - 3
the f(^) can be approximated by lower derivatives to within
0
terms of 0(u ) , I > 0 .
Hence cp must depend explicitly on
1
f ,f', . .. ,f ^P"" ^ which completes the proof.
The restriction to one-point I.F. is essential
as the following considerations show.
f
[
V ( x S
X
)
3
=
V >
x
u 2
( >
x
+
Observe that
°-tu (x) ] .
3
Recall that
2
E ( x ) = x - u(x) - A ( x ) u ( x ) .
3
2
Since
cp(x)
= x - u( ) -
f
[
X
\ : $
x
)
]
and E^ differ only by terms of 0 [ u ( x ) ] , cp is third order.
3
That is, the second derivative appearing explicitly in
been approximated in cp.
has
Observe that cp uses information at
x and at x - u(x) and is therefore an example of the multipoint I.F. which are studied in Chapters 8 and 9 .
Such
approximation of derivatives is impossible for one-point
We turn to the case of multiple zeros.
I.F.
COROLLARY.
Let m be arbitrary and known.
There
exist cp of all orders, with cp depending explicitly on m, such
that EPF(cp) = 1 and for all cp, EFF(cp) £ 1 .
depend explicitly on the first p - 1
ALTERNATIVE FORMULATION.
known.
Moreover cp must
derivatives of f.
Let m be arbitrary and
Then for all p there exist cp depending explicitly on m
such that cp e I
p
p
and if cp e
then d ^> p.
must depend explicitly on the first p - 1
PROOF.
zeros and F ^
1//m
Define F = f
.
depends only on f ^ ,
Moreover, cp
derivatives of f.
Then F has only simple
I <; j.
An application
of Theorem 5-3 completes the proof.
Note that this corollary assures us of the existence
of optimal I.F. whose order is multiplicity-independent.
A
basic sequence of such I.F. will be explicitly given in
Section 7-3.
Observe that if we define G = f (
m - 1
) and insert
G into any optimal one-point I.F. of order p, we obtain a
one-point I.F. with informational efficiency equal to unity.
^
This approach leads to I.F. which depend
f
( m - l ) .(m)
f
explicitly on
(m+p-2)
A case of greater interest Is-when m is not known. i^Note
that u = f/f' has only simple zeros.
/-^
Replacing f by u in any
optimal one-point I.F. of order p leads to an I.F. which is
of order p.
We conclude that there exist I.F. of all orders
such that EFF(cp) = p/(p+l) for zeros of all multiplicities.
These I.F. do not contain m explicitly.
taken up in greater detail in Chapter 7.
These matters will be
r
5.5-1
5-5
The Coefficients of the Error Series of E
We saw in Section 5 . 1
^
E
that
- a = Y (a)(x-a)
g
s
+ 0[(x-a)
s
g
s + 1
].
(5-31)
In certain special cases, an explicit expression may be found
for the error of E . Thus for the calculation of square
p
roots, f = x - A and
4
2
E
2 "
a
=
e
2(a+e)'
a
=
A
>
e = x - a.
In general, the expression for E
- a, is an infinite series
s
whose leading term is given by the first term on the right
side of ( 5 - 3 1 ) •
E
is^ ( a ) / J ! -
A
The coefficients of the infinite series are
recursion formula for these coefficients will
be found which does not involve differentiation.
An interest-
ing property of the coefficients will be proven in Section 5 . 5 2 .
5 . 5 - 2
5.51
A recursion formula for the coefficients.
We recall the definitions of the following symbols which will
be used frequently;
( \
f(x)
u ( x ) =
^ ,
f
7
. (v
f
a
x
)f
(
j
)
=
_( x^)
.
A
. (,
f
_ |( x^ )
f
x
)
=
(
j
)
e-x-a.
The expansion of u(x) into a power series in e which
will be needed below is derived now.
Define
by
00
u(x) = f(x)/f'(x) =
£
l
ve.
( 5 - 3 2 )
t
1=1
Since
00
f(x) =
£
a ^ ,
f'(x) =
J=l
£
J
a
j
e
J
_
1
'
j=l
and u(x)f'(x) = f ( x ) , we obtain
I
a
l
=
I
V
q
( t + 1
q
a
- ) t l-q'
+
1
-
1 * 2 , . . . ,
q=l
or
t
A
l
=
Z
q=l
V
q
( t + 1
-^Vl-q'
1 - 1 , 2 , . . . ,
( 5 - 3 3 )
5.5-3
and finally
V
l
=
h
"
V
Z
t + 1
"
q )
Vl-q'
t=l,2,...,
(5-34)
q-i
as a recursion formula for the calculation of the
= 1.
In ( 5 - 3 3 )
with
( 5 - 3 4 ) . as throughout this section, the
and
functions which occur are evaluated at a unless otherwise
Indicated.
An explicit formula for the
may also be obtained.
It is not difficult to prove that
l
V j
J-l
where r =
integers
a
i
a
n
d
w
h
e
r
I
e
t
W"-
h
such that
e
J
i=l
"
^
inner sum is taken over all
or ( 5 - 3 5 ) ,
the first few v
be calculated as
v, = 1,
2
v
v
4
3
(5-35)
ia^ = j .
From either ( 5 - 3 4 )
V
V \
=
A
~ 2'
= 2A
=
-
2
4A|
- 2A ,
3
+
7A
2
A3
-
3 A
4
.
l
may
We are ready to turn to the problem of finding the
Define t »
coefficients of the error series.
v
by
a
jS
00
(x) =
£
T
t
/
e
B
1
(5-36)
' .
1=0
= 0 , for 0 < I < s, and
•s
This may be proven directly by induction on s.
Since E (x) e I , we expect x,
0
t
S
o, s
= a.
Let s = 1 .
S
o
C
Then
E ( x ) = x « a + (x-a) = a + e.
1
Now assume t .
a
= 0 , 0 < I < s, t
Ks } S
= a.
a
Substitute
(5-36)
O, S
into the formula of Lemma 5 - 3 ,
«
«
s
W
E
=
.
W
-
to find
00
00
00
l=S
t=0
l=s
which completes the induction.
Substituting ( 5 - 3 6 )
00
V
t=0
into ( 5 - 3 7 )
00
^
V
t=0
yields
00
^
.
u(x)
V
1=0
#
^t-i
A
f
!
r
5.5-5
and using the expansion of u(x) given by ( 5 - 3 2 ) ,
multiplying
t
the series, and equating the coefficient of e
to zero yields
t-1
ST
t,s+l
+
^- W,s
s
I
t + l - r r , s - °r=l
+
rv
may be considered known, ( 5 - 3 8 )
Since the
(~ )
T
5
38
can be used to
determine the t * .
A number of i9
are given in Table 5 - 3
c, s
c, s
These results are summarized in
0
THEOREM 5 - 4 .
Let
00
E
1=0
1=1
Then
t-1
,
s +
l
+
U-sK,s
+
I
r v
T
t + l - r r , s - °'
r=l
with t
= a, T-, -1 = 1 , t .
a
O, S
for
0
<
l
< s and s >
n
v,-L
JL,J_
1 .
= 0 for I > 1 ,
and t , „ = 0
C,S
5.5-6
a
TABLE 5 - 3 -
FORMULAS FOR t .
•I, s
t
i I
0
1
1
a
2
3
a
a
a
1
2
A
3
- 2A + 2A
h
k
2
2
4A| - 7A A
2
3
3
+ 3A^
2 A
2 "
A
3
- 9 A | + 12A A
2
3
-3A
U
5A^ - 5 A A
2
3
+ A^
5 . 5 - 7
Using Table 5 - 3 we find
E
2
( x )
-
(x)
- a
=
( x )
- a
= (5A ,
E
E
4
a
= A
2
e
(2k*
3
2
+ (-
2k\
-
A
-
5A A
3
) e
2
3
3
+
2 A
+ (-
+ A
3
) e
9A
4
) e
4
3
+ (kk\
+ 12A A
3
£
+
-
3
T A ^
-
3A^)e
+ lk^
U
+
k
+ 0 ( e
5
) ,
0 ( e
5
) ,
( 5 " 3 9 )
0(e ).
5
Observe that for the cases worked out above, the coefficient
of e
s
in the expansion of E (x) - a is Y ( a ) , as expected.
s
s
(See Table 5 - 1 for the formulas of Y .)
s
r
5.52
A theorem concerning the coefficients.
The
following theorem may be used to check tables of the t „ _
and has a somewhat surprising corollary.
5-5
THEOREM
I
s=l
Note that since t ,
PROOF.
t « _ = A,.
is equivalent to Z
S —JL
on t.
Z
o^i
For t =
t„
„ — A
v, S
T
1,
. for
r
-
The proof is by induction
v
T
^
= 0 , for I < s, (5-40)
a
= i
s
1
,
2
1
,
w
=
i
1
A
= i«
t
h
Let
I > 2.
Sum
recursion formula for t» _ to obtain
v , S
£
*C
't^-l
t + l - r
s=l
s=l
s=l
s=l
r = l
or
I
I
H,s
s=l
r
l-l
-
I
r=l
r v
t l-r
+
I
s=l
^s'
l
r , s
the
5.5-9
where we have used the fact that t
= 0 for r < s.
An
r, s
application of the inductive hypothesis yields
I
I
l-l
I
t,s *
s=l
T
I
t+l-r r "
r=l
r v
A
I
^
q=l
+
1
" ^
v
q
V l - q
"
t v
l
Finally, the recursion formula for the v^, (5-34), yields
s=l
which completes the proof.
COROLLARY.
Let k be an arbitrary positive integer.
Then
s=l
PROOF.
J^-
s=l
X\.
00
s=l t=0
k
k
oo
k
4=1
8=1
t=k+l
8=1
5.5-10
Since
I
s=l
s=l
an application of Theorem 5 - 5 yields
k
£
k
E ( x ) = ka +
g
£
A^e* + 0 ( e
k + 1
)
1=1
s=l
The fact that
1=1
1=1
and hence that
t=l
yields the corollary.
Taking the limit as k
oo of ( 5 - 4 l )
yields a special
case of the general theorem that if a series is convergent,
then it is also Cesaro summable.
6 . 0 - 1
CHAPTER 6
ONE-POINT ITERATION FUNCTIONS WITH MEMORY
Two classes of one-point I.F. with memory are:
studied:
Interpolatory I.F. and derivative estimated
These I.F. are always of nonintegral order.
The structure
of the main results, given by Theorems 6 - 1 to 6 - 4 ,
remarkably simple.
I.F.
Is
6.1-1
6.1
Interpolatory Iteration Functions
In Chapter 4 we developed the general theory of
I.F. generated by direct or inverse hyperosculatory interpolation; such I.F. are called interpolatory I.F.
If n * 0 ,
the interpolatory I.F. are one-point I.F,; if n > 0 ,
are one-point I.F.'with memory.
they
The reader is referred to
Theorems 4 - 1 , 4 - 2 , and 4 - 3 for results concerning the convergence and order of interpolatory I.F.
The generalization to
multiple zeros is handled in Chapter J\ • In this section
we limit ourselves to comments and examples.
the same as in Chapter 4 .
The notation is
6 . 1 - 2
6.11
Comments.
Observe that Interpolatory one-
point I.F. with memory use s pieces of new information at x^
and reuse s pieces of old Information at the n points
x
i-l' i-2> * ' i-n
x
#
#
x
Thus their informational usage is s.
#
Their order is determined by the unique positive real root
of the equation
n
t
n + 1
- s
)
t
J
= 0 .
( 6 - 1 )
j=0
As was shown in Section 3 . 3 ,
bounds on this root are given by
Hence bounds on the informational efficiency are given by
1 < EFF < 1 + i
s
Furthermore,
f
n i ~*
n
1
P
oo
n+l,s
=
3
+
1
9
Theorem 5 - 3 states that the informational efficiency of any
one-point I.F. is less than or equal to unity.
Hence the
increase in informational efficiency for interpolatory
I.F.
with memory is directly traceable to the reuse of old information.
On the other hand, we conclude from ( 6 - 2 )
that the
old information adds less than one to the order of an Interpolatory
I.F.
6.1-3
The
from
Table
order
a
3-1
w i t h
approaches
f u n c t i o n
case
dependence
of
s o l v e d
of
most
k
i t s
n;
t h i s
p a r t
of
p r e c i s i o n
g e n e r a t e d
degree
of
-
d e g r e e
S e c t i o n
an
r
1
a
on
=
v a l u e ,
i s
s
n
n
s .
p a r t i c u l a r l y
and
may
Observe
•+
1,
f o r
1.
seen
the
r a p i d l y
l a r g e
Then
be
t h a t
q u i t e
true
=
s
s .
(6-1)
as
The
may
be
must
2
i n t e r p o l a t o r y
A
may
s o l v e d
f o r
a l s o
i n
a v a i l a b l e
S e c t i o n
of
i s
t o
w i t h
be
used
a
i t e r a t i o n .
o n e - p o i n t
10.21.
f o r
i s
at
A
I . F .
I . F ,
t h a t
l e a s t
I . F .
p o l y n o m i a l
f o r o n e - p o i n t
f o r
memory
i n t e r p o l a t o r y
t h a t
each
demonstrated
I . F .
have
drawback
i n t e r p o l a t i o n
be
g i v e n
(s +4s)^].
+
a r i t h m e t i c
d i r e c t
i s
i s
and
i n t e r e s t
of
t e c h n i q u e ,
5.33,
example
i s
c a l c u l a t i o n .
by
1
l,s""
drawback
the
+
order
and
p
m u l t i p l e
n
the
l i m i t i n g
p r a c t i c a l
e x a c t l y
A
=
of
of
r e d u c t i o n
in
w i t h
memory
6 . 1 - 4
6.16
Examples.
The reader is referred to Appendix A
for the interpolation formulas used in the following examples.
The notation is the same as in Chapter 4.
We shall not give
the conditions for convergence; such conditions may be found
in the examples of Section 4.3.
The first three examples use
inverse interpolation; the next three examples use direct
interpolation.
We take y^^ = f
1
EXAMPLE 6 - 1 .
= f ( x ^ ) ^ !J
1
- ^(y^),
n ~ 1 , s * 1 , (secant I.F.).
Newtonian formulation,
f
=
x
* ± ~ yi'^i^i-i] - i -
In the Lagrange-Hermite
f t x
1
>
i
x
1
,
1
] '
formulation,
f
x
l
x
i i-l" 'i^l l
e
|Y (oi) P
a
e
"
1
P «•
,
i(l+v/5)
i
EFF -
£
-v
1 . 6 2 .
-
1 , 6 2 .
In the
II
6.1-5
n » 2, s = 1
EXAMPLE 6-2.
ff
cp
2
+
,1 "
4
1
j-*
r
i
[f Lx ,x _ J
+
i
i
i
i
" f Lx^x^g]/'
1
1
- ^ f t — |Y (a)|^P- ),
l'
3
p
„ !.84.
e
1.84.
=
EPF
n = 1 , s = 2.
EXAMPLE 6-3.
' 1 , 2
=
'0,2
+
f
H
i '
a,
-X
^
'o,2 - i " T '
I
x
7
I
H
.
