AN ABSTRACT OF TRE THESIS OF
for
(Degree)
(Naine)
(Majcr)
Date Thesis
T i tie
IJ;
J
1'1Q
9
.....
Funct IOnS U
Abstract Approved.
(Major
Professor)
Many formulas for inechanial quadrature of differential
equatIons and continuous functions have been developed hut
very little has been done toward irrational functions where
the integrand becomes infinite or vanishes1
The object of this thesis is to develop formulas for
mechanical quadrature of irrational functions of the followwhere
\,íO(J()
The function f(x)
Ing three types: Type A
function
The
the
first
order,
Type
B,
has
a
role
of
0(x)
f(x)
vG(x) where G(x) has a zero of the first order.
The function f(x) = -/G(x) where 0(x) has two poles
Type C,
first
order,
of the
Althou.gh the Newton-Cotesfbrmulas can he applied to
these types of functions after making a transformation, the
present formulas are more convenient because they are
developed for these particular cases where the infinite
ordinate or vertical tangent lie at either or both ends of
the interval.
FORMULAS FOR MECRANICAL QUADRATURE
OF IRRATIONAL FIJNCTIONS
by
MABEL ELLEN YOUNGBERG
A THESIS
submitted to the
OREGON STATE AGRICULTURAL COLLEGE
in partial fulfillment of
the requirements for the
degree of
MASTER OF SCIENCE
July 1937
APPROVED:
Head of Mathematics Department, In Charge of Major
TABLE OF CONTENTS
Page
Chapter
Introduction
I.
i
Chapter II. Development of Formulas
Section
I.
3
Formulas of Newton-Cotes
3
Section II. Derivation of New Formulas
8
Type A, f(x) = /(x) where 9(x) has a pole of
io
the first order,
1,
Four-point Formulas
10
2,
Five-point Formulas
12
3,
Six-point Formulas
12
4,
Seven-point Formula
12
5
Eight-point Formula
13
Type B, f(x) =
/9(x) where 9(x) has
a zero
of
the first order
13
i,
Four-point Formulas
13
2,
Five-point Formulas
14
3,
Six-point Formulas
15
4.
Seven-point Formula
15
Type C, f(x) = V9(x) where 9(x) has two poles
of the first order
15
1,
Three-point Formula
15
2,
Five-point Formula
16
3.
Seven-point Formula
16
Chapter III, Comparison of Results Obtained by Formula
and by Integration for Certain Examìles
Appendix,
Table of Formulas
Bibliography
17
i
vii
ACKNOWLEDGIVIENT
The writer wishes to express her sincere
thanks to Dr, W, E
Mime, Head of the
Mathematics Department, who suggested this
topic and directed the work done on it,
FORMIJLAS FOR !V1ECHPNICAL QUADRATURE
OF IRRATIONAL FOiTCTIONS
CHAPTER
I
INTRODUCTION
The problem of mechanical quadrature or nwnerical
Integration has become one of great interest today in many
fields of science on account of Its practical and time
saving value1
Often it is desirable to find the integral
of a ftriction by nwiierical integration because eIther, the
general formula can not be found;
difficult to apply,
or,
if found,
it is
very
The first attempt to answer the prob-
lem of numerical integration was made by Sir Isaac Newton1
(3)
He followed through the developments and applications of the
subject to a degree of detail which has been difficult to
surpass,
Therefore, his work has been the basis for many
of the later developments1
The formulas for mechanical quadrature are usually
developed by eIther of the following methods:
(1)
the
(4)
method of differences, or (2) the method of ordinates1
Although these methods are equal in importance, they differ
greatly In appearance1
The latter has an advantage in that
the calculation of a difference table is not required,
There aro many formulas and methods which have con-
tributed to this specialized field of mathematics,
Ariong
2
those who have directed their interest toward the numerical
(2)
