ON
~lOLTIPLE
POINT
TAYLOR SERIES
EXPANSIONS
by
CHESTER WAYNE DYCHE
A THESIS
submitted to
OREGON STATE COLLEGE
in partial fulfillment of
the requirements tor the
degree of
MASTER OF ARTS
June 1956
APPAOVED
Redacted for Privacy
Araoclatg Profeasor of hdathematlss
In
Chargc of MaJor
Redacted for Privacy
&ot1ng Shalrman
of Departmant of
Mathsmatloa
Redacted for Privacy
Chalrnan
of
Sohool" Srailuata 0onmlttea
Redacted for Privacy
Dean
of
Orad,uatc Sohoo1
prcscntcd
Data thosl.s
tr
fypcdt by lY.
M. $ltone
s
ACKNOWLEDGMENT
The author is indebted to Professor
w.
M. Stone whose
encouragement and assistance led to the completion of this
thesis.
TABLE OF CONTENTS
PAGE
CHAPTER
I.
II •
III.
IV.
INTRODUCTION
1
THE TWO POINT THEOREM
3
SPECIAL CASES
10
NUMERICAL EXAMPLES
13
BIBLIOGRAPHY
17
ON MULTIPLE POINT TAYLOR SERIES EXPANSIONS
INTRODUCTION
The rapid advance in the electronic digital computer
and its application has brought forth many advances in
numerical and approximation methods.
The literature pub­
lished in this field has increased and work is being done
in many areas.
One such area is the evaluation of a function in
terms of its derivatives at one or more points.
Such
formulas have been developed and used by many authors al­
though no general approach has been made.
Boola (2, 10?­
109) used a semblance of this idea to solve a special
problem.
Milne (10, 53?-542} appears to have priority
with a method for the numerical integration of different­
ial equations which depends on the values of the first
three derivatives.
Salzer (12, 103-106) has developed
integration formulas using both the function and its first
derivative.
Luke (9, 298-30?), Hammer and Hollingsworth
(4, 92-96) and Fettis (3, 85-91) obtained various modified
forms of the trapezoidal formula.
Householder (5, 232-240)
discussed several types of quadrature formulas.
Hummel and
Seebeck (6, 243-24?) developed a treatment for the evalua­
tion of a function without regard to quadrature methods.
2
Lotkin (7, 171-179) developed a specialized version of
the first integral of the function.
Andress (1, 394­
396) developed essentially the same method on the complex
plane.
The bibliography here only indicates the numerous
approaches that have been made to this problem.
The objective of this paper is to generalize Taylor's
theorem of the expansion of a function about a single
point to an expansion about two points .
multiple point expansion are:
The uses of the
to obtain generalized
formulas f or mechanical quadrature; generalization of the
Milne-Lotkin approach to the numerical integration of dif­
ferential equations; a representation of functions by ra­
tional fractions which is of interest in the construction
of tables.
3
THE TWO POINT THEOREM
The following assumptions will be made:
tinuous in an interval which contains
derivatives up to and including the
tinuous and the
(m+.n+P+l)th
The letters
negative integers.
noted by
( ~)
are con­
n
must be of exponential
are defined to be non­
The binomial coefficients will be de­
q < 0
t
or
q > p.
will mean to sum over
n.
(m+n+p)th
x; all the
with the understanding that the expression
is zero for
and
and
m
and
0
derivative is sectionally
continuous in that interval; F(x)
order.
F(x) is con­
Also
A function
from
k
t
G(x,p}
0
o if
=
(I<)
The sy.mhol . L f(m,n,k)
k=O
to the larger of
p <
m
o.
is defined by the definite inte­
gral
X
(2.1)
G(x,p)
=
f
.
(mill-tp-f:l.)
F
m n
(t) (x-t) t dt.
0
Application of the standard one-sided Laplace transforma­
tion to both sides yields
(2.2)
g(s,p) =
d
n
ds
n
4
m+n+p
L
m+n+p-k
{k)
s
F
(O)J.
k=O
Because of the Leibniz formula (2.2) becomes
n
n
(-1) g(s,p} = m!