—
f
Xi
1
f
i- i-iif;
| e
1
,]
^I^T;
V l
fl_ .
(f.-f,.,)
1
^±|i- l Y ^ a ) ! * ^ " ) ,
e '
p = l
p
• 1
EFF - -| ^ 1 . 3 7 .
+ v
2
t ;
f
/
3
_ _ 1 _
f
2
; _ / ^ ^ i
^ 2.73.
6 . 1 - 6
EXAMPLE 6 - 4 . n = 1 , s = 1 , (secant I.F.).
In the
Newtonian formulation,
P
l,l
( t )
=
f
i
+
(t-x )f[x ,x _ ],
±
1
1
1
x
" i ' fLx ,x _ J
1
In the
1
1
Lagrangian, formulation,
f
<D
1,1
X
f
X
= l i-l" i-l l
i i-i
f
_ f
Observe that for the case n = 1 , s = 1 , the I.F. generated by
direct and inverse interpolation are identical.
only case, for n > 0 , where this is so.
e
! i+l
e
P
-
p
|A (a)| '
1
p
=
|Yja)|P-\
~
1 . 6 2 ,
2
• 1
p « |(l+
v/5)
EFF =
~
s
1 . 6 2 ,
This is the
6 . 1 - 7
EXAMPLE 6 - 5 . n = 2, s = 1 .
The second degree
polynomial equation
P ^(t) = f
2
h
=
x
±
+ (t-x )f[x ,x _ ] + (t-x )(t-x +h)f[x ,x _ ,x _ ],
±
x
f [ x
i " i-l*
1
1
1
±
i' i-l' i-2
x
x
1
±
"
]
x
1
1
1
i" i-2
x
must be solved for t - x .
i
i f ^ ^ l M a ) ! * ^
1
p . 1.84.
. ) ,
EFP = f ~ 1 . 8 4
10,21
This I.F. is discussed in Section
EXAMPLE 6 - 6 , n = 1 , s = 2 .
The third degree poly-
nomial equation
0 = P
f
H =
x
x
2
(t) - f
+ (t-x )f' +
±
f
x
i - ^ i^ i-l^
x
±
+
x
i" i-l
must be solved for t - x
(t-x )
i
±
+
f
2
(t- ) H,
X l
i-l "
2 f
x
x
t i* i-i
±
(x -x _ )
1
j L
1
2
]
2
6 . 1 - 8
e
i4-1
i ± l i -
| A , ( a ) p
(
p
_
EPF = ^
l
)
,
P
1.37
=
1
+ v / 3
-
2 . 7 3
r
6 . 2 - 1
6.2
Derivative Estimated
One-Point Iteration Functions With
Memory
6.21
The secant iteration function and its
generalization.
We have used direct and inverse interpolation
to derive one-point I.F. with memory.
We observed that the
secant I.F. could be generated from either direct or inverse
interpolation.
We now give two additional derivations of the
secant I.F.; the method of derivation suggests a second general
technique for generating one-point I.F. with memory.
The secant I.F., together with slight modifications
thereof, must share with Halley s I.F. (Section 5 . 2 )
1
the dis-
tinction of being the most often rediscovered I.F. in the
literature.
Discussions of the secant I.F. may be found in
Bachmann[6.2-1],
Jeeves
Collatz
Ostrowski
[ 6 . 2 - 4 ] ,
[ 6 . 2 - 2 ,
Chap. I l l ] , Hsu
Chap.
[ 6 . 2 - 5 ,
3 ] ,
[ 6 . 2 - 3 ] ,
and Putzer
Its order seems to have been first given by Bachmann
Let
( 1 9 5 4 ) .
3
be the inverse to f.
[ 6 . 2 - 6 ] .
[ 6 . 2 - 7 ]
One way of writing
1
Newton s I.F. is
1
Newton s I.F. is second order and uses two pieces of Information.
We estimate 3 ' ( y ) by %
1
where
"3(y)
Q
1 / L
(t) - *(y)
+ (t-y)
-
s ( y i _ i ) '
y-yi_
i-l
6 , 2 - 2
is the first degree polynomial which agrees with S at the
points y and y^.^*
It is convenient to use the symbols y
and y^ and the symbols x and x
3'
by
i
interchangeably.
Replacing
^ in (6-3) leads to
x
which is the secant
x
i" l-l
I.F.
On the other hand, we may write
* 2
x
f '(x)
and estimate f'(x) by differentiating the first degree polynomial
P ^ ( t ) m f(x) + (t-x)
1
1
f(x) - t(x y
x-x i-1
which agrees with f at the points x and X j ^ -
lmml
Again the secant
I.F. is generated.
In the following sections we will deal with two
broad generalizations of these ideas.
a.
Rather than estimating the first derivative of an
optimal I.F. of second order, we estimate the
(s-l)st derivative of an optimal I.F. of order s.
b.
Rather than estimating the first derivative from
two values of the function, we estimate the
(s-l)st derivative from n + 1 values (one new and
n old) of the first s - 2 derivatives.
In general the estimation of f
2r
1
leads to different I.F.
in Section 6 . 2 2
1
1
and the estimation of
We investigate the former
and the latter in Section 6 . 2 3 .
After study-
ing the estimation of derivatives in the optimal basic
sequence E , we will be able to deal with the case of arbitrary optimal I.F.
The one-point I.F. with memory thus gen-
erated will be said to be derivative
estimated.
r
6 . 2 - 4
6.22
Estimation of f f
.
We first develop the
theory of one-point I.F. with memory which are generated by
estimating f
v
' in the I.F. E ; the corresponding theory
s
o
for arbitrary optimal I.F. then follows easily.
Section
From
5-11,
s-1
E
s
= x +
>1
( 6 - 5 )
H
y,(x) _
= -(—- i ) »
s
f
(,y )
J![3'(y)]
3
E
( j )
U
9
~
J J .
J
y-f(x)
uses s - 1 derivatives and hence s pieces of information,
s
1
We estimate f ^ ^ ^ ( x ) from f ^ f x ^ ) ,
1
and I
=
0 , 1 , . . . , s - 2 .
changeably.
with j *
The symbols x and x
i
0 , 1 , . . . , n ,
will be inter-
The I.F. so generated use s - 1 pieces of new
information at the point x^ and reuse s - 1 pieces of information from the previous n points.
Hence s - 1 will be of basic
importance and we define
S - s -
1 .
( 6 - 6 )
In place of r = s(n+l), we introduce
R
«
S(n+1).
( 6 - 7 )
The advantage of using S and R instead of s and r will become
evident below.
r
6.2-5
Let P
Q
P ^S
i-p
( x
n
f
(
" ^
( t ) be the polynomial such that
O
y
Ti
x
i-j)'
J »- 0,l,...,n,
I - 0,1,...,S-1.
(6-8)
pj; I(x)
IJ y O
S
We estimate f^ ^(x) by
.
S
f
* n
S )
<
x )
~
Let
P
n ! s
(
x
)
-
(
6
_
9
)
As shown in Appendix A,
f
(S)
where ^
.
( x )
.
f
U )
(
x
=
)
| j
f
( R )
(
|
i
n
jj
)
lies in the interval determined by x
Let * E ^ o be generated from E
s
f^(x)
by * f ^ ) ( x ) .
,
c
n
(6-10)
3
(x-x^j) ,
i
, x
i
. .. j X ^ .
by estimating
A new approximation to a is defined by
x,., = * E _
i+l " ' n , S ^ i i - l ' * * " i - n ^ *
k
j 5
x
j x
x
We derive the error equation for * E
S
is independent of f^ ^ for j < S.
from Y
Let * ^
S
g
by estimating f^ ^ by * f ^ .
n
g.
n > s
Recall that
be generated
Then
Sri
_
* -n,S" =
EJ
-v
x
_l_
+
I
V*
J-l
+
X Y
n,S
u S
E
- S
+
Y
* n,S
u S
*
( 6
'
1 ; L )
On the other hand,
a = E
+ Y u
s
S
+
s
«(
S + l )
S 4
: L
(© )f ' ,
( 6 - 1 2 )
1
where 0^ lies in the interval determined by 0 and y^. Hence
E
* n,S "
a
"
uS
I* n,S-V
Y
i
"
8
f^TTT
( S + 1 )
(«i)f
S + 1
-
(s)
Since by corollary d of Theorem 5 - 1 *
S
as (-l) A , A
s
*
s
f^ V(S!f')* ^
S
=
S
E
a
_ (-D u
An application of
*E_n , S„ - a = -
depends on f
+x 1
r. (S) _ ( S ) l _ (-l)°;
f
7
only
conclude that
S
S
v
f
g
( S (Sfl),
+l)
( 0
f) l f
N-S+l
yields
( 6 - 1 0 )
R ! f
J
Let
e
x
a
i-j - l-j ' >
e
i+l
=
x
i+l "
a
=
E
* n , S " °"
7 - F'TV ! = g 7 ^ - p
6
±
where T| lies in the interval determined by x
1
P
l
= f (Tl ) .
1
Then
i
and a and
6.2-7
-
*
M
i
e
i
S
(
e
R
*M, -
-
i - r
e
i
)
S
*
+
N
i
e
i
+
1
-
( R )
(-i) f
(^)
"
q l i ' [f'CnJ] ,
v
-
/
(
1
_
1
)
S
+
3
i ( s i )
5
+
(
*N, = ( S + l ) ! [ * ' (
1
P
i
n
(6-13)
}
S+l*
This is the error equation for
„; we use It to
nf o
ume
derive the conditions for convergence and the order. Ass
that e
x
x
is nonzero for all finite i.
i
o' l' *
# #, x
n
a
r
e
s u f f
i
c : J
-
e n t l
y
°l
o s e
We shall show that if
to a, then
e ^ ^ + 0 .
Let
|x-a| £ r j .
Let f
v
' be continuous on J and let f
be nonzero on J.
f map the interval J into the interval K,
R ^ S + 1 ,
on K.
Hence we can conclude that g (
Let
•(H)
Let
Since n > 0 ,
s + 1
)
is continuous
6 . 2 - 8
for all x e J.
Let
v
1 1
v
S
l 3
S+l*
v
2
Let
3(8+1)1
for all y e K; that is, for all x e J.
*N =
Let x .x,,,..,x
e J.
o 1
n
Let
x
Then
max[|e |,|e |,...,|e |] £ r .
o
1
n
By an inductive argument analogous to one employed in
Section 4 . 2 1 ,
it can be shown that if
2
then x
±
R
"
S
e J for all 1.
* M r
R
"
+ *NT
1
Hence
| * M
S
±
£
I,
1*^1
| £ *M,
^ *N for all i
Then
n
5
1+1
< *
M 5
i
II
j=l
6
( l-j
+ 5
i)
S
1
+ ^sf ,
b
±
_ 5= le _j I .
±
An application of Lemma 3-2 shows that if
+
then 6
1
4
-*0.
* N r
<
S
1,
Hence e. -* 0 .
i
Observe that
*M
n
±
-+ - ( - l ) A ( a ) ,
R
*N. ±
Y
s + 1
(a).
All the conditions of Theorem 3-4 now apply and we conclude
that
(6-14)
where p is the unique real positive root with magnitude greater
than unity of the equation
n
t
n + 1
- S
j=0
We summarize our results in
THEOREM 6 - 1 .
Let
r
6.2-10
Let R m s ( n + l ) , n > 0.
nonzero on J.
x
Let
R
Let f^ ^
x
0
* i>*">
be defined as follows;
x
n
be continuous and let f
e J and let a sequence
Let
be defined by (6-8)
be
{x^
and
(s)
(6-9).
Let * E
by
•
n
be generated from E
g
s
+
by estimating f
1
v
'
Define
* * n,s( i' i-l''' " i-n)'
x i+1
E
x
x
x
x
Let
~ i - j " °"
T
s
+
D
for all x e J and let
v
Assume that e
1
1
A
X
2
2
is nonzero for all finite i.
Suppose that
S
a^^Mr *- + *NT < 1.
Then, x
±
e J for all 1, e
±
iy*)l ~
( p
-+ 0, and
1 ) / ( R
1 )
~ .
6
< -
15)
6.2-11
where p Is the unique real positive root of
n
t
n + 1
- S
>
t
= 0,
J
J«0
and where A
f^/iRlf).
=
R
(S)
We turn to the case where f
arbitrary optimal one-point I.F.
v
' is estimated In an
It will turn out that the
order and asymptotic error constant of the I.F. so generated
(S)
are identical with the case where f
' is estimated in £ 5 + 1 •
v
From Theorem 2 - 1 0 ,
* s + i
-
V
i
+
U
f
S
+
<
1
is the most general I.F. of order S + 1.
optimal one-point I.F.
(S)
Let *cp
n s
S
estimating f ^ ' by * f £ )
v
analogously.
in
'
cp
g+1
and let * U ^
S+l
a
+1
"n,S
J
S
"
be an
be generated from c p
Then
n,S ~ " n,S "
Let cpg
6
s + 1
by
be defined
1
6
)
6 . 2 - 1 2
Hence
*<p
Y
.
n , S
- a - *E
q
TI
n,S
- a + *u
f
s+i
n,fa
,
n
*cp
n,S
Q
-
a
*
e . ^
l+l
y
= *M.ef
i i
*U
(e.
< - e . )
l-J
1'
[f
ii-
S
+ (*N.
t
+
S+X
[S'(p.)]
1
—1
J
l
(6-17)
where
were defined In ( 6 - 1 3 ) .
and
We can proceed as
before to arrive at
THEOREM 6 - 2 .
Let
|x-a| £
x
R
Let R = S(n+l), n > 0.
nonzero on J.
Let f^ ^
Let x , x ^ , . . . , x
Q
ry.
be continuous and let f
e J and let a sequence
n
be
{x }
i
be defined as follows$' Let * f £ ) be defined by ( 6 - 8 ) and ( 6 - 9 )
s
Let <P
e
t>
Slfl
ar
* arbitrary optimal one-point I.F.j let
S+1
cp
s + 1
m E
g
+
1
+ Uf
S
estimating f^ ^
.
Let *cp
be generated from c p
n g
by
in < p
s+1
; then
Define
x
i+l
=
c
( x
* Pn,S i
;
x
x
i-i""' i-n^
s+1
by
6.2-13
Let e
1-J
= x1-J
Let
- a,
v
R1
for all x e J.
1 |f'I >
3
v ,
2
Let
q
V V
n
0
+
S+'l*
v
Assume that e
2
H-S
# M r
R-l
±
Nr
A
2
Is nonzero for all finite 1.
* ^.
+
.S+l*
2
<
Then, x
±
x >
e J for all 1 , e
1 ^ ± 4 -
±
-»• 0, and
(p
IVa)l -
l)/(R
l)
- ,
where.p is the unique real positive root of
t
n+l _
s
n
\
j-0
and where A
R
=
Suppose that
/(Rlf)
.
t
J ,
0
j
( 6 - 1 8 )
6.2-14
6.23
1
Estimation of g^"" ^.
We turn to one-point
f s-l)
I.F. with memory which are generated by estimating £
in
v
y
the I.F. E ; the corresponding theory for arbitrary optimal
s
I.F. then follows easily.