solution of differential equations are Runge, Heun, Kutta,
Nystrori, Moulton, and J,
C1
Adams,
Still others have
directed their attention to quadrature formulas for continuo's functions by one of the above mentIoned methods,
Some
of the outstanding accomplishments in this branch of numerical
integration are due to Newton, Cotes, Euler, Gregory and
(1)
Steffensen1
Up to the present time, however, very little
has been done toward the numerical integration of irrational
functions where the integrand becomes infinite as in the
case of i// x
or vanishes as
in
/x
at x = O
It
is the
purpose of this thesis to develop mechanical quadrature
forrriulas for
such functions,
3
CHAPTER II
DEVELOPMENT OF FORMUIS
L
Section
Formulas of Newton-Cotes
Before taking up the
ruietIod
T
3f development of the
let us consider the basis for
formulas of this thesis,
the Newton-Cotes formulas
Let F(s) denote a given continuous function of
s
in
the interval (a, b), which may he expressed In this manner,
(b
\
F(s)ds,
tía
Although Newton-Cotes formulas can be applied to any
closed interval (a, b), it is convenient to make a change
of variable so that the relationship between the original
variable
s
and the new variable x is given by the expres-
sion
s(b-a)x+a,
(1)
By direct substitution of the original limits (a, h)
in (1) we have the new limits
(1)
that ds =
(b - a)clx,
i),
(O,
and F(s) =
FÒ
It also follows
-
a)x +
a]
from
which we
shall call 1(x) for conveniences
Therefore, we have
(2)('b
F(s)ds
\
Ja
('i
-.7
F[(b
(b - a)
-
a)x + aJ
d.x
(b - a)
JO
I
JO
f()dx.
4
N
We will now derive the forimila for'
Ch
\
Ja
f(x)dx since
Jo
F(s)ds can be found by (2),
A change of variable does not affect the continuity of
a function,
since f(x) is continuous, let us
Therefore,
assume that it can be approximately represented by a poly-
nomial in z of cicgroe n, expressed as P(x)
By the rules
of integration, the desired integral or the function f(x)
with respect to z in the interval
(0,
1)
approximately
is
equal to the integral of the approximating polynomial with
Hence we have
respect to x in the interval,
(i
(3)
Ci
P(x)dx
f(x)clx
\
Jo
=
+ Remainder,
o
where the right hand part of the equation may he approxi-
mately represented in this manner
('i
(4)
In
t
P(x)dx
(4), Y. is
val (O, 1)
+ Remainder
+ A1Y1 + -
the ordinate corresponding to x
(y1 = f.(xj), where i
1,
2
-
A1 is the undetermined coefficient of Y1
(0,
1)
whence
is
f
From
\
-
.
V
-
in the inter-
n), and where
Also the interval
subdivided into n equal segments of length h,
i/n,
(3)
and (4), we have by direct substitution
('i
(5)
-
-
f(x)dx = A0Y
+
AY1
+
A
Jo
Y -----+
+
By letting f(x) in the above equation take on the
following values
i,
x,x2,
-
-
-
-
x', a set of n siimiltaneous
5
equations in A are now obtained by direct integration1
(6jdz1=A0+Ai+A+A3+ -------1x
= 1/2 =
c
+
+
= 1/3 = h2A1 +
2hA
Jo
-----
+
3I
+
3h2A3
+
-
-
+
+n2h2Ant
2.
xndJ
Jo
n+1
=
hA
+
2hA
+
3hA3
+ --
z,
Before solving these simultaneous equations Cor the
A's we will
eliminate
the factor h from each equation, and
at the same time substitute for it,
its value 1/n,
We then
obtain
(7)
1A0+A1+A+ ------------A1 + 2A
A1 + 22A
+ 3A3 +
-
32A3 +
---------
n2A
3
A1 +
2A
+ -
n
+ flnA
-
In order that the A's in these n simultaneous equations
may be uniquely determined, the determinant
dents
of
the A's must not equal zero,
this determinante
of
the
coeffi-
Let us next study
(8)
D
1
1
1
o
o
O
i
2
12
22
i
2
1-1
3-n
32_
-------
in
O
33
n3
3r
nfl
We will evaluate D by the first column,
Since all of
the elements are O except the first, we have
(9)
D =
i
1
2
12.