L
k:O
n
k
P+k
(m+n +p+l)!
s
r
(k}
(s)
(m+p+l+k) !
(2.3)
(k)
m+P
- .t!l!
~
(m+n+p-k)!
( 0)
F
sk+1-p
(m+p-k)!
- p -1
= mi
L
k =O
n
k
(m+n +p+l) !
n
(m+n+p+l) !
(k)
p+k
t
s
(s )
(m+p+l+k)
n
+ m!
L
k=-P
m+p
- m!
~
k=O
k
(k)
p+k
s
(s)
f
(m+p+l+k) !
(k)
(m+n+p-k)!
F
(m+p-k):
s
(0)
k+l-p
•
The inversion formulas
(2.4)
1
- 1{ sp+k (k)
t
k
(s)} = (-1)
k+p
_!.
d~+p
k
X
F(x)
p <
0,
5
and
=
(2.5)
k
X
lead to the form
(m+n +p+l)!
(m+p+l +k )!
~
-1
p+k (k)
{s
f
(s)}
The formula for interchange of order of summation,
(2.7)
n
k
k=O
j=O
L .L
n
f(k,j)
=
n
LL
j
=0
f(k,j)
'
1:1
p+k (k)
f
( S) }+
k =j
yields
(m;n +p +l)!
{ m+p+~+k} !
(2.8)
{S
5
n ) (m+n +p+l)! j-p (j)
(
m!
j
::O
x
j-p
F
j:
+
~
k-p
c n-j +p) (-1} . k!
(x)
k= j
k-j
(m+k +l):
-p-1
(k)
(m-tn +P+l) !
= m! L
k=O
f
( s)}
(m+P+l +k)!
~
k =O
To eva luate the second
s~~ation
(with respect to
r) it
is necessary to itera te t he integra tion of both sides ot
the identity
n~
+
( -1)
(2.9)
r =o
m+l
r c n-j +P)
r
X
r +j
j
=
x (1-x)
times, the last integration being between
That is,
n-j +p
0
and
1.
7
>
n-j +p
£__
(
-1)
r c n-j +p )
r=O
1
-------­
(r+j+l}····(m+r + j +1)
r
(2.10}
1
= 1
m!
J
o
j
m+n-j +p
x (1-x)
dx
j I (m+n-j +p }!
=
m! (m+n +p+1)!
and (2.8) takes the form
n
(-1} G(x,p)
> (~)
-p-1
=
.£_
A
(m+p+1+k)!
k=O
(m-m +P+l}!
m%
l
l
- ·
{s
p-+k (k)
r
(s)}
The theorem of the mean for integrals gives as an estima te
of G(;x,p),
m-tn+l
G(x,p) = x
(m+n +p+1)
F
1
r
mn
(X) J (1-u) u du
0
(2.12 )
m+n+l
=
ml n! x
(m+n+p+l)
F
(X}
•
(m-tn +l)!
where
0 < X < x.
The final form of the two point Taylor
8
expansion is, therefore,
~ (~)
--1'-1
i_
ml
k=o
p+k
{s
t
(k)
(s)J
(m+p+l+k)!
(2.13)
~(
k)' Gm ) F (k) (0)- (-1) k-pIn
=L
_ m+n+p- • {
k=o {m+n+p+l)! k-p
~ k-p
)F(k) (x)J x k-p
+ R(x,p) ,
where
n
(2.14)
R(x,p) =
(-1)
m!
n! x
m+n+l
(m+n+p+l)
F
(X)
•
(m+n+l)!(m+n+p+l)!
The Boole (2, 107-109) development is quickly dupli­
cated in modern notation by using the two-sided Laplace
An obvious identity,
-sh
-sh
1 - e
(1 - e
)f(s) = ----sh (1 + e -sh )f(s)
transformation.
1 + e
(2.15)
= {~ 2
00
~
= ~
(sh)
24
-
3
•••• J
-sh
(1+ e
)f(s)
n
2n+l
-sh
(-1) En(sh)
(1 + e
)f(s) ,
n=O
where the
En
are the Bernoulli numbers, is written down.