We again take S = s - 1 . R = S(n+l); the
symbols y and y^ are used interchangeably.
Let
g be the
polynomial such that
(6-19)
We estimate ^
S
oj^ (y)
\ y ) by
l
.
s
{ 3)
z (y)
s
n
Let
^ (y).
(6-20)
s
As shown in Appendix A,
n
s (y) - ^
(s)
where 6
±
g ^ ) ( ) by
y
( y ) =fTSf
( R )
II
(ei)
1
1
E
n
s
be generated from E g
3^ ^(y).
S
x
i+l ~
+ 1
(6-21)
S
(y-y -j) >
±
lies in the interval determined by y^Y^i>
Let
s
s )
•••
>y±-n'
by estimating
A new approximation to a is defined by
l E
x
n,S^ i
j
x
,x
i-l'*'* i-n^
r
6.2-15
I
We derive the error equation for E
_
_
n,S -
iE
=
a -E
x
S-l
Vv L D l
+
x
L
+
E
g
B
31
(J) J
f
f
S
^
s
+
^ )f
.
g
We have
1
J b | ^ ^S) S
+
N
Izlif
S!
+
y
s) s
f
tf
n
f
( 6
.
2 2 )
^ he )f ,
s
+1
+
2+1
±
(6-23)
where 8
i
(6-21),
lies in the Interval determined by 0 and y^. Prom
(6-22),
(6-23),
and
k
S+l
Since
V j T
where T l
P
i + 1
lies in the interval determined by x
f
i + 1
a
= ^i+l^
w
e
c
o
n
c
l
u
d
e
1
+
and a and
1
that
n
y± i -
X + i ^
+
M
i+1
=
-
II
(yi-j-yi)
R!g'(p
( - i )
L
N
i+1 ~ ~
8
1 + 1
*
)
1
s
+
V ^ f " -
>
^
(SH)J*'(P
1
^ )
i + 1
) '
1
r
6 . 2 - 1 6
Because of the dependence of (6-24) on
Y „y
we
±
shall focus our attention on H,
H =
Let 3 ^ )
A}-
be continuous on H and let 3 ' be nonzero there.
R
Let y y , . . . , y
o J
|y| |y| £
1
e H.
n
through the parameter
Since M
P
i+1
5
depend on
y
we cannot prove that all
y
1 + 1
and N
1 + 1
a manner analogous to the proof of the last section.
we assume that y^^ e H for all I and that y
finite i.
i
i
1 + 1
e H in
Instead
Is nonzero for all
Let
R!
for all y e H.
^
3
V
l 'I^V
(s+1)! ^
V
Let
V
*2
Then
|y
i + 1
l <; lM\y±\s
n
II (| _jl
yi
An application of Lemma 3 - 2
2
S ,
|y J) +
+
s
±
shows that if
R - S ljy^R-1
+
i
N A
S
<
l f
1
1S+1
k\j \
±+1
6 . 2 - 1 ?
then y
A
-*• 0 .
Hence e
±
-*• 0 .
Observe that
S
a
( - 1 ) % ( 0 ) ,
R
(y)
=
(R)
f r ^ { f }
All the conditions of Theorem 3-4 are now satisfied and we can
conclude that
where p Is the unique real positive root of
n
.n+
T A1
„
t"
- S
V
^
4-J -
t
= °-
j=0
Since
e
where
lies in the interval determined by y ^ j
have that
J > l i - | y
R
( „ ) | ( P - l ) / ( » - l ) .
We summarize our results in
a
n
d
a
*
w
e
6.2-18
THEOREM 6 - 3 .
Let
H =
Let R = S(n+l), n > 0 .
nonzero on H.
iy I <; a }
Let
be continuous and let
Let x , x ^ , . . . , x
Q
{x } be defined as follows:
and ( 6 - 2 0 ) .
3
^
by
35 ^ .
i
I
1
Let
S
I
E
n
g
finite i.
i
S
^ ^
be defined by
be generated from E
s
+
1
by estimating
_
x
"n,S^ i
;
x
x
i-l'''*' i-n)'
e H for all 1 and that y
±
±
is nonzero for all
Let
l *
(
R
(6-19)
Define
i+1
Assume that y
be given and let a sequence
n
Let
±
be
)
s
l
R!
,
K
H
s
*
1S
for all y e H and let
\
Suppose that 2
R
_
S
1
Mr
R _ 1
+ W
l
A
2
S
< 1.
2
( S + 1 )
1
,
6.2-19
Then y
-• 0 and
±
^ i ± l l . |
0 ) | ( P - l ) / ( H - l ) ,
V
(6-25)
where p Is the unique real positive root of
n
t
n
+
i
-
i
s
y
j
t
=
o,
and where
(R)
Furthermore,
K±i"_l
V a )
|(p-i)/(H-i),
(6-26)
where
Y (x)
=
.
(-D
R
g
(
R
)
(y)
R
R! [ 3 ' ( y ) ]
The case where %
/—
K
J
y=f(x)
' is estimated in an arbitrary
optimal one-point I.F. may be handled in a fashion which
should by now be familiar.
We summarize the results in
6 . 2 - 2 0
THEOREM 6 - 4 .
Let
H = {y| |y|
Let R = S(n+l), n > 0 .
nonzero on H.
Let 3 f (
Let x ,x^,.,.,x
{x } be defined as follows:
Let cpg
( 6 - 2 0 ) .
s+1
*
E
+
+ f
1
estimating 0
be defined by
( 6 - 1 9 )
be an arbitrary optimal one-point
+1
u
s
) be continuous and let 3 ' be
be given and let a sequence
S+1
let c p
R
A}.
Let
±
and
1
I.P.;
1
•
<P 5 be generated from c p
Let
^ by
n
in c p
s+1
s+1
t»y
; then
Define
x
Assume that y
±
finite i.
i+l "
± c p
x
n,S^ i
;
x
,x
i - l ' * * * i-n^
e H for all 1 and that y
is nonzero for all
±
Let
\^
S + l )
\
(S+1)!
I s
.lu
*
V
I
V s
1
*
K
6,2-21
for all
y
e H.
Let
X
1
R
Suppose that 2 "
Then y
S
X
MA
R _ 1
3
V
+
K
+ ^NA^ < 1
-*• 0 and
i
-
^
4
-
l ^ o ) ! ^ "
1
' ^ - ! ) .
where p Is the unique real positive root of
n
.n+1
^
t
- S > +.tJ _= 0.
1 1 T X
J
Q
j=0
Furthermore,
6,2-22
6.24
Examples.
The reader is referred to Appendix A
for the approximate derivative formulas used in the following
examples.
We shall not give the conditions for convergence.
The notation is the same as in Sections 6 . 2 2 and 6 . 2 3 .
n ~ 1, S = 1 , (secant I . F . ) .
EXAMPLE 6-7.
E
* i,i
^ ± 4 -
=
x
Tp-
i '
1
lAgCcOp- ,
P - i(l+V*>) -
1.62.
n = 2, S » 1 .
EXAMPLE 6-8.
*f' - f [ x , x _ ] + f [ x , x _ ] - f [ x _ , x _ ] ,
1
1
1
±
* 2,i
E
-
x
i
±
2
"
1
1
;7'
*2
I ^ ± 4 - .
e
' i
lAgCa)!*^"^.
P . 1.84.
1
2
r
6 . 2 - 2 3
EXAMPLE 6 - 9 .
n = 1 , S =2 .
* l " x p f ^
{
f
2 f
I
^
+
'
3
f
C x
X
]
i' i-l }'
r
2
1
l A ^ C a ) ! ^
EXAMPLE 6 - 1 0 .
5
"
1
5
U
i
,
f
"
P
-
1
+ 7 3 -
i
2 . 7 3
n = 1 , S = 2 . We estimate f" by * f
in Halley's I.F. rather than in
.
u
Thus
i
- *i " 1 - I ( u X j <
|A (ct)
4
e
i'
| * ( P - D ,
p
.
i
+
v
/
3
^ 2 . 7 3 .
1
6.2-24
EXAMPLE 6 - 1 1 .
n = 1 , S = 1 , (secant I.F.).
1 , '
* 1
-
f [ x
1
, x
±
1
-
1
1
l , l
E
f
[ x
i
1
, x
V
p =
| ( 1 +
,
_
1
1
J
+
[ x
1
, x
2,l
b±ij^,y
l
-
1 . 6 2 ,
1
1
_
J
2
1
E
v/5)
1
f
1
e
'
i
f
n = 2 , S - 1 .
1
*
~
p
• U '
2
i
x
2
EXAMPLE 6 - 1 2 .
3
"
|Y (a)| -\
I * ± i i -
J '
1
"
(o
X
l
"
f
l
8
),i(P-l),
"
f
L x ^ ^ x ^ J '
'
2'
p . 1.84.
6 . 2 - 2 5
EXAMPLE 6 - 1 3 .
1
n = 1 , S = 2
"
5
i
T
f -f
1 _
,
+
-3
f [ x
f, ,
M-l
i
x
i' i-l
J
l
S , 2
^
±
4
| . , I
P
X
- i
l Y . f a ) ! ^ "
"
4
U
" i
1
) ,
+
A
P
=
1
+
7 3
^
2, 7 3
6.3
Discussion of One-Point Iteration Functions With Memory
6,31
A conjecture.
The theory of interpolatory
I.F.
and derivative estimated I.F. has been developed in
Sections 6 . 1 and 6 . 2 .
first s - 1
In the case of interpolatory I.F.,the
derivatives of f are evaluated.
Hence s pieces
of new information are required for each iteration.
For the
case of derivative estimated I.F., the (s-l)st derivative of
an optimal one-point I.F. is estimated from the first
s - 2 derivatives at n + 1 points.
If we set S - s - 1 , then
S pieces of new information are used for each iteration.
For
interpolatory I.F.,, r » s(n-fl) represents the product of the
number of new pieces of information per iteration with the
number of points at which information is used; R = S(n+l) plays
the corresponding role for derivative estimated I.F.
at Theorems 4 - 1 ,
4-3,
6-1,
6-2,
6-3,
and 6 - 4 ,
A glance
reveals a remark-
able regularity in the structure of the results.
The only
parameters which enter are p and r or p and R; p depends only
on s and n or S and n.
When we deal with inverse
interpolation
the asymptotic error constant depends on Y^; when we deal with
direct interpolation the asymptotic error constant depends
on A .
r
From Theorems 6 - 2 and 6 - 4 we can conclude that the
asymptotic behavior of a sequence generated by a derivative
estimated I.F. is independent of the optimal one-point I.F. in
which the derivative is estimated.
r
Values of p for different values of n and s or n and
3-1,
S may be found In Table
The effect of estimating the highest derivative of
optimal one-point I.F, has been studied.
The idea of estimat-
ing the two highest derivatives suggests itself.
Calculations
of orders of such methods indicate that such a procedure would
not be profitable.
We have proved, for the case of interpolatory and
derivative estimated I.F,, that the old information adds less
than unity to the order.
We conjecture that this is true no
matter how the old information is reused.
CONJECTURE.
388
cp
x
cp[x^; i - i
Let
, x
i - 2 ' ** * l-n^
9
x
« G ^JL*^ ^'^i * * ^ * 1
1
f
U - i )
i~l
#
0 #
v
x
,
f
f'
f
I
I
I
new information is used only at x^.)
.+
1.
#
U - i ) ~
'•*" l-n' i-n' i-n""' i-n
be any one-point I.F. with memory.
p < I
,p,
^i-l'^l—l ^i l'" * *
(The semicolon shows that
Let cp e ^I^.
Then
In particular, we conjecture that it is impossible
to construct a one-point I.F, with memory which is of second
order and which does not require the evaluation of derivatives.
The fact that neyf information is used at only one point is
critical; in Section 8.6 we give I.F. which are of order
greater than two and which do not require the evaluation of
any derivatives.
Those I.F. are, however, multipoint
I.F.
with memory.
It must be emphasized that the conjecture does not
apply to the case where the sequence of approximants generated
by an I.F. is "milked" for more information.
See Appendix D
for a discussion of acceleration of convergence.
6.32
Practical considerations.
One-point I.F, with
memory typically contain terms which approach o/o as the
approximants approach a.
Consider, for instance, the I.F. of
Example 6 - 9 :
# f
l
" x p i j ^
{
2
f
i
+
f
i-l " 3 f [ x ^ x , ^ ] } ,
(6-29)
2
1 i
U
* l,2- i- i-57* iE
x
u
f
x
In theory, *f- - * f " ( a ) .
i
In practice, it approaches a o/o form
L
which naturally poses computational difficulties.
Note that
2
*f-" is multiplied by u£
which goes to zero quite rapidly.
L
(6-29)
also that the last term of
tlon term to x^
accurately.
u^.
Note
may be regarded as a correc-
Hence *f^ need not be Imown too
It might be worthwhile to do at least part of the
computation with multiple precision arithmetic; this matter
has not been fully explored.
In order to use an I.F. such as C
P
, n + 1 approxin,s
Y
mants to a must be available.
This suggests using
which requires but two approximants, followed
by <Po
a
i?Q
o'*»«'9v*
a*
a
t
t
h
e
g
,
successively
beginning of a calculation.
6 . 3 - 5
6.33
Iteration functions which do not use all
available information.
All the I.F. studied in this chapter
use all the old information available at n points.
It is
possible to construot I.F. which use only part of the old
information available at n points.
This may lead to simpler
I.F. but ones which are not of as high an order.
For example,
the simplest estimate of f" is
•
1
Define
+
E
1
2
r
x
x
i - i
•
by
2
+
»
i
.
2
-
»
i
-
»
i
-
a
?
t
f
i
-
(
6
"
3
0
It may be shown that the indicial equation associated with
this I.F. is t
2
- 2 t - 1 - 0 , with roots 1 ± y / 2 .
Thus the
4-
order of
E^
2
is 2 . 4 l .
On the other hand. * E
estimates f" from f ^ f ^ ^ f ^ f ^ p
evaluation of f and f
1
2
, which
is of order 2 . 7 3 -
If the
is expensive, it is preferable to use
the slightly more complicated *E-^ g.
)
r
6 . 3 - 6
6 . 3 4
S e c t i o n
t h e
5 . 5 a
of
E
worked
l e a d i n g
*E 1
, 1
f o r
term
"
*
a
of
t h e
G e n e r a l i z a t i o n
The
out
term
e q u a t i o n
c a l c u l a t i o n
.
a t t e m p t e d .
a d d i t i o n a l
d i f f e r e n c e
r e c u r s i v e
s e r i e s
An
f i r s t
a
of
A
two
number
t h e s e
2 ^
a
^
e
i
e
I . F .
i - 1
t o
"
was
e r r o r
d e r i v e d
I . F .
of
I . F .
of
which
e
(a)
e r r o r
has
s e r i e s
by
not
have
b e l o w .
g i v e n
been
been
The
Theorem
s e c a n t
-e
i " i - r
In
p e r m i t t e d
the
memory
g i v e n
c o u r s e ,
e q u a t i o n .
of
e r r o r
are
-
^(a)
w i t h
the
and
i s ,
the
c o e f f i c i e n t s
terms
of
i n
6 - 1 .