22
i
2
3-------n
32
33
-----
n3
This cofactor,D, is a well known determinant of Vander-
monde type and can be expressed in this manner,
(10)
D,
Because D,
(n
0,
-
1);
-----
2
14,
and therefore by (9) D
0,
the A's can he
determined,
It will he convenient to write
(6)
in the following
manner,
(11)
1A0+A+A+ -----------= x0A0 +
1
_L
xA1
+
x2A
+
x2A
+
XnÌA
+
x1A1
+
wherex00x,x
------------x2A
xA
+
-
+
X
riA
+
------
---------
-j.
Let us assume that
P(x) =
x0)(.x
-
(
P,(x)
=
(x
-
x,jx
-
x1)(x
-x) ----- (x -x),
X
+
=
a10x1
P(X)
P1(x)
,
X
(x
-X)
x0)(x
-
------
+
-
-
-
+
a°
-x),
-(x
or
or
'i'
a1x'
+
P(x)
=
azoXl2
+
xc).
-
(Lx
- ----
=
(x
1n
+ a
aa'.
)(x -
-
-
o
-
-
or
x
n
+a
flfl-i
-----------
+
n
-i-a.
We will now multiply the left hand members of (6) by
in reverse order,
the coefficients of P0(x)
(12)
an°
This gives us
---------------------fa°
n3
2
n-1
on the left hand side, while on the right we get
A0P0(x0)
+
A1P0(c1)
which reduces to A o P,(x o
),
+
-
-+ AnP(Xn)
-
since P o (x1) = O, P o
()
O.- -,
in the equation for P0(z), we have
By setting x
P0(x)
=
(x
-
x1)(x0
-
x1)
-----------
Therefore from (i2t, we obtain
a°
a1
A0
+ -
-
-
x1)(x0
(x
O
-x)
-
+
fl+]
2
.
-
-
-(x0-x)
(x0
-
Similarly, we get
Aafl
i
+a_1+
i
:i.
P1(x1)
Afla'1+a1
,+ -----
i
n
P
(xc)
The exact value of the A's can now be found by sub-
stituting in the values X0 = O, x1
,
-
-
-
-,
x
1,
and the values of the a's1
These values of the A's are then substituted in the
integral formula (5), which gives the desired formula for
the numerical integration of the function1
By this procedure the following Newton-Cotes formulas
are developed:
r2h
+ 4Y1 +
=
o
jf(x)dx
4h
('4h
f()cIx =
(7Y
+ 32Y1 + 12Y' + 32Y
+ 7y4),
o
('6h
f(x)dx =
6h
840
(41Y0 + 216Y1 + 2'7Y
+ 272Y3 + 27Y4
S
+ 216Y5 +
Section IL,
4iY)1
Derivation of new formulas,
In this section,
formulas for mechanical quadrature
of irrational functions, where the integrand becomes infinite
as
(a)
in the case
of i/i/i.x2 at
in v'
at
.x
O,
=
of i/1/
1,
at x = O,
and (b)
in the case
and where the integrand vanishes as
are developed for the interval O
nh,
The Newton-Cotes formulas will fail for tIese three
irrational functions unless a transformation is made, because
in case (a)
there is an infinite ordinate,
two infinite ordinates and in (c)
in (b) there are
one vertical tangent,
To bring out the difficulties encountered when using
a
Newton-Cotes formula for one of these irrational functions,
io
fi
we will apply it to case
i
T
(a),
j
cix
o v
Expressing x as x 9)(x), where
.i(x)
is
a
holomophic
Ci
function, we have
dx,
Let us now make the trans-
)o\/xq)(x)
52
formation x
then dx = 2sds,
2sds
We get
Jo/szÇ's2.j
¡1
2ds
¡
=
/
t
J 0V
1?