9
The inversion yields
(2.16}
F(x) - F(x-h)
00
~
= L _ ( -1 )
n=O
n
2n+1
En h
{ F
(2n+1}
(2n+1)
( x) + F
(x-h)} ,
which is equivalent to the original form of Boole.
10
SPECIAL CASES
The re placement of
p
by
0
in the general formula
of (2.13) yields
(3.1)
+ R(x,O)
or
F(x) - F(O)
(3.2)
=
~ (m-+n-k)!
L_
k =l
(m-tn)!
()
{ mk
F
(k)
k
{0)- (-1) ( nk )F
(k)
+(m+n+l) R(x,O) ,
where
( 3 . 3)
R(x,O)
=
n
m+n+l
(-1} m! n! x
{(m+n+l)!}
2
F
(m+n+l)
(X)
This formula is that of Hummel and Seebeck (6, 243-247),
who state that many special cases arise from variations
of
m and
n.
If
n =0
e quation (3.2) becomes the Tay­
lor series with remainder.
Since
(3.4)
J F (t) dt
X
f -1{ f(s) }
~
s
=
0
k
(x) }x
11
the choice of
p =-1
reduces (2.13) to
X
I
+L
F( t) dt
0
=
R(x,-1)
(3 .5)
(m+n-1-k)!
(m n)!
----{
k=O
m)F (k) (0)- (-1) k+lGn J F (k) (x)}xk+l•
0
k+l
k+l
where
(3.6)
R(x,-1)
=
n
m+n+l (
)
F m+n (I)
(-1} m! n! x
(m+n+1}! (m+n) !
Here again varied formulas are obtained by adjusting m
and
n.
For
x
Jo
m = n,
n-1
F(t) dt
~
= L_
k=o
)
t
(Zn-l-k ·( n \p!'
(2n)! \ k+~
(k)
(0} - (-1)
k+l (k)
k+l
F
(x)Jx
(3.7)
+ R(x,-1) ,
where
(3.8)
R(x,-1) =
n
2
2n+l
(-1) (d!)
(2n)
F
(X)
•
( 2n+l ) I ( 2n ) !
Thus (3.5) is a generalization of Lotkin's (8, 29-34)
formula since he limited his study to the case
m = n.
In their papers on the numerical solution of differ­
ential equations Milne (10, 537-542) and Lotkin (8, 29-34)
have used the two point expansion formula, n = m = 3, as
12
a "corrector", with the Taylor formula used as a "predict­
orn.
Lotkin's proof of convergence for such a method may
be generalized without difficulty.
The transform pairs
X
J
(x-t)F(t) dt
0
and
-1
L {f
(3.10)
reduce (2.13) for
X
(1)
( s)
s
p
= -2 to
X
J
F(t) dt - (m-+n)
0
(3.11)
~(
2 k) '
+L _ m+n- - ·
k=o
(m+n-1)!
= -ftF(t) dt
0
X
mx
}
J
t F(t) dt
0
{G m \ F(k) (0)k+2 )
=
R(x,-2)
k
( k)
k+2
(-1) / n ) F
(x)}x
•
\k+2
Probably a better approach to finding a formula for
X
I
-p+l
t
F ( t) dt
0
in (3. 5).
would be to replace
F(x)
by
-p+l
F(x)
X
13
NUMERICAL EXAMPLES
To show the value of the two point formula it is
first applied to the Bessel differential equation,
(2)
xy
(x) + y
(1)
(x) + xy(x)
0,
=
(4.1)
= 1,
y(O)
and
y(x) =Jo(x}.
y
Taking
(1)
y(x)
y(x,l) =
2
12 - X
2
2(6 +X )
(4.2)
(6 -
2
n =m
(1)
and
=
the solution
z(x)
in terms
1, 2 and 3 are
,
2
y(x,2) =
- X )
2
4
2(240+ 17x + x )
y(:x,3) =
201, 600 - 41, 280x + 1134x - 3:x
X
= 0,
( 0)
= z(x)
(x)
of the two equations defining
of their values at zero for
y
)( 80
2
2
4
4
6
6
8( 25 .zoo + 1140x + 33x + x )
with appropriate remainder terms.