I . F . ,
( 6 - 3 1 )
*E 2 , 1
*E
1,2 •
n
a
-
°
*
-
A
3 ( ° 0
* V ) i i-i
a
e
e
e
i
+
e
i _ i
e
i - 2
+
A
2 (
a
)
1 |( ) - V )
2A
a
a
e
i >
e{.
( 6 - 3 2 )
( 6 - 3 3 )
7.0-1
7
CHAPTER
MULTIPLE
In
S e c t i o n
7.2
we
ROOTS
show
t h a t
a l l
E
are
S e c t i o n
7-3
we
of
l i n e a r
s
o r d e r
f o r
b a s i c
sequence
Our
nonsimple
r e s u l t s
e x t e n d e d
t o
{ g
on
s
}
I . F .
t h e
c a s e
a
whose
of
o r d e r
i s
by
n e c e s s a r y
I . F .
t o
be
of
I . F .
of
incommensurate
order
r o o t s
and
f o r
o r d e r
an
i n t e r p o l a t i o n
i n
S e c t i o n
s u f f i c i e n t
r o o t s
i s
s t u d y
o p t i m a l
m u l t i p l i c i t y - i n d e p e n d e n t .
d i r e c t
m u l t i p l e
7-6
second
In
g e n e r a t e d
Theorem
An
g i v e s
z e r o .
of
s t u d i e d
7*5.
c o n d i t i o n
a r b i t r a r y
i n
are
f o r
an
m u l t i p l i c i t y .
S e c t i o n
7.8.
7 . 1 - 1
7 . 1
I n t r o d u c t i o n
A
number
m u l t i p l i c i t y
are
Three
of
d e f i n i t i o n s
g i v e n
i n
z e r o s
t h a t
u
has
p r o v i d e d
o n l y
t h a t
the
I f
d e r i v a t i v e s
i t s
d e r i v a t i v e s
t o
s i m p l e
b y
u
in
i n
any
z e r o s
N e w t o n ' s
s i m p l e
G
known
be
z e r o s ;
of
a
f
then
t h e n
the
i f
=
f
( 7 - 1 )
lA.
and
G
by
e n t i r e
If,
we
t o
i n t e r e s t :
P
have
o n l y
m u l t i p l i c i t y
r e p l a c e d
a p p l i e d .
I . F . ,
has
p r i o r i
are
o r d e r
p a r t i c u l a r
p
I . F . ,
may
of
G = fi^ ),
z e r o
must
and
be
are
r e l a t e
1 . 2 3 .
1
m u l t i p l i c i t y
f
S e c t i o n
f u n c t i o n a l s
v-frs
Observe
which
f o r
or
F
u,
G,
are
or
t h e o r y
example,
m.
t o
F
The
be
used.
and
t h e i r
which
we
s i m p l e
p e r t a i n s
r e p l a c e
f
g e n e r a t e
u
9
w h i c h
i s
second
o r d e r
f o r
- X -
z e r o s
cp-a
n
/
o f
u"(
2u'
2
( x - a )
I t
o r d e r
p o i n t s
f o r
out
i s
a l l
w e l l
known
nonsimple
a l l
#
1
Newton's
r o o t s .
and
a)
a)
v
t h a t
m u l t i p l i c i t i e s
E.
I . F .
S c h r o d e r
i s
of
[ 7 . 1 - 1 ,
l i n e a r
p .
3 2 4 ]
t h a t
cp =
x
-
mu
( 7 - 2 )
7.1-2
i s
of
been
a l l
second
o f t e n
E
are
s
S e c t i o n
w h i c h
7-3
(7-2)
o r d e r
f o r
z e r o s
r e d i s c o v e r e d .
of
l i n e a r
we
w i l l
i s
t h e
In
o r d e r
m u l t i p l i c i t y
S e c t i o n
f o r
c o n s t r u c t
f i r s t
of
7.2
nonsimple
an
member.
o p t i m a l
we
m;
t h i s
w i l l
z e r o s .
b a s i c
f a c t
prove
has
t h a t
In
sequence
of
7.2-1
7.2
The
Order
E
of
3
c a n ,
under
g
of
was
t h e
E
s
o
d e r i v e d
a s s u m p t i o n
n e v e r t h e l e s s ,
a p p l i e d
t o
the
i n q u i r e
7-1.
z e r o s .
t h e
t h a t
as
c a l c u l a t i o n
THEOREM
nonsimple
from
of
The
T a y l o r
s e r i e s
f
has
o n l y
t o
the
b e h a v i o r
m u l t i p l e
o r d e r
simple
of
E
i s
n
a - * - ) .
z e r o s .
of
z e r o s .
g
e x p a n s i o n
E
We
l i n e a r
We
when
s
prove
f o r
a l l
Moreover,
E_ ,
•
n
- a
S
i
/
-
-
^
1
3
PROOF.
-i \ s
I
(5-l6)
From
-
( 7 - 3 )
t = i
m
and
(5-17),
s
E
s
+
1
( x )
=
x
-
£
Z j ( x ) ,
Z j ( x )
=
Y
j
( x ) u
J
( x )
j = l
Hence
E
g
+
1
( x )
-
a
-
x
-
a
-
^
Z ^ ( x ) .
J - l
D e f i n e
7 ^
by
Z j ( x ) = £ 7 (*-a) .
1
tt3
t = l
(7-4)
7.2-2
Hence
s
E
g
+
( x )
1
-
a
=
-
1
S i n c e
j (
Z
x
)
J Z j ( x )
s a t i s f i e s
-
( j - l ) Z
_
J
1
t h e
( x - a )
I
d i f f e r e n c e
( x )
+
u ( x ) Z j _
1
-
( x )
+
0 [ ( x - a )
d i f f e r e n t i a l
=
0,
Z
( x )
1
] .
2
(7-5)
e q u a t i o n
=
u ( x ) ,
(7-6)
we
f i n d
u(x)
=
on
s u b s t i t u t i n g
( x - a ) / m
l9i
M
0 [ ( x - a )
- ( M ) 7
I7
S e t t i n g
+
=
(7-4)
l / m
l
] ,
2
j
H
i n t o
(7-6)
and
n o t i n g
t h a t
+ £ 7
1 )
y
1
- 0,
y
= 1.
1 } 1
(7-7)
y i e l d s
,1-1-M
'1,J
71 , J - 1 '
7
1 , 1
=
M,
Hence
(7-8)
E
g
+
1
where
C[M,J]
(7-8)
i n t o
(x)
-
a
=
d e n o t e s
(7-5)
(x-a)
a
b i n o m i a l
c o e f f i c i e n t .
S u b s t i t u t i n g
y i e l d s
£
J=0
(-D C[M,J] + 0[(X-AF] =
J
(x-a)(-l) C[M-l,s]
s
+
0[(X-AF],
r
7.2-3
where
we
have
used
t h e
£
w e l l - k n o w n
(-l) C[M,j]
J
I d e n t i t y
-
(-1) C[M-1,b].
8
J=0
Hence
E
s
+
1
( x )
-
=
a
( x - a )
+
0[(x-a)
] .
-t=i
S i n c e
i f
m
and
i s
a
o n l y
p r o d u c t
p o s i t i v e
i f
m
=
c o m p l e t e s
1;
which
i s
a
of
(7-3).
out
c o e f f i c i e n t
l/m
from
of
each
( x - a )
term
i s
of
z e r o
t h e
p r o o f .
7-1.
( x )
-
a
=
w e l l - k n o w n
The
s i d e
2
the
f a c t o r i n g
t h e
EXAMPLE
E
i n t e g e r ,
( l
( x - a )
+
0 [ ( x - a )
e r r o r
c o n s t a n t
i s
have
COROLLARY.
G ( M , S )
2
]
r e s u l t ,
a s y m p t o t i c
We
-
Let
=
L
=
I
L
s Jm
[J
1=1
( i - t m )
g i v e n
by
the
r i g h t
7.2-4
Then
Q(m,s)
PROOF.
<
1 ,
S i n c e
l i m
Q(m,s)
m —•• oo
G(m,s)
may be
=
1,
w r i t t e n
a s
s
G(m,s)
=
II
( l
-
t = l
the
r e s u l t
f o l l o w s
I f
by
x
±
+
=
1
E
2
m >
( x
x
S i n c e
5
t h e
±
1
) ,
i + l
and
i f
a
sequence
C
"
m )
"
a
=
1
c o n v e r g e s
(Appendix
a*
D)
=
and
a *
as
an
x
may
an
be
used
t o
c a l c u l a t e
i n t e g e r - v a l u e d
I . F . ,
cp =
x
-
i + l
-
(
x
f o r
"
a
an
approximants
i s
formed
a
)
x
2
we
a
i + 2 ~
i + 2 "
0 [ (
+
x
X
i
- a )
can
2
] .
(7-9)
a p p l y
A i t k e n ' s
b y
x
i + l )
i + l
+
x
2
i
*
a,
w
C
1
-
m )
e s t i m a t e
v a r i a b l e .
mu m a y b e
i "
x
e s t i m a t e
x
e s t i m a t e
(
l i n e a r l y ,
i+2
Using
of
t h e n
sequence
formula
i m m e d i a t e l y .
u s e d .
Once
m
(
i "
x
f o r
i s
a
)
m.
known,
Note
t h e
t h a t
m
second
i s
order
7.3-1
The
7.3
B a s i c
Sequence
g
s
7.31
f o r m u l a s
were
sequence
f o r
o r d e r
E
of
a
s
I n t r o d u c t i o n .
d e r i v e d
simple
i s
i s
not
i s
c o n s t r u c t e d
f o r
z e r o s .
l i n e a r
b a s i c
E
below
t h e s e
I . P .
S e c t i o n
m
t h e
>
5*13*
S e c t i o n
nonsimple
f o r
f o r
;
In
f o r
sequence
g
In
7.2
we
z e r o s
1.
An
case
form
an
i s
b a s i c
t h a t
the
hence
t h a t
{E
and
m
o p t i m a l
showed
o p t i m a l
where
e x p l i c i t
b a s i c
s
)
sequence
a r b i t r a r y
but
known.
S i n c e
a
p r i o r i ,
problems
the
are
t h e o r e t i c a l
We
w i l l
These
r e s u l t s
i n t e r e s t
t h a t
on
i n
m
may
Let
of
h ( x )
has
a
Then
p r o c e e d i n g
t o
a
are
I . F .
found
numbers
d e r i v e
be
a
simple
as
i n
of
z e r o
j=0
i s
o f t e n
v a l u e
as
f a r
i s ,
however,
the
terms
w i t h
the
d e s i r e d
new
a t
f i r s t
and
I . F .
by
m u l t i p l i c i t y
m
a.
5.11
of
of
t h e i r
not
as
m.
p r a c t i c a l
r e s u l t s .
E
g
by
c e r t a i n
p r o p e r t i e s .
c o e f f i c i e n t s
second
t h e
known
c o n s i d e r a b l e
s u r p r i s i n g
of
the
of
S e c t i o n
s-1
z e r o
some
= f^ (x)
z e r o
a
e x p l i c i t l y ;
t h e s e
h(x)
Then
s t u d y
l e a d s
t o
of
l i m i t e d
The
and
l e a d s
S t i r l i n g
One
t e c h n i q u e .
are
m u l t i p l i c a t i o n
p o l y n o m i a l s
depend
m u l t i p l i c i t y
c o n c e r n e d .
f i n d
p o l y n o m i a l s
t h e
k i n d .
f o l l o w i n g
Let
=z .
Let
i t
H
i s
be
t h e
c l e a r
i n v e r s e
t h a t
t o
h .
7.3-2
i s
of
o r d e r
s
f o r
a l l
m.
In
p a r t i c u l a r .
.pl/rri/
cp
2
=
H(z)
-
z H ' ( z )
=
x
-
h
\
'(x]T
=
X
"
m
u
(
) >
x
( 7 - H )
cp
3
cp
2
=
H(z) -
was
z H ' ( z )
known
Bodewig
t o
[7-3-2]
R a t h e r
I . F .
of
of
v i e w .
t h i s
+
E.
and
| z
H " ( z )
S c h r o d e r
=
we
u s i n g
a t t a c k
x
-
(1870).
'[7.3-3,
(7-10)
t h e
-
|m(3-m)u(x)
[7-3-1]
O s t r o w s k l
t h a n
t y p e ,
2
t o
Chap.
See
2
A
2
( x ) u
2
( x )
a l s o
8].
g e n e r a t e
problem
m
from
h i g h e r
o r d e r
a n o t h e r
p o i n t
.
T
4
7*3-3
7.32
e x t e n d
as
the
The
s t r u c t u r e
n o t a t i o n
p a r a m e t e r s .
f o r
Thus
E
a
+
1
I . F .
we
of
so
=
x
I t
t h a t
f
-
^
i s
and
(5-17)
r e p l a c e
( x , f , l )
.
g
a d v a n t a g e o u s
m
a p p e a r
t o
e x p l i c i t l y
by
Z j ( x , f , l ) .
L e t
F = f
Observe
t h a t
a
z e r o
m u l t i p l i c i t y
1
of
b a s i c
sequence
r a t h e r
t h a n
F.
f o r
E
of
l
/
m
.
(7-12)
m u l t i p l i c i t y
C l e a r l y ,
a l l
m.
( I t
t h r o u g h o u t
t h i s
{E
m
1
i s
of
f
i s
( x , P , l ) )
i s
c o n v e n i e n t
s e c t i o n . )
a
t o
z e r o
an
of
o p t i m a l
use
E
g
+
1
Let
s
Z j ( x , P , l )
Then
(5-18)
=
W j ( x , f , m ) .
becomes
J W j ( x , f , m )
-
( j - l ) W
J
_
1
( x , f , m )
+
m u ( x ) W j _
1
( x , f , m )
= 0 ,
(7-13)
W ^ ( x , f , m )
w h i l e
Lemma
5 3
_
becomes
=
rau(x),
7.3-4
LEMMA
e
s+1
7-1.