(s
2
which is continuous within the closed interval
because of the definition ofI4(x), which becomes ÇJ(s2)by
the transformation,
Still greater difficulties, however, arise
culations because x1 must be determined from
in the formulas, Section I,
It
is due
to the
5
n the cal-
before Yj
can be found,
inconvenience of the Newton-Cotes
formulas, for these three mentioned types of functions that
the writer of this thesis has derived the following formulas,
Only the first formula in each type will he completely
derived due to similarity in development of the rest,
Type A1
f(x)
v'e(x)
where 8(x) has a pole of the first
order,
i,
Four-point formula,
Let f(x) represent a function in x where lin / x f(x)
C
X
in the interval O to h,
(In case the
not be at an end of the interval,
make
a
it
infinite ordinate should
would he necessary to
change of variable, Section I, i, before completing
the derivation),
11
We now assume that
Ti
f(x) may he approximately rep-
resented within the interval by a polynomial in x of degree
3,
thus
\/
P3(x) + Remainder1
f(x)
Then it follows by Section
I,
and (4) that
(3)
"h
+ Remainder
f(x)dx
(1)
Ap0
where
Ap1
+
Ap
+ A3q3
f(x) and A1 the undetermined coefficients,
(x) = -/T
Replacing
+
by
cp(x)
1, x,
z2, x
respectively, in
(1)
we obtain by direct integration the following four siniul-
taneous equations which must he satisfied by the A'S,
-
Ihcìx
2V h
Jovx
('h
(h
xdz
jOV'x
-
.h
/h
h2
=
A0 + A,
+ A
O + hA1 +
= O +
hA1
+ A3
2hA
+ 3hA3,
4h2A
+
g
5
9hA3,
and
('h
jo
= ..h
= O + h 3A1 + 8h 3A
+ 2 7h .&3
1
x
After the elimination of a factor h, h2, and h3, respectively, from the last three equations above and the
from all four equations, it is found
315
that the determinant of the coefficients of the A's is not
common factor
equal to zero by Section I,
(8)
and (9)
,
Hence, there is
12
one solution which uniquely determines the A's,
These A!S
can be found by solving the 4 simultaneous equations in (2)
the method used by the writer
Section
(li) through
I,
-
or by the method given in
(1.3).
A360
A322
= -108
It follows
-
A0
356
by the substitution of the above values of
the A's in (1) that
Ch
I
(356p0 + 360Pa - 1O8
f(x)dx
+ 22cp3),
Jo
which is the desired four-point formula of degree
3
for the
interval O to h,
We also have by the above method formulas where the
0-2h
intervals are
Type A,
2,
and 0-*3h.
See Appendix, Section II,
1.
Five-point Formulas.
The five-point formulas for the intervals O to nh,
where n
1,
II, Type A,
3,
2,
,3,
4 are listed
in the Appendix, Section
2.
Six-point Formulas.
The six-point formulas for the intervals O to nh, where
n = 1, 2, 3, 4, 5 are listed in the Appendix, Section II,
Type A,
4,
3.
Seven-point Formula.
The seven-point formula for the interval O Lo 6h is
given in the Appendix, Section II, Type A, 4.
13
Eight-point Formula,
5,
The eight-point formula for the interval O to 7h is
given in the Appendix, Section II, Type A,
5,
f(x) = VG(x) where 9(x) has a zero of the first
Type B,
order,
Four-point formula,
1,
Let f(x) represent a function in x where
hm
xC
f(x)/'/i
-
C
in the interval O to h, where
the interval
The vanishing ordi-
is the result of a change of variable,
nate of the function then lies at one end of the interval, O,
may he approximately repre-
We now assume that f(x)/
sented withiñ this interval by a polynomial in x of degree
3,
thus
=
P(x)
Remainder,
+
and by the rules of integration
rh
(1)
f(x)d.x
JO
=
N \/5
ReplacIng
by
cp(x)
+ Remainder
+ A1p1 + A
= A0p0
where p(x) is equal to
P(x)dx
+
Ap
f(x)//,
1,
x',
x,
x3 respectively, we obtain
by direct integration the following four simultaneous equations which must he satisfied by the A's,
rh_
i/.xdx
(2)
-
= 2hJh
.10
3
2h/
:x\/i-dx =
j.
=
=
A0 + A
+ bA,
+ A2 + A3
-f.