is less than
By taking
2•10
m
-6
=
The error in
•
n
;=
2
and setting up the equations
for a stepwise procedure the representations of
z(x)
y(l,3)
in terms of values at an arbitrary
x0
y(x) and
become
14
(4.3)
y =Yo + {x-xQ)
2
2
(x-x
)
(y~ + y' ) +
0 ( y t t _ y I t)
0
12
and
2
{4.4)
z = Zo + (x-xQ) (z'+z')
+ (x-xQ) (z''-z'') •
0
0
12
2
To reduce these to solvable equations in terms of
and
z(x)
the values of
differentiating (4,1).
z'
and
z''
y(x)
are obtained by
Finally
X
(4.5)
and
y (x-x0 ) (7x-x 0
)
+ z{l 2 +
2
6(x-x ) + (x-x 0 ) (2-x )
2
~
X
X
J
(4.6)
= 12y6
+ 6(x•x 0 )y~' + (x-x 0
and successive appliea tiona for
to the table
x0
= 0,
2
)
y
t t t
0
1, 2; • · · •• leads
15
J~(x)
X
y(x)
.To(x)
z(x)
1
.7655
.7652
-.4399
-.4401
2
. 2246
. 2239
-.5770
-.5767
3
-. 2596
-.2601
-.3400
-.3391
4
-.39?8
-.3971
.06503
.06604
5
-.1791
-.1776
.32?7
. 3276
6
.1496
.1506
. 2781
.2?67
7
.3007
. 3001
.00643
.00468
8
.1613
.1?1?
-.2535
-.2346
9
-.1109
-.0903
-.2461
-. 2453
10
-.2467
-.2459
-.02758
-.04347
Table 1.
That is, J (x) has been rather quickly estima ted from
0
0
to
10
with sufficient accuracy for drawing curves.
The values of
J 0 (x)
.T~(x)
and
have been taken from
Watson (13, 666-685).
A form of the error function serves as a second
numerical example.
For
2
(4.7)
y(x)
=
e
X
f
2
00
e
-t
dt
X
the two point formul a for
m
= n = 4 and x 0 = 0 reads
17
BIBLIOGRAPHY
1.
Andress, w. R. The expansion of a function in terms
of its values and derivatives at several points.
American mathematical monthly 60: 394-396. 1953.
2.
Boole, George. A treatise on differential equations.
4th edition. New York, G. E. Stechert, 1877.
235p.
3.
Fettis, Henry E. Numerical calculation of certain
definite integrals by Poisson's summation formula.
Mathematical tables and other aids to computation,
9. 85-91. 1955.
4.
Hammer, Preston c., and Jack w. Hollingsworth. Trape­
zoidal methods of approximating solutions of dif­
ferential equations. Mathematical tables and
other aids to computation 9: 92-96. 1955.
5.
Householder, Alton s. Principles of numerical analy­
sis. New York, McGraw-Hill, 1953. 268p.
6.
Hummel, P.M., and c. L. Seebeck, Jr. A generalization
of Taylor's expansion. American mathematical
monthly 56: 243-247. 1949.
7.
Lotkin, Mark. A new integration procedure. Journal of
mathematics and physics 32: 171-179. 1953.
8.
Lotkin, Mark. A new integrating procedure of high ac­
curacy. Journal of mathematics and physics 31:
29-34. 1952.
9.
Luke, Yudell L. Simple formulas for the evaluation of
some higher transcendental functions. Journal of
mathematics and physics 34: 298-307. 1956.
10.
Mi lne, William Edmund. A note on the numerical inte­
gration of differential equations. NBS journal
of research 43: 537-542. 1949.
18
11.
Rosser, J. Barkley.
z
J
-x2
e
0
dx
and
Theory and application of
Jz
0
- p 2Y2
e
JY
dy
e
-x2
dx.
Brook­
0
1yn, N. Y,, Mapleton House , 1948.
l92p .
12.
Salzer, Herbert E. Osculatory quadrature formulas .
Journal of mathematics and physics 34: 103-106.
19!55.
13.
Watson, G. N. A treatise on the theory of Bessel
functions. 2nd edition. Cambridge, England,
Cambridge University Press, 1944 . 804p .