( x , f , m ) = fi (x,f,m) -
We
e
s e e k
s + 1
c o e f f i c i e n t s
(x,f,m)
= x
p
-
e^(x,f,m).
s
s
a
w
..(m)
£
p
s
^
J
such
( m ) Z
j
t h a t
( x , f , l ) ;
(7-14)
P
and
such
t h a t
{ £
s
+
1
s
^ j ( m )
( x , f , m ) }
S u b s t i t u t i n g
s
=0,
i s
(7-l4)
an
< j
,
o p t i m a l
i n t o
t h e
b a s i c
s e q u e n c e .
formula
o f
Lemma
7-1
y i e l d s
x
-
^
p
s
^ ( r a ) Z
J
( x , f , l )
J=l
s-1
s-1
s
3=1
u ( x )
'
v
1
-
y
p
B
.
l
i
J
( m ) Z j ( x , f , l )
We
(5-18)
use
X
t o
Z j U ^ l H -
sp
e l i m i n a t e
S
.(m)
j
u ( x ) z l ( x , f , l )
and
f i n d
sp _ (m)
+
s
+ m J p
1}i
B
.
l
j
J
. ( m )
1
-
n J p
j=l
where
we
have
c o e f f i c i e n t
s p
w i t h
p
s , o
i n i t i a l
s
t a k e n
of
x
^ ( m )
1
( m j - s ) p
f o r
s
c o n d i t i o n s .
c a l c u l a t i o n
of
the
S
9
^(m)
J
i s
a
s
_
which
e q u a l
1
^> 0
and
p
^(m).
p
may
t o
^ j ( m )
-
s ,
be
c o n s i d e r e d
u n i t y .
m J p
. (m)
j
(7-15)
E q u a t i o n
S
p
_(m),
o
s ,
(7-l4),
i n
+
=
(m)
p
s
-
=
1
t h e
Then,
^
-
:
0,
L
( m )
f o r
p e r m i t s
E q u a t i o n
as
s
t h e
(7-15)
=
0,
<
j
(7-15)
as
r e c u r s i v e
shows
t h a t
J
9
p o l y n o m i a l
i n
m;
an
e x p l i c i t
formula
i s
a.
by
d e r i v e d
b e l o w .
D e f i n e
the
a s s o c i a t e d
f u n c t i o n s
^9
W
t
( x , f , m )
=
^
a
t
^
j
( m ) Z
J
( x , f , l ) ,
. (m)
J
t
<
j
.
(7-l6)
j = l
The
^(m)
were
i n t r o d u c e d
by
[7.3-4].
Z a j t a
s
e
s
+
1
( x , f , m )
=
x
-
^
s
W
t
( x , f , m )
=
x
1=1
^
t = l
s
= X
j = l
-
l=i
S i n c e
t
^
j = l
a ^ f m )
Z^ ( x , f ,
l )
8
.
l
j
J
( m ) ]
r
7.3-6
we
have
s
P
s
^ ( m )
=
£
a
t , j
(
m
'
)
(
"
7
1
7
)
1=3
To
f i n d
(7-16)
a
r e c u r s i o n
i n t o
(7-13),
formula
f o r
and
(5-18)
use
t h e
j (
t o
m
)
w
s u b s t i t u t e
e
e l i m i n a t e
u ( x ) Z ^ ( x , f , 1 ) .
Then
1^
Zj(x,f,l)[ta^j(m)
-
U - l ^ ^ j C m )
-
m
«J0^_i^j_i( )
m
+
m
J
a
t - l , j (
m
^
°*
=
J=l
Hence
(t+l)a
w i t h
I
a
<
a.
j
.
,
(m)
as
i s
C a l c u l u s
Jordan
Our
a
t + 1
^(m)
=
1,
i n i t i a l
+
a,
(mj-t)a ^(m)
(m)
=
p o l y n o m i a l
i n
We
b r i e f l y
of
F i n i t e
[7-3-5]
n o t a t i o n
i s
o
r
0
for
c o n d i t i o n s .
d i g r e s s
t
>
0,
E q u a t i o n
and
a
(7-18)
(7-18)
= 0
Am)
=
shows
for
0
t h a t
m.
t o
l i s t
' D i f f e r e n c e s .
Riordan
n o t
- mja^^Cm)
t
q u i t e
[7*3-6]
The
f o r
s t a n d a r d .
some
d e f i n i t i o n s
r e a d e r
t h e
We
i s
r e f e r r e d
standard
d e f i n e :
from
t o
t h e o r y .
t h e
7.3-7
i-i
[ x ]
=
t
J[
( x - i )
" P a l l i n g
( x + i )
" R i s i n g
F a c t o r i a l "
1=0
l-l
p
[ x ]
-
t
F a c t o r i a l "
1=0
C[x,l]
[x],/ll
=
C _ [ x , t ]
Ax],/II
=
[ x ] ^
=
^
" R i s i n g
S^
B i n o m i a l
C o e f f i c i e n t "
F a c t o r i a l
C o e f f i c i e n t "
" S t i r l i n g
Numbers
of
the
F i r s t
Kind"
" S t i r l i n g
Numbers
of
the
Second
second
kind
are
o f t e n
example,
by
J=0
I
x
" F a l l i n g
l
Kind"
j=0
S t i r l i n g
by
two
numbers
d i f f e r e n t
w i l l
use
S
and
w i t h
the
u s u a l
We
g e n e r a t i n g
of
t h e
t y p e s
T,
f i r s t
of
There
s ;
i s
and
f o r
no
symbol
f o r
a
c o n t i n u e
our
s t u d y
f u n c t i o n
f o r
d a n g e r
of
Chebyshey
the
of
j (
hj(x,m) = ^
t=0
m
c o n f u s i n g
.(m)
^
a
y
be
l
o ^{m)x .
t
and
S.
the
We
l a t t e r
p o l y n o m i a l .
p
)
s
denoted
and
a
p
d e r i v e d
^(m).
by
A
d e f i n i n g
7 . 3 - 8
I t
f o l l o w s
from
( 7 - 1 8 )
( l - x ) h j ( x , m )
where
m
I s
a
t h a t
+
mjhj(x,m)
p a r a m e t e r .
A
i s
not
d i f f i c u l t
t o
s a t i s f i e s
-
m j h
s o l u t i o n
h j ( x , m )
I t
h j ( x , m )
=
[1
v e r i f y
-
J
of
1
( x , m )
( 7 - 1 9 )
( l - x )
t h a t
_
the
m
]
J
-
0,
( 7 - 1 9 )
Is
.
f u n c t i o n s
k-
^9
. (m)
which
J
s a t i s f y
00
[1
-
( l - x )
m
]
=
J
£
1=0
a l s o
s a t i s f y
(7~l8)
and
I t s
I n i t i a l
c o n d i t i o n s ,
and
hence
00
[1
O b s e r v i n g
-
( l - x )
m
]
=
J
a
£
l
l
f
i
(m)x .
t h a t
m
1
-
( l - x )
m
=
(-D^Cim.rW
£
r = l
and
a p p l y i n g
the
a
t
j
J
m u l t i n o m i a l
( m )
=
J . ( - l ) < -
theorem
+
J
I
y i e l d s
n
r = l
•
( 7 - 2 0 )
7 . 3 - 9
w i t h
t h e
sum
t a k e n
o v e r
a l l
n o n n e g a t i v e
i n t e g e r s
a
r
such
t h a t
L
ra
r
-
l
>
I
r = l
On
t h e
o t h e r
[1
-
a
r
=
J '
r = l
hand,
( l - x )
]
m
=
J
£
C [ j , r ] ( - l )
( l - x )
r
r
m
r=0
r=0
t = j
Thus
a
t , j
(
m
)
=
1
( -
1
)
r
c
[ J ^ H - l )
t
C [ r m , t j .
(7-21)
r^O
Since
an
C [ r m , t ]
e x p a n s i o n
i s
of
a
p o l y n o m i a l
cj^
j(m)
i n
t h e
i n
m
of
degree
p o l y n o m i a l s
r=0
k-0
k=0
r=0
I, ( 7 - 2 1 )
C [ r m , t ] .
e x h i b i t s
S i n c e
r
7.3-10
Using
J
r=0
y i e l d s
v
a Jm)
t
(7-22)
E q u a t i o n
i s
n o t
r
[ m x , t ]
( - l )
e x h i b i t s
d i f f i c u l t
C
-
t o
-
C
r
+
a.
show
£
t
I
#
J
,(m)
a s
S
a
hence
T
k , J
m
k
^-22)
'
p o l y n o m i a l
i n
m.
I t
t h a t
[ x , j ] ( - 1 ) ^
J |
j=0
and
t , k
X
\ , k
T
k , Jm
k
k = j
t h a t
C
r
[ m x , t ]
=
^
a
t
^ ( m ) C
r
[ x ,
j ]
J=0
Thus
C
t i v e
t o
we
have
[mx,t]
t h e
i s
base
a
g e n e r a t i n g
f u n c t i o n s
e q u i v a l e n t l y
C
f u n c t i o n
[ x , j ] .
f o r
Since
t h e
a.
C [x l]
s
t h a t
V
C [ m x + t - l , t ]
=
^
j=0
a
t
^ ( m ) C [ x + J - l ,
j ] .
.(m)
=
r e l a -
C[x+l-l l]
9
7.3-11
We
now
d e r i v e
an
e x p l i c i t
f o r m u l a
f o r
t h e
p
. ( m ) .
t h a t
1=3
and
u s i n g
(7-22)
y i e l d s
s
k=j
t=k
S i n c e
t
!
"*,,k
~
s
!
S
s + l , k + l '
t=k
k
Our
r e s u l t s
are
summarized
i n
R e c a l l i n g
7.3-12
THEOREM
fi (x,f,m)
= x
s+1
D e f i n e
a
^ 9
W
t
7-2.
. (m)
J
( x , f , m )
=
-
Define
^
p
s
^ ( m ) Z
p
J
,(m)
0
by
( x , f , l ) ;
P
^ (
s
m
)
= °;
f
o
r
s
<
J«
by
£
a
t
^ ( m ) Z j ( x , f , 1 ) ;
cr^j(m)
= 0 .
f o r
l<
j
.
J - l
Then
00
[1
-
( l - x )
m
]
J
=
£
CT
(7-24)
t,j( ) ^
m
x
t=0
j
= ]T
( - l )
r
C [ j , r ] ( - l )
t
(7-25)
C [ r m , t ] ,
r=0
,j(m) =
C
r
( - D
[ m x , t ]
=
t
+
J
Y
ff
a
t
£
S ^
^ ( m ) C
r
k
T
k
^ m
k
(7-26)
,
(7-27)
[ x , j ] ,
j=0
P
8
j
J
( » )
-
( - D
J
+
S
f t
I
S
s
+
l ,
k
+
l
T
k
, J
m
k
'
^
'
7.3-13
The
p a r t s
of
f o l l o w i n g
Theorem
p r o p e r t i e s
of
7-2.
j (
COROLLARY
d e g r e e
PROOF.
—
a
n
Is
not
j (
d
a.
j(m)
q
m
from
a
complete
a
p o l y n o m i a l
t h e
l i s t
v a r l o u
of
the
) *
Is
i n
m
of
b .
^(m)
f o l l o w s
(7-26).
from
l
=
m.
(7-26),
from
s i n c e
_ -1
PROOF.
PROOF.
m j ,
t h i s
c.
1
T h i s
COROLLARY
>
f o l l o w s
T h i s
COROLLARY
I
)
T h i s
COROLLARY
For
m
T h i s
f o l l o w
I.
PROOF.
T
c o r o l l a r i e s
f o l l o w s
d.
For
( m )
j(m)
I
f o l l o w s
<
j
,
from
(-l)
=
from
=
l + 1
C[m,I]
(7-25).
0,
^(m)
(7-24).
f o r
l
<
j
and
was
d e f i n e d
as
I
>
m j .
z e r o .
r
7.3-14
COROLLARY
e .
00
t=o
t=j
PROOF.
S e t
COROLLARY
PROOF.
x
=
1
f.
Set
COROLLARY
j=o
PROOF.
( 7 - 2 4 )
j ( l )
m
=
1
=
&
t
and
j
use
C o r o l l a r y
( K r o n e c k e r
d.
d e l t a ) .
(7-24).
In
g .
«t
^
In
00
a ^ j ( m )
X
=
£ , j (
m
)
=
t
C
m
+
t
"
1
' ^ *
J-0
'
Set
COROLLARY
a
h .
x
=
1
In
The
(7-27).
l e a d i n g
c o e f f i c i e n t
( - l ^ C i y i ] !
1
of
a.
,(m)
.
1=0
PROOF.
T h i s
f o l l o w s
i=0
from
(7-26)
and
n o t i n g
t h a t
i s
r
7.3-15
COROLLARY
1.
p
,(m)
j
0
s ,
d e g r e e
a
p o l y n o m i a l
i n
m
of
s .
PROOF.
T h i s
COROLLARY
PROOF.
PROOF.
f o l l o w s
J .
T h i s
COROLLARY
p
T h i s
p
(m)
f o l l o w s
£.
The
=
m
from
, (m)
o
(7-28).
from
f o l l o w s
k.
COROLLARY
g i v e n
i s
=
1
from
l e a d i n g
S
.
(7-28).
+
( - l )
s
+
(7-23)
1
C [ ' m - l , s
and
c o e f f i c i e n t
by
.8
4 ^
^
£
( - l ^ C t j , ! ] !
3
.
1=0
PROOF.
T
s,J
T h i s
=
f o l l o w s
^ i V - I
1=0
from
(7-28)
and
3
(-D^tj,!]! .
] .
C o r o l l a r y
of
p
c
,(m)
i s
r
7 . 3 - 1 6
COROLLARY
ra.
p
n
s ,
PROOF.
and
T h i s
.(m)
j
f o l l o w s
=
1
from
f o r
e v e r y
(7-23)
m
and
such
t h a t
C o r o l l a r i e s
e .
COROLLARY
n .
p
s
PROOF.
T h i s
COROLLARY
, ( l )
> J
f o l l o w s
=
1.
f^om
C o r o l l a r y ( ( 7 - 2 8 ) .
o .
s
p
s
^ ( m ) ( - l )
J
+
1
C [ l / m , j ]
J=l
PROOF.
s
J=l
= 1 ,
s
=
1 , 2 , 3 , . . .
.
s
^
d
mj
r
7 . 3 - 1 7
i s
t o
P(x)
f o r
h o l d
=
s
f
=
Z j [ x , x
1
/
i
f o r ^ a r b i t r a r y
n
=
x
and
1 , 2 , 3 , . . .
, l ]
m
( x )
-
.
( - l )
J
+
1
f.
Take
hence
I t
i s
E
s
not
+
f ( x )
=
( x , x , l )
1
m
.
=
d i f f i c u l t
C [ l / m , j ] x .
x
Then
g
t o
s
+
1
( x , x
show
m
, m )
=
0 ,
t h a t
Thus
s
0
=
-
x
£
p
^ ( m )
s
i
( - l )
J
+
1
C [ l / m , j ] x ,
J-l
and
t h e
r e s u l t
f o l l o w s .
C o r o l l a r y
n
shows
t h a t
f o r
m
=
1 ,
s
e
r e d u c e s
as
s
+
1
( x , f , m )
e x p e c t e d
E
g
+
.-
x
-
£
p
s
^ ( m ) Z
J
( x , f , l )
t o
1
( x )
=
x
-
^
Z j ( x , f , l ) .
J-l
Thus
e
g
+
1
( x , f , l )
-
C o r o l l a r y
( 7 - 1 4 )
and
E
g
e
+
1
( x ) .
l e a d s
t o
an
i n t e r e s t i n g
r e s u l t .
From
( 7 - 2 3 ) ,
s
e
s
+
1
( x , f , m )
=
x
-
^
j = l
s
^
t = j
a ^ j ( m ) Z j ( x , f , l )
.
7 . 3 - 1 8
Then
00
l i m
S
s
+
1
( x , f , m )
=
x
-
> —+ 00
00
V
V
t—i,
. _
a
t
j
j
( m ) Z
J
( x
J
f , 1 )
.