2hA,
+ 3h&3
14
x2/ix
2h3/E
2h\/h
= O + h2A
= O +
hA1
+ 4h2A
+ 9h2A3
+ 8h3A, + 27h3A3,
From the last tiwee equations above the factor h, h2,
and h3, respectively, are removed and a coimnon factor
2hV
945
from all four equations1 The determinant of the coef-
ficients of the A's is not equal to zero since it is of the
Vandermonde type, Section
i,
(9),
Hence the A's are uniquely
determined by either method mentioned in the development of
previous formula,
A
= 13, A
= -66, A1 = 282, A0 = 86,
These values
substituted in equation (1), gives
2hV'
jf(x)dx
860
+
-
+ l3p3),
which is the desired four-point formula for the interval
O
to h,
In a similar manner the complete list of formulas for
this type are developed,
1,
See Appendix, Section II, Type B,
for the four-point formulas for the interval O to nh,
where n =
2,
1,
2, 3,
Five-point Formulas.
The five-point formulas for the intervals O to nh where
n = 1, 2, 3, 4 are listed in the Appendix, Section II, Type B,
2,
15
3,
Six-point Formulas.
The six-point formulas for the interval O to nh where
n = 1, 2, 3,
4,
5
are listed in the Appendix, Section II,
Type B, 3,
4. Seven-point Formula1
The seven-point formula for the interval O to 6h is
listed in the Appendix, Section II, Type B,
Type C,
4,
9(x) where e(x) has two poles of the
1(x)
first order,
Let us represent 1(x) by a function in x where
in the interval (-1, 1),
1(x) =
In case the infinite
ordinates do not lie at the ends of the interval,
a
change
of variable must be made.
1,
Three-point formula,
Let us assume that f(x) may be approximately represented
by a polinomial of degree 2 in x,
pN)
Remainder,
+
We will now take the integral between (-1, 1), which gives
(x)d
(
J
P(x)d
+ Remainder
J-i.
Ap(_1)
(1)
+ A0p(0) + A1p(1), where
the p's are the values of p(x) at the middle snd ends of the
interval, and the A's the undetermined coefficients,
Replacing
p(.x)
by 1, x', x
respectively we obtain by
16
direct integration the following three simultaneous equations
which must he satisfied by the A's,
pi
A_1 + A0 + A1
dx
j-i Vii'i
J-1
x
Vi-x
ri
clx
j1
\/1X
A_1+O
=
2
+ A1
That the determinant of the coefficients cf tho A's is
not equal to zero, can be easily shown by expanding the
determinant by the element in the first row and second column,
Therefore, the A's can be uniquely determined by the method
of elimination which gives us
A -1 -'
A0 -
-7t
A
By substituting these values of the A's in (1) we get
(x)
dx
(-1)
+
2(0)
+
The development of the five-point and nine-point formu-
las is similar to the above steps,
in the Appendix,
Type C, 2 and
3,
These formulas are listed
17
CHAPTER III
COMPARISON OF RESULTS OBTAINED BY FORMULA AND
BY INTEGRATION FOR CERTAIN EXAMPLES
To illustrate the value of these formulas, let us apply
them to examples of each type considered in Chapter II,
Section II.
In all of the examples except the last we will apply
the Newton-Cotes five-point formula for that part of the
interval which is closed because we know that it is accurate
to the seventh decimal place for these examples, and the new
formulas for the ends of the interval where the infinite or
vertical tangents occur,
Type A.
cl
Example
To find(
I.
j-i
____
,
where there is an infinite
Vi-s
ordinate at each end of the interval.
By direct integration, we get
1,
1
3.1415926
arc sin x
-
J-1
-
1
2.
By formula, where f(x) =
Vi-x2
Vi
ç(x) =
Vi-x2 =
1
Vi+x
and the interval is divided
'
in this manner we get, by using the
formulas
Sec.
Sec. III
Sec.
Type A
2
j
Sec,
I
2
II
Type A
I
2
2,
-.g
-1
Sec
I
o
indicated in each subdivision of the interval above,
the following result:
cp'S
.7071068
.7254763
.7453560
.7669649
.7905694
.1
.9
.8
.7
.6
Multipliers
250
416
24
224
31
2/T
.9272945
=
x
x2
fo
.6
.5
.4
.3
.2
1.2500000
1.1547005
1.0910894
1.0482848
1.0206207
7
32
12
32
7
.442146432
x
x2
.2
.1
0.