00
= X
-
£
j = l
Z j ( x , f , l )
=
l i m
3
E
0 0
s
+
1
( x , f , l )
7.3-19
7.33
a„
. (m)
Formulas
e x p r e s s e d
i n
f o r
terms
e
of
s
.
T a b l e
C [ r m , t ] ,
7-1
l i s t s
w h i l e
some
T a b l e
7-2
o f
t h e
l i s t s
J
^9
some
o f
t h e
a,
. (m)
e x p r e s s e d
i n
powers
of
m.
T a b l e
7-3
l i s t s
^90
some
o f
t h e
p
^ ( m ) .
o
9
S
Z j ( x )
i n
e
s
=
T a b l e
+
1
T a b l e
7~3j
r e c a l l i n g
t h a t
J
Y j ( x ) u ^ ( x ) ,
5~1*
Using
and
e n a b l e s
u s i n g
us
t o
t h e
e x p r e s s i o n s
c a l c u l a t e
a
f o r
number
of
Y
j (
x
)
g i v e n
t h e
( x , f , m ) ;
S = x - mu(x),
2
£ = x - mu(x)| (3-m) + mA(x)u(x)
2
Q
r
P
= x - mu(xWg (m -6m+ll)
Pi
3
+
2
2
m(2-m)A(x)u(x) + m 2A|(x) - A (x)u (x)k
£
2
3
2
S = x - mu(x)|- ^ (m-10m+35m-50) + ~ m(7m-30m+35)A(x)u(x)
5
2
1m"(5-3m)
2
2Ag(x) - A (x)u (x)
2
3
+ nr 5A^(x) - 5A(x)A(x) + A (x) u(x)j
3
2
J
3
4
T
7 . 3 - 2 0
J
o
+
M
i—i
O
O
o
H
I
oo
o
+
O
l
oo
s
H
I
OJ
OJ
o
oo
o
oo
OO
-3-
o
oo
o
oo
oo
n
t
CVI
OJ
o
oj
o
+
+
OJ
OO
-3-
o
OJ
o
OJ
o
H
OJ
oo
-3-
O
o
o
o
OJ
oo
OJ
7.3-21
H
I
oo
CO
oo
OJ
OO
CO
OJ
OJ
OJ
H
l
0
3
OJ
OJ
I
9
OJ
H
oo
I
0
OJ
I
3
OJ
I
-3OJ
OJ
OO
TABLE
7-3.
p
.(m)
* s , j
'
J
?
i
=
v
( - l )
'
S
+
\
J
si
K
S
,. . ,.T.
.m .
s+l,k+l
k , j
k
2
i *
k
3
3
1
m
2
-
m
(m/6)(m -6m+ll)
3
-
2
4
-
From
Corollary"
p
(m/2)(m-3)
s
, i
(
m
(m/24)(m -10m +35m-50)
3
)
k,
2
the
-
1
n
1=1
3
m (m-2)
2
m
(m /l2)(7m -30m+35)
expressions
+
2
2
in
2
t h i s
column
can
I -
be
(m /2)(3m-5)
3
replaced
by
m
k
r
7.4-1
7o4
The
C o e f f i c i e n t s
The
S e c t i o n
e r r o r
5 . 5 .
c o e f f i c i e n t s
We
of
w i l l
be
t h e
s e r i e s
now
the
R e c a l l
which
of
f o r
d e r i v e
e r r o r
the
used
E r r o r
an
B .
L e t
of
u(x)
i n t o
i s
d e r i v e d
e v a l u a t e d
a
a
now.
a t
a
n
g
s t u d i e d
a l g o r i t h m
f o r
c a l c u l a t i n g
d e f i n i t i o n s
of
t h e
e ( x , f , m ) .
s
o
the
f o l l o w i n g
symbols
»
v - ' - ^ -
a
power
=
-
w
(x)
z e r o
of
s e r i e s
m u l t i p l i c i t y
in
e,
t h i s
which
s e c t i o n ,
i n d i c a t e d .
(x)
o ) ( m ) e
£
t
4
The
w i l l
o t h e r w i s e
=
m.
be
a l l
e x p a n s i o n
needed
f u n c t i o n s
D e f i n e
00
oo
= ] [ ae,
r
r
f'(x)
=
£
r=m
a)^(m)
b e l o w ,
are
by
(7-29)
.
S i n c e
r=m
^
A . ( x ) .
Throughout
u n l e s s
,
1=1
f(x)
i n
f r e q u e n t l y :
(x)
be
£
was
of
a . .
t h a t
of
E ( x , f , l )
s
s e r i e s
* M - m >
Observe
S e r i e s
r a
p
e
r - 1
3
r
7 . 4 - 2
and
m u ( x ) f
,
( x )
=
m f ( x ) ,
we
o b t a i n
l+l-m
Y
C
q (
D
m
) (
^
+
"
1
l )
(
a
t + l - q
,
1
=
m
'
r
a
+
1
>
q=l
or
l-l
q=l
( 7 - 3 0 )
S i n c e
m
=
^
1 ,
=
and
A^,
hence
I t
f o r m u l a
we
f o r
i s
o b s e r v e
t h a t
n o t
oo^(m)
^ ( l )
- - B
t
t
>
B
+
( 7 ~ 3 0 )
=
as
d i f f i c u l t
i s
g i v e n
r^
» c-)
t h a t
t o
r e d u c e s
prove
t h a t
an
t
-
3 4 )
when
e x p l i c i t
by
[(m+i)B,_,,
J
B
( 5
e x p e c t e d .
1
- 1
t o
.
J
>
m
n
r
^ ( - D
r
!
j = i
3
—
7
j
1
"
^
1
-
i = i
( 7 - 3 1 )
where
r
=
>
and
where
t h e
i n n e r
sum
i s
t a k e n
o v e r
a l l
A
n o n n e g a t i v e
i n t e g e r s
a
±
such
t h a t
)
i a
±
=
as
v
j .
Observe
t h a t
1 = 1
o)
p
t
(m)
i s
the
e x c e p t
f o r
( 7 - 3 0 )
or
same
f u n c t i o n
c o e f f i c i e n t s
( 7 - 3 1 ) *
the
of
which
f i r s t
the
B.
l
f >
depend
few
u^(m)
m
0
i s
of
the
t
on
m
niay
o n l y .
be
Prom
c a l c u l a t e d
A.
i
e i t h e r
as
7.4-3
cu^m)
=
_
( m )
=
( m
co (m)
=
-
3
4
We
the
B
2,m'
=
c
of
1,
( m
t u r n
e r r o r
l ) B ^
+
l )
+
t o
-
m
B ^
2
t h e
s e r i e s .
2 B ^
m
(
+
,
m
S
problem
D e f i n e
r
n
of
^
P
^
^
f i n d i n g
(m)
7\
B
-
3
B ^
the
m
.
c o e f f i c i e n t s
by
Ks $ S
oo
e
s
( x , f , m )
=
^
^ t ,
s
(
m
)
e
t
(7-32)
-
1=0
S i n c e
e ( x , f , m )
e
I
.
o
t
L e t
=
s
ex.
=
T h i s
1.
e x p e c t
t»
assume
A
may
be
i
( x , f
,m)
=
a
and
A,
i n t o
the
f o r m u l a
(m)
U, o
S u b s t i t u t e
S
„
=
0
f o r
0
<
<
I
s,
and
C, S
p r o v e n
d i r e c t l y
by
i n d u c t i o n
on
s .
Then
e
Now
we
S
( 7 - 3 2 )
B
+
1
=
( x , f , m )
=
C
x
=
a
+
( x - a )
(m)
v •S
( x , f , m )
-
=
of
0,
=
a
f o r
Lemma
.+
0
e.
<
I
<
s
e
7 - 1 ,
c ' ( x , f , m ) ,
(7-33)
t o
f i n d
t=0
w h i c h
c o m p l e t e s
00
00
t=s
t=S
the
i n d u c t i o n .
(7-32)
S u b s t i t u t i n g
(7-33),
i n t o
u s i n g
oo
mu
(x)
= Y
<» (m)e ,
1=1
%
l
t
i
and
e q u a t i n g
t o
z e r o
THEOREM
the
7-3.
c o e f f i c i e n t
Let
the
of
e
,
we
m u l t i p l i c i t y
of
00
e
a
( x , f , m )
be
a
a t
m.
L e t
00
l
-K (m)e ,
1=0
'
=
a r r i v e
^
mu(x)
l)S
=
l
as (m)e .
^'
t
1=1
Then
l-l
s
\ , s + l (
m
)
+
C ^ s J ^ s ^ )
^
+
r
a
3
t + l - r (
m
^ r , s
=
° '
r = l
(7-34)
w i t h
A
t
^
g
^
0
( m )
j
S
( m )
=
0
=
f o r
a,
0
^ ^ ( m )
<
I
<
=
s
A ^ - ^ m )
1,
and
s
>
1.
=
0
f o r
I
>
l ,
and
7.4-5
1
PQ
+
B
OO
<N
Q
g
oo cvi
pq
pq^
OO
OJ
+
+
-=J-
«
s
OO
pq
I
oo
°°
<N
oo oj
pq
B
oo
pq
^
a
q
pq
OJ OJ
pq
00
+
a
1
OJ
^
H
i
pq
oo
«—riOJ
r-rjCM
OO
CvJ
i
B
pq
oo
+
B
B
OO
oo
+
B
CvJ
OJ
pq
^
*»
OJ oj
OJ
PQ
-3+
OO OJ
pq
OJ
r—1
^
Q
H
O
H
CvJ
OO
^j-
7.4-6
S i n c e
can
be
used
t o
the
o)^(m)
d e t e r m i n e
may
the
be
7v„
c o n s i d e r e d
as
(m) .
t h a t
Note
known,
A-
v yS
same
f u n c t i o n
of
a),,(m)
as
t«
v
are
l i s t e d
e (.x,f,m)
2
i n
T a b l e
-
= B
a
7-4.
U s i n g
+
(m+l)B
(m+l) B^
2
| ( ^ 3 ) B ^
m
-
2
e^(x,f,m)
-
a
=
•Km
i s
of
v . .
Some
3
-
i s
the
t h i s
2^m
+
t a b l e
we
f i n d
2B-
-
(3m+4)B
+
3(m+3)B
(m+4)B
Q
B_
A-
v^S
p
B-
+
3B,
B_
-
3B,
o
+6m+8)B^
of
C
B
|(m+l)(2m+7)B
(m)
t h e
S
V/S
e
g
_
(7"3^)
+
B,,
^
(m)
7.5-1
7.5
I t e r a t i o n
In
F u n c t i o n s
t h i s
of
m u l t i p l e
r o o t s .
i n
t h e
of
c a s e
a n a l y s i s .
s e c t i o n
We
s i m p l e
G e n e r a t e d
we
By
D i r e c t
g e n e r a l i z e
have
t o
assume
r o o t s
i n
o r d e r
Theorem
s t r o n g e r
t o
I n t e r p o l a t i o n
c a r r y
4-3
t o
t h e
c o n d i t i o n s
t h r o u g h
our
c a s e
t h a n
7.5-2
7.51
n
+
n
1
o
s
j
The
e r r o r
9.pproilim^iitB
b
e q u a l
e
t
h
p o l y n o m i a l
e
t o
t h e
a
new
D e f i n e
t o
f i r s t
s
We
r e p e a t
assume
(7-35)
c h o s e n
.
t h a t
P„
n
-
1
b y
i '
x
^ ( t )
l - l ' • '
n , s (
x
i
+
some
b y
l )
-
a
o f
°
t h e
r e a l
J^ ^_IJ•••' iX
x
m u l t i p l i c i t y
s
a
u s i n g
number
-
1
f
a t
x
j ^
x
+
i - n '
i n
t
'
t h e
a
n
d
n
1
i n t e r v a l
w
h
e
r
e
r
i _ i *
• • • '
p o i n t s
o f
x
P
r o o t s
s ( n + l ) .
^
+
^
i ^ *
x
w h i c h
n, s
9
,
one
o f
* • '
f
x
( i i) =
+
—
n
( t - x ^ )
,
3
b y
S e t
t
+•.
=
=
3
v
x
n
(-i+i-i-j) *
J=o
i -
n
*
x
i - n + l
s a t i s f i e s
them
Then
]
x
(7-35)
d e t e r m i n e d
f^'te
e
a r e
-
n
=
b
n
L e t
d e r i v a t i v e s
of
r o o t
r e a l
m.
c r i t e r i a .
-
l i e s
x
f i r s t
t o
f i n d
a
o f
x
have
^ )
x
h a s
s
0
9
c a n
a
L e t
d e r i v a t i v e s
p r o c e d u r e
we
a s
We
where
z e r o
approjtimaht
t h i s
I f
a
whose
P
Then
e q u a t i o n *
i + 1
i s
7-5-3
x
where
£±( ±+i)
=
(
f
where
T|^
L e t t i n g
e
l i e s
+ 1
j__j
=
i n
x
m
e
=
i + l )
"
n
\ r
i n t e r v a l
a
*
w
ml
^
r T
e
a
r o o t s .
B e f o r e
of
an
(7-36)
We
i s
assume
a n a l y z i n g
i n d i c i a l
the
r
r
i
<
x
i
+
l -
d e t e r m i n e d
v
e
a
the
a
)
by
v
U
i + l
s
l
i + 1
rr
y
;
m u l t i p l i c i t y
of
e q u a t i o n
x ^
+
^
and
t h i s
m
=
e r r o r
/
xs
r
t
11
^
f o r
the
c a s e
of
e
e q u a t i o n .
•
„
i - j -
e
i
+
m u l t i p l e
(7-37)
s ( n + l ) .
e q u a t i o n
l )
a.
t h a t
r
i s
'
j=0
e r r o r
<
a
t
~JjnT7Z
1
E q u a t i o n
S i n c e
ml
=
the
i - j
/
i + l
x
•
we
i n v e s t i g a t e
the
r o o t s
7.5-4
On
7.52
S e c t i o n
the
we
3.3
t h e
r o o t s
i n v e s t i g a t e d
p o l y n o m i a l
of
an
t h e
I n d i c i a l
p r o p e r t i e s
e q u a t i o n .
of
t h e
In
r o o t s
of
e q u a t i o n
k-1
g
under
t h e
k
>
a
a s s u m p t i o n
( t )
=
t h a t
t
k
f o r
-
a
£
k
>
1,
t
J
=
(7-38)
0,
k a > 1.
The
i s
o r d e r
of
t h e
d e t e r m i n e d
by
I . F .
t h e
w h i c h
are
r o o t s
of
(7-39)
b e i n g
t h e
s t u d i e d
i n
S e c t i o n
7.5
e q u a t i o n
n
m t
w h i c h
i s
of
t h e
form
n
+
-
1
s
(7-38)
Y
we
Theorem
demand
3-2
t h a t
becomes
m
<
r
=
j
=
0,
(7-40)
w i t h
k = n + 1,
S i n c e
t
a =
s ( n + l ) ,
(7_4i)
(7-39)
i s
s a t i s f i e d
and
7.5-5
THEOREM
7-4.
L e t
n
I f
n
=
0,
Assume
r e a l
n
t h i s
J>
1
a
e q u a t i o n
n
p o s i t i v e
(t)
n + l , s / m
s
m
d
<
r
s i m p l e
=
has
=
1
±
s_
m
the
P
^
m
s
n
+
^
1
t
i
r e a l
s ( n + l ) .
r o o t
max
t
n-fl
r o o t
Then
S
/
/
=
J
0
^
t h e
=
s/m.
e q u a t i o n
has
one
and
m
<
^
^ n + l , s / m
m
+
1,
F u r t h e r m o r e ,
j
*
m
+ 1
e
"
v
m
where
e
s/m
d e n o t e s
+
o t h e r
t h e
r o o t s
are
T a b l e s
a
=
s/m,
f o r
i d e n t i c a l
c o m p a r i s o n
the
m
=
w i t h
c o n d i t i o n
ka
b a s e
of
^
P
—•
7-5,
1,
•
^
s ,
m
m
_
-i
n+1
n + l , s / m
s i m p l e
7-6,
3,
3-1
i s
7-7,
and
and
o t h e r
1
n a t u r a l
=
=
l o g a r i t h m s .
^
m
+
+
1
Hence
1.
oo
2,
the
>
n + l , s / m
m
a l s o
T a b l e
w i t h
o
p
1
n
A l l
.
^
n+l
4
i s
t h r e e
v i o l a t e d
and
have
and
7-8
m o d u l i
g i v e
r e s p e c t i v e l y .
r e p e a t e d
here
t a b l e s .
The
are
l e f t
l e s s
v a l u e s
t h a n
of
T a b l e
t o
c a s e s
b l a n k .
one.
p.
7-5
i s
f a c i l i t a t e
i n
w h i c h
r
7.5-6
TABLE
"a
1/1
1
7-5.
VALUES
OF
P
k
2/1
3/1
V i
2.000
3.000
4.000
2
1.618
2.732
3.791
4.828
3
1.839
2.920
3.951
4.967
4
1.928
2.974
3.988
4.994
5
1.966
2.992
3.997
4.999
6
1.984
2.997
3.999
5.000
7
1.992
2.999
4.000
5.000
7.5-7
TABLE
7 - 6 .
VALUES
OP
0.
K, d
"a 1 / 2
i
3/2
4/2
1.500
2.000
1.618
2.186
2.732
2/2
k
1
2
3
1.234
1.839
2.390
2.920
4
1.349
1.928
2.459
2.974
5
1.410
1.966
2.484
2.992
6
1.445
1.984
2.494
2.997
7
1.466
1.992
2.498
2.999
7.5-8
TABLE
"a 1 / 3
i
7-7.
VALUES
2/3
OP
p.
3/3
4/3
k
1
1.333
2
1.215
1.618
2.000
3
1.446
1.839
2.210
4
1.126
1.552
1.928
2.284
5
1.199
1.604
1.966
2.313
6
1.243
1.631
1.984
2.325
7
1.271
1.646
1.992
2.330
I
7 . 5 - 9
TABLE
a* 1 / 4
7-8.
VALUES
2/4
OF
P
k
&
3/4
4/4
1.319
I.618
\ k
1
2
3
1.234
1.548
1.839
4
1.349
1.648
1.928
5
1.079
1.410
1.697
1.966
6
1.130
1.445
1.721
1.984
7
1.163
1.466
1.734
1.992
1
7.5-10
7.53
e r r o r
The
o r d e r .
We
r e t u r n
t o
the
a n a l y s i s
of
the
e q u a t i o n ,
n
e
T i
=
+
M
±
n
(
e
i . j -
i
e
+
i )
s
>
(7-42)
^
f
We
s h a l l
assume
e
e^
r
U
)
)
t h a t
-
Hence
(
i + l
T
^
-
(7-^3)
-
0
0.
We
may
(7-42)
r e w r i t e
as
n
'i+1
J=0
where
(7-45)
Prom
(7-43),
v a n i s h
f o r
we
any
c o n c l u d e
f i n i t e
a
i
=
l
n
t h a t
i .
5
i
A
1.
±
Assume
t h a t
e
±
does
n o t
Let
=
l n | e
±
| ,
T
±
=
l n | T
1
| .
(7-46)
r
7.5-11
Prom
(7-44),
n
mo
T
i+1
±
+
s
' 1 - J '
j=0
or
n
T
^
m
'i+1
Now,
J
±
(7-47)
and
s/m
i s
i d e n t i c a l
r e p l a c e s
T
Observe
In
±
(7-47)
^
m
(3-34),
w i t h
s .
+
e x c e p t
t h a t
T
±
/ m
r e p l a c e s
t h a t
mi
f
f
(
W
r
( a )
)
( a )
whereas
J
F u r t h e r m o r e ,
(3-43)
i s
±
-•> I n
r e p l a c e d
K
.
by
n
m
x
p - i
I
J=0
By
one
methods
may
a n a l o g o u s
show
t o
t h o s e
used
i n
the
proof
t h a t
( p - l ) / ( r - m )
'i+1
We
summarize
our
r e s u l t s
m i
i n
f ^ ( c t )
of
Theorem
3 - 3
7.5-12
THEOREM
L e t
7-5.
J
where
a
i s
assume
be
a
t h a t
n o n z e r o
z e r o
m <
on
of
r .
J .
=
| x
m u l t i p l i c i t y
L e t
L e t
f (
x
r
{ x ^
p o l y n o m i a l
a r e
P
n , s
p o i n t s
x
f o r
f
e q u a l
i ^
number,
a s
x
JL+1
£
#
Let
e
a n y
f i n i t e
±
_ j
=
such
t h e
f i r s t
t h e
x
i
±
_ j
i - n '
"
a
and
t h a t
e
i
P
x
i
1
/
j _
e
an
s ( n + l )
and
l e t
d e f i n e
and
f (
) f (
m
r
)
a
i n t e r p o l a t o r y
s - 1
t h a t
(
s
N
5
d e r i v a t i v e s
t h e r e
i + i )
x
i - l '
x
assume
+
and
=
d e r i v a t i v e s
Assume
° n , s (
J
be
g
N
s - 1
t h a t
l + l
-
b u t
x
f i r s t
r
n
P
such
x
e
1*
t h a t
L e t
c o n t i n u o u s
L e t
t o
J
be
m.
f o l l o w s :
i » i > • • • >
x
)
rj*
^
- x . . , . . .
o
sequence
| x - a |
t h a t
#
e
- '
0
±
x
f
e x i s t s
° *
=
of
D e f i n e
i - n )
does
o f
a t
t h e
a
<&
r e a l
n
g
b y
#
n o t
v a n i s h
f o r
~* ° «
Then
( p - l ) / ( r - m )
i+ll
P
m l
;
r
l i
f
f <
I
e
where
p
i s
t h e
unique
r e a l
(
( a )
r
)
m
> ( a )
p o s i t i v e
(7-48)
r o o t
of
n
n+1 - 2m
Y
/_j
j=0
t
J
=
0 .
7 . 5 - 1 3
Note
the
case
m
e r r o r s
were
assume
e
u i
=
1,
t h a t
we
(7-48)
showed
s u f f i c i e n t l y
/e,
0 ,
which
r e d u c e s
t h a t
t o
e^
s m a l l .
i m p l i e s
0
For
e.
(4-30)
i f
p r o v i d e d
the
- * 0 „
case
m
=
the
m
>
1 .
For
i n i t i a l
1,
we
7.54
7*53
S e c t i o n
a p p e a r
i n
p r o d u c t
number
of
r e d u c e
w h i c h
s
as
~
p o i n t s
the
t h e
a t
o n l y
f o r
-*
on
n = l ,
m
=
2.
00
i s
S e c t i o n
o r d e r ,
r a t i o
s
1,
=
m
m
=
2,
t o
of
=
s i m p l y
5.32
1
+
7-2.
f o r
t h e
s
1
F u r t h e r m o r e ,
t o
n
Thus
I . F . ) ,
l i m i t i n g
i n
and
as
v a l u e
the
w i t h
t h e
the
f i x e d ,
the
t h a t
o r d e r ,
m u l t i p l i c i t y
For
m.
t h e
u s e d ,
l i n e a r l y
( s e c a n t
the
i s
of
p a r a m e t e r s
i n f o r m a t i o n
the
s/m.
r e s u l t s
m u l t i -
i s
t o
e q u a t i o n
the
o r d e r
o r d e r
i s
f o r
=
of
t h e
n
t h e
the
1,
o r d e r
s/m.
s
=
case
e
I f
a p p e a r s
are
of
of
The
o n l y
7~5
p i e c e s
e f f e c t
w h i c h
the
the
i n f o r m a t i o n
The
s ,
e x a m p l e s .
Theorem
new
w h i c h
the
EXAMPLE
i n
of
of
z e r o .
f a c t o r
and
s a t i s f y i n g ;
number
d e t e r m i n e s
2,
n
v e r y
c o n c l u s i o n
t h e
of
depends
same
are
t h e
of
p l i c i t y
D i s c u s s i o n
i
3,
n
of
=
0.
s i m p l e
T h i s
I . F .
r o o t s .
I f
was
m
d i s c u s s e d
=
1,
r
7 . 5 - 1 5
EXAMPLE
i n
S e c t i o n
1 0 . 2 1
7 - 3 .
f o r
' i + 1
s
t h e
-
1 ,
case
n
of
|A.(a)p
-
2 .
T h i s
s i m p l e
( p _ l )
I . F .
r o o t s .
m
=
2,
p - 1
'i+11
1
f'"(a
3
f"(a
I f
p ^ 1.84,
,
1*1
I f
i s
p
<v
1 . 2 3 .
d i s c u s s e d
m
=
1 ,
T
7 - 6 - 1
7 . 6
O n e - P o i n t
A
number
I . F .
w i t h
t h i s
s e c t i o n .
e a r l i e r
I . F .
memory
f o r
we
I . F .
0.
n
>
bounds
on
the
Memory
t e c h n i q u e s
the
case
t h e s e
s h a l l
The
i f
of
^ S i n c e
themes
memory
With
o r d e r
p a s s
a r e
max
S i n c e
t h e o r y
which
d e r i v a t i v e s
c o n s i d e r
=
f / f '
p e r t a i n s
by
u
d i r e c t
c o n c l u s i o n
u
of
and
t o
i t s
t h a t
4-3
by
<&
( u )
n, s
Q
u.
be
t h e
I . F .
are
v a r i a t i o n s
on
7.5
<
are
r
=
o n e - p o i n t
s ( n + l ) ,
w i t h
t h e n
—
has
o n l y
s i m p l e
s i m p l e
z e r o s
d e r i v a t i v e s .
s t u d i e d
s t a t e s
z e r o s ,
by
r e p l a c i n g
As
i n
we
an
can
f
e x a m p l e ,
S e c t i o n
a p p l y
and
i t s
we
4.23.
The
t h a t
( p - l ) / ( r - l )
g e n e r a t e d
from
0
n,
s
by
r e p l a c i n g
( p - l ) / ( r - l )
u <
r
r lu'
p
i n
l i g h t l y .
Then
x - a |
g i v e n
by
.(r)
L e t
m
o n e - p o i n t
M
i n t e r p o l a t i o n
Theorem
i f
z e r o s
are
them
S e c t i o n
g i v e n
1,
L
m u l t i p l e
o v e r
i n
showed
p
of
c o n s t r u c t i n g
t e c h n i q u e s
s t u d i e d
We
f o r
\ a
( a
f
t h e
I
7 - 6 - 2
In
S e c t i o n
7 . 4 ,
co, ( m )
was
d e f i n e d
b y
00
mu
(x) = Y
e
l
u We >
=
x
-
a.
t
1=1
Hence
\x-a\
In
y
p a r t i c u l a r ,
x
l , l
(
u
-
)
i
X
"
i
U
u
i s
of
o r d e r
t o
show
g i v e n
* L , l
(
u
§ ( 1 + ^ 5 )
t h a t
t h e
^
1 . 6 2
f i r s t
two
f o r
a l l
terms
of
i "
x
i - l
i "
u
i - l
( 7 - 5 0 )
m.
I t
i s
t h e
e r r o r
n o t
d i f f i c u l t
s e r i e s
a r e
by
)
"
a
~ -
B
2 , m
(
a
)
e
i
e
e.e*
i
i - l
B.
j.m
9
- J t e i ,
ma
m
( 7 - 5 1 )
=
B
j,m
a
J+"i-l
ma
m
a
9
a
=
j
f
( j )
JT
9
T
7.6-3
^
i n f o r m a t i o n a l
t h e
drawback
a p p r p a c h e s
has
i n f o r m a t i o n a l
e f f i c i e n c y
t h a t
a
an
o/o
as
of
the
form
.81
usage
f o r
a l l
a p p r o x i m a n t s
w h i c h
may
of
m.
two
*]_
c o n v e r g e
n e c e s s i t a t e
and
^
s
u
t o
f
an
f
e
a,
r
s
f
r
o
m
u
m u l t i p l e
p r e c i s i o n
a r i t h m e t i c .
I f
a v a i l a b l e .
w h i c h
m
i s
S i n c e
p e r t a i n s
t o
known,
a
f^
has
1
7
" )
1
1
s i m p l e
number
o n l y
z e r o s
g e n e r a t e d
by
r e p l a c i n g
f
t a g e
no
o/o
o c c u r ;
t h a t
r a t h e r
h i g h
d e r i v a t i v e s
We
d e r i v a t i v e
* F
can
t h a t
t h e
a l s o
t o
*f
by
of
f
on
n [
-
^
1
have
(See
t o
I . F .
t h e
are
t h e
t h e o r y
t h e
I . P .
have
t h e
a d v a n -
d i s a d v a n t a g e
t h a t
b e ' u s e d .
by
new
z e r o s ,
Such
#
t h e y
f
t e c h n i q u e s
a p p l i c a b l e
must
the
.
m
o t h e r
s i m p l e
i s
f (
r e p l a c e
e s t i m a t i o n
a n a l o g o u s l y
n
forms
of
F
=
f ^
1
I . F .
1 1 1
For
.
can
example,
6.22.)
S e c t i o n
We
I t
i s
perform
d e f i n e
c l e a r
I . F .
* n
1
E
(
F
)
=
x
"
( "
7
5
2
)
*n
has
o r d e r s
r a n g i n g
As
on
t h e
second
a
from
f i n a l
o r d e r
^(1+^/5)
e x a m p l e ,
we
2,
t o
perform
I . F . ,
f
£
2
*b
x
-
as
m
a
f u n c t i o n
d e r i v a t i v e
of
n.
e s t i m a t i o n
7.6-4
D e f i n e
* n
1
£
=
"
x
^ "
m
7
5 3
)
n
I t
may
be
^ ( l + v / 5 )
l i n e a r
shown
as
o r d e r
a
t h a t
f u n c t i o n
f o r
a l l
b e t w e e n
(7-53)
and
s i d e
(7-53)
i s
of
whereas
l i n e a r
t h e
the
of
(7-52)
a
n
of
and
n o n s i m p l e
i s
l i n e a r
d e n o m i n a t o r
c o m b i n a t i o n
o r d e r
of
on
f ^ j ,
t h i s
m.
t h a t
The
the
r i g h t
^
j
^
of
s i d e
n.
from
?±_y
of
0
1
I . F . )
e s s e n t i a l
denominator
c o m b i n a t i o n
0
r a n g e s
( s e c a n t
z e r o s .
t h e
I . F .
(7-52)
i s
of
d i f f e r e n c e
on
<
t o
the
j
i s
r i g h t
£ n,
a
1!
7.7-1
7.7
Some
G e n e r a l
The
we
have
f o l l o w i n g
o b t a i n e d
a.
E
R e s u l t s
.
s
i n
s t a t e m e n t s
t h i s
the
r e s u l t s
which
f o r
non-
c h a p t e r :
2,3*....,
=
t y p i f y
i s
of
l i n e a r
i s
o p t i m a l ;
order
a l l
s
simple
b .
S
.
s
z e r o s .
2,3,....,
=
i t
^
s
on
c.
A
m
and
i t s
o r d e r
n o n o p t i m a l
g e n e r a t e d
from
$
,
n,
m < r
F i r s t
we
d
=
e
which
and
by
by
i t s
r e p l a c i n g
f
and
may
be
i t s
d e r i v a t i v e s .
d i r e c t
i n t e r p o l a t i o n ,
i s
m u l t i -
does
p
e
n
d
e
n
t
^
b
u
t
not
of
l i n e a r
o r d e r
i f
s(n+1).
s e c t i o n
the
not
we
s h a l l
make
some
g e n e r a l
remarks,
f o l l o w i n g
CONJECTURE.
I . F .
u
g e n e r a t e d
t h i s
s u g g e s t
^
by
I . F .
L P .
s
p l i c i t y
In
any
e x p l i c i t l y
m u l t i p l i c i t y - i n d e p e n d e n t .
m u l t i p l i c i t y - i n d e p e n d e n t
d e r i v a t i v e s
d.
i s
depends
I t
depend
m u l t i p l i c i t y - i n d e p e n d e n t .
i s
i m p o s s i b l e
e x p l i c i t l y
on
t o
m
c o n s t r u c t
and
which
an
i s
o p t i m a l
7.7-2
We
of
b
of
and
we
have
have
e x c l u d e d
r e s t r i c t e d
f o l l o w i n g
i l l u s t r a t e s
t h e
of
r o o t s .
m u l t i p l e
r e q u i r e
cp(a)
n e c e s s a r y
S i n c e
f ( a )
t h a t
t h e
m u l t i p l e
p i t f a l l s
=
which
Bodewig
we
a,
H(x)
=
o p t i m a l
>
1.
t h a t
cp/(a) = 0
= 1 -
f ' ( a )
a r e
second
and
t h i r d
r o o t .
This
a t
x
=
=
1
Taking
B o d e w i g ' s
a.
none
H(x)
x
one
=
In
-
may
m
because
I . F .
as
by
because
[7.7-1]
Bodewig
e n c o u n t e r
f o l l o w s :
f (x)H(x)
cp b e
of
i n
the
Since
s t u d y
we
b o t h
z e r o
terms
of
t h e
c o n c l u s i o n
or
t h a t
m
I . F . ,
>
1,
H(x)
cp'(a)
=
^
(7-5*0
i f
t h a t
f o r
r e a s o n s
a
i s
H(x)
has
a
w h i l e
v a n i s h
when
l / [ f ' ( x ) u ' ( x ) ]
0
a
example,
(7-5*0
of
t h a t
Bodewig
v a n i s h
(l-m,)7m
=
i s
f ( a ) H ' ( a ) .
p o s s i b i l i t y
terms
m / f ' ( x )
order
(7-54)
Newton's
f ( a ) H ' ( a )
-
f o r
i n
t h e
.
second
f'(a)H(a)
i g n o r e s
Hence
and
d i s c u s s i o n
a r g u e s
=
t h a t
l / f ' ( x ) .
f ' ( a ) H ( a )
f o r
m u l t i p l e
shows
r o o t s
i n c o r r e c t .
The
c o n d i t i o n
cp(a)
above
=
be
of
0 [ ( x - a )
d i f f i c u l t y
by
r e a s o n i n g
a,
which
l e d
i m p l i e s
t o
d i f f i c u l t i e s
f ( x ) H ( x )
0,
because
p e r m i t s
t h e
H(x)
m
"1
t o
t o
on
t a k e
c o n d i t i o n
and
s i n g u l a r i t y
i s
o u r s e l v e s
i n c o r r e c t
cp(x)
m
dependence
c .
The
A
e x p l i c i t
"
] .
t a k i n g
Since
u ( x )
=
0 ( x - a ) ,
we
can
a v o i d
t h e
7 . 7 - 3
cp(x)
=
x
-
u ( x ) h ( x )
.
Then
<p(x)
-
a
=
x
=
L e t
of
h
be
second
c o n t i n u o u s
o r d e r
i f
-
a
p a r t i c u l a r ,
we
+
( x - a )
i n
a
and
m
c l o s e d
o n l y
h ( x )
In
-
h '
i s
i s
e q u i v a l e n t
c o n t i n u o u s ,
-
t h e n
m
=
h ( x ) 0 [ ( x - a )
about
a.
O ( x - a ) .
=
( 7 - 5 6 )
2
] .
Then
cp
i s
( 7 - 5 5 )
ra.
i m p l i e s
( 7 " 5 5 ) .
( 7 - 5 6 )
E q u a t i o n
( 7 - 5 5 )
t o
(
x
u
summarize
i n t e r v a l
] | h ( x )
r e q u i r e
h
We
+
2
i f
h ( a )
I f
0 [ ( x - a )
our
r e s u l t s
|
i n
x
j
m
- D ^ O .
( 7 - 5 7 )
7.7-4
THEOREM
r o o t
v a l
of
m u l t i p l i c i t y
a b o u t
t h e n
a,
L e t
7-6.
cp
m.
i s
I f
of
h '
h ( a )
i s
=
c o n t i n u o u s
L e t
=
x
d e r i v a t i v e s
e x p l i c i t l y
f u n c t i o n
f
o u s l y
of
i n
t h i s
on
of
z e r o s
and
f'
-
c o n t i n u o u s
-
cp
i s
-
u ( x ) h ( x )
o r d e r
m
second
/
n
of
i f
i n
and
„
Let
a
a
be
c l o s e d
o n l y
a
inter-
i f
n
second
f
m.
x .
have
h i g h e r
L e t
Then
t h a n
h ( x )
i t
f o r
d i f f e r e n t i a b l e
o r d e r
h
a.
a
the
i f
and
C o n j e c t u r e
e q u i v a l e n t
does
f i r s t
n o t
t o
o n l y
and
does
w h i c h
t h e
depend
c o n t i n u o u s l y
i m p o s s i b l e
we
w h i c h
h
t h e
be
i s
7-8
a t
i s
Where
m u l t i p l i c i t y
S e c t i o n
and
o r d e r ,
s e c t i o n
u ( x ) h ( x ) ,
of
In
on
I . F .
e a r l i e r
cp(x)
whose
h
i s
x
i f
m.
For
s t a t e d
t h e n
=
second
h ( x )
I f
cp(x)
f o l l o w i n g
upon
n o t
we
any
depend
d i f f e r e n t i a b l e
t h a t
h ( a )
an
w h i c h
=
m
f o r
a l l
f
m.
c o n s t r u c t
- * m .
But
h
t h i s
h
i s
depends
n o t
o n l y
c o n t i n u -
7 . 8 - 1
7 . 8
An
I . F .
Of
Incommensurate
LEMMA
hood
of
a
Then
h ( x )
z e r o
7 - 2 .
a
of
Let
f (
Order
)
be
m u l t i p l i c i t y
m.
m
+
1
c o n t i n u o u s
i n
t h e
n e i g h b o r -
Let
-»• m .
PROOF.
L e t
f ( x )
=
( x - a )
g ( x ) .
m
( 7 - 5 8 )
Then
g ( x )
- g ( a )
-
f
(
m
m
}
,
(
a
t
)
0.
L e t
a ( x )
Then
G(x)
=
O ( x - a )
.
( x - a )
and
u(x)
From
( 7 - 5 8 )
and
,
w
x
^ f | f .
=
* = g .
( 7 - 5 9 )
( 7 - 5 9 )
_
l n l f ( x ) I
_
m
"
l n | u ( x ) I
~
l n l x - a
l n l x - a l
-
+
l n | g |
ln|m+G|
_^
m
'
7.8-2
LEMMA
uh'
f '
x
L e t
f
and
h
be
as
i n
Lemma
7-2.
Then
0.
-*
are
!
7-3.
PROOF.
We
p o s i t i v e ;
the
s h a l l
assume
r e s u l t
i s
f o r
t r u e
s i m p l i c i t y
t h a t
g e n e r a l .
Then
i n
f
and
f o r
£ a,
h
,(„)
-
<*>
~
1
u
,
_
m
l n [ u ( x ) j
{
f ( x ) u ' ( x )
l
n
[
u
(
x
)
]
)
2
u
(
x
)
'
Hence
(
\
» ( ) h ' ( x )
-
1
l n [ u ( x ) ]
h
l / m ,
x
The
f a c t
t h a t
LEMMA
-+ m,
u'
7-4.
h ( x ) u ' ( x ) '
l n f u ( x ) j -
"
l n [ u ]
-oo
Let
f
and
h
be
cp(x)
=
x
u ( x ) h ( x ) .
c o m p l e t e s
d e f i n e d
as
In
the
p r o o f
Lemma
7-2.
Let
Then
cp' ( x )
-
(7-60)
0 .
PROOF.
For
x
^
a,
h
and
hence
1
-
u ' ( x ) h ( x )
cp a r e
d i f f e r e n t i a b l e
and
q)'(x)
Note
t h a t
h
-+ m
u'
l / m ;
the
from
=
Lemma
c o n c l u s i o n
7-2,
f o l l o w s
uh'
-
u ( x ) h ' ( x ) .
0
from
I m m e d i a t e l y .
Lemma
7-3,
and
1
7.8-3
We
and
f ,
hope
has
t o
c a s e .
of
have
the
may
Theorem
t h a t
p r o p e r t y
c o n c l u d e
I t
shown
be
7-6
t h a t
shows
t h a t
cp
shown
h ,
i s
depends
- * m .
S i n c e
second
t h a t
t h a t
h
which
o r d e r
(h-m)/u
cp i s
-+
not
b u t
oo.
o n l y
f
cp'
0,
one
might
t h i s
i s
not
the
Then
second
on
an
o r d e r .
a p p l i c a t i o n
I n s t e a d
we
have
Lemma
7-2
THEOREM
7-7.
and
cp =
l e t
Let
x
-
h
and
uh.
f
s a t i s f y
the
c o n d i t i o n s
of
Then
1/m.
-
PROOF.
(
m
)
( q )
(7-61)
m •
S i n c e
cp =
we
ln( m
f
x - u h
=
x
-
u
If
I n
l n | u | '
h a v e
cp-q
x - q
R e c a l l
I n I x - a
=
I n
I x - q I
1
_
I n l f
I n
u
x - q
t h a t
n
_
l n | f I
l n | u |
m
m
I n
I n I x - q I
x - q |
+
-
l n l g l
ln|ra+G
x - q
'
u
"
m+G
7.8-4
where
f
The
f a c t
=
( x - a )
m
g ,
G
m i
&
t h e
c o m p l e t i o n
T h i s
theorem
m e n s u r a t e
w i t h
t h e
t h a t
i s
i f
( x - a )
^ .
t h a t
g^i^fll,
p e r m i t s
=
e
an
of
—\
t h e
shows
order
Q . o ( x - a )
^
p r o o f .
t h a t
s c a l e
a r b i t r a r y
/
the
o r d e r
d e f i n e d
by
p r e a s s i g n e d
of
cp
(1-14)
p o s i t i v e
i s
.
incomObserve
number,
t h e n
cp-a
x - a
We
can
(7-61)
w r i t e
9
x - a
1+e
as
1/m
f
ln^m
cp-q
C
x - a
Thus
cp
c o n v e r g e s
g o e s
l o g a r i t h m i c a l l y
of
x
In
p a r t i c u l a r ,
and
1
C
g e n e r a t e d
<
1
i f
l i n e a r l y
by
l e t
| x - a |
cp
t o
m
)
( a )
mi
-
l n | x - a |
and
i t s
z e r o .
I t
i s
c l e a r
i f
x
i s
c o n v e r g e s
f ( x )
<
=
(
=
l / m .
( x - a )
m
.
a s y m p t o t i c
Q
Then
e r r o r
t h a t
the
s u f f i c i e n t l y
C
=
-
c o n s t a n t
sequence
c l o s e
l n ( m ) / l n | x - a |
t o
f
7-8-5
The
R e c a l l
f o l l o w i n g
g e n e r a l i z a t i o n
=
l
n
1
f
In
where
ln|m+G|
a c c e l e r a t e
-»
t h e
X 1
more
In
S i n c e
l
n
l
l n | u |
+
-+ m,
h
ln|m+G|'
t h i s
t o
m
1
1
| ^
n
s u g g e s t s
by
h
l n | h
l n l g l
-
t h a t
we
can
d e f i n i n g
-
m l f
"
I n j u r
l
l n | f
h
i + l
i n
"
l n j u j
t u r n ,
+
l n j l i j '
s u g g e s t s
^
For
of
f
v i e w .
=
We
l n | u j
=
( x - a )
have
"
1
i m p l i c i t l y
m
+
n
,
(7-62)
u
=
( m u )
and
h
of
f
+
l n l x - a l
l n | x - a |
g e n e r a l l y
n
T h i s ,
m.
=
2
m
=
u|
c o n v e r g e n c e
h
p o i n t
i t s e l f .
t h a t
h
or
s u g g e s t s
m
0
d e f i n i n g
l n | i i |
f
may
( x - a ) / m
=
> - ^ - - - ;
( x - a )
h
h.
-
0
L
by
(7-62)
•
be
d e r i v e d
from
a n o t h e r
and
m
=
f,
h e n c e
m
~
The
a n a l y s i s
cp =
x
-
uti
of
has
the
n o t
o r d e r
been
In
l n | u |
of
1f1
+
I . F .
c a r r i e d
l n
nT
of
t h e
o u t .
form
cp =
x
-
u h
±
and
7.8-6
A f t e r
e x p e n s i v e
f i n d i n g
A f t e r
l i m i t
t h e
and
f'
c a l c u l a t e
r o u t i n e
the
i n t e g e r ,
t o
f
h
have
=
been
l n | f | / l n | u | .
i t
m i g h t
be
of
h
has
been
can
s w i t c h
r o u t i n e
c a l c u l a t e d ,
w o r t h w h i l e
In
t o
d e t e r m i n e d
t o
<5
2
=
-
i s
g e n e r a l
a l w a y s
t o
x
a
i t
the
mu.
n o t
r o o t -
c a l c u l a t e
n e a r e s t
h