-.1
1.0206207
1.0050378
7
1.0050378
32
12
32
"06207
7
1,
J.
'_d...
-4
rT,
-
i
.40271068
-
.Z
I
'I
The fir3t two intervals are multiplied by 2 because
the function is symmetrical,
methods differs by .0000001.
The result found by these two
19
Find the value of the elliptic integral
xample II
ri
Kl
,wherek22
V(1-2)(i-kx2)
,Jo
From Die elliptischen Funktionen von Jacobi,
1,
Mime-Thompson, we have
K = 1.8540747
i
By formula, where f(x) =
2.
p(.x)
V(i2)(1-k2x)
'
and the interval is
-
sub-
y'(1+)(i-k4x)
divided so that from
.4,
O -
we will use Newton-
Cotes formula, 2, Section I, and frein .4-
will use formula
4,
1., we
Type A, Section II in reverse
ordinate is at the upper
order, since the infinite
limit,
f(:x)
Xj
o
i.
.1
.2
.3
.4
Multipliers
'7
1.0075518
1.0309826
1.0726983
1.1375393
32
12
32
77
ç
(
.4
.5
.6
.7
.8
.ç_:,
i.
.417340582
(X)
.8811343
.8728716
.8730378
.8826775
.9038769
.9405128
1.
=
4,764
36,936
-11,610
79,440
-23,895
93,096
46,494
27
X
i
.436737284
225,225
Total
1.854077866
20
The results found by these two metbods differ by
.000003 16.
Type
B.
('1
Example
I.
y1_x2
To find
clx,
J
where there is a
.1_1
vertical tangent at each end of the interval.
1.
By integration, we obtain
(il
X
i
\/1_x2dxVi_x+arcsjnxI
j-1
J-1
1.57079633
2.
By formula, where f(x) =
Vi-x2,
(x)
i/i-t-x
and the interval is divided in this manner
we get, by using the indicated formulas in each
subdivision of the interval the following result.
21
'ultipliers
p(x)
1.41421.356
1.
.9
.8
.7
70
1.37840488
1.34164079
1.30384048
1.26491106
.6
Product
864
552
1568
411
"-V
10,395
.22364840597
x 2
f(x)
.6
.5
.4
.3
.2
.80000000
.86602540
.91651514
.95393920
.97979590
7
32
12
32
7
4
=
.363088014
=
.397313088
f(x)
.97979590
.99498744
1.
.99498744
.97979590
.2
.1
0.
.1
.2
7
32
12
32
il
X
90
Total 1.57078593
The first two intervals are
multiplied by
2
because
function is syl!mletrical,
the
The results
of the two methods differ by .0000104.
Type
Example
1,
I
Find
1
Ç
_____________
V(1-x)(1-k2x)
1
, when k2 =
By Mime-Thompson's table, we get
fi
-
/(1_
)(1k2x)
2K =
3.7081494
22
By formula 3, Type C, Section II where
2,
i
= v(1_kdx2)
(p(x)
p1
-1.
-
.75
-
.5
-
25
0.
.25
.5
.75
i.
1.4142135
1.1795356
1.0690449
1. 01600 10
1.0000000
1.0160010
1.0690449
1.1795356
1.4142135
,
we obtain this result,
Multipliers
69
176
-126
336
-280
336
-126
176
69
Total = 3.70863184
These results differ by .00048244
By the comparison of the results from the two methods
we seo that the results are very similar,
The amount of work required in using those formulas
may be greatly simplified by using a calculating machine
and by tabulating the results in a manner similar to that
above,
i
APPENDIX
TABLE OF FORMFJLAS
Section
I
Newton-Cotes Fon-nulas
1,
Three-point Formula
(;
2,
+ 4Y1 +
Y).
Five-point Formula
'4h
f(x)ftx =
3,
(7y0
+ 32Y1 + 12Y
+ 32Y3 +
Seven-point Formula
r6h
6h
f(x)d.x =
840
(41Y
+ 216Y1 + 27Y
+ 272Y3 + 27Y4
+ 216Y5 +
41Y).
ii
Section II
Derived
Fornn.ilas
e(x), where 9(x) has a pole of the first
Type A, 1(x)
order,
1,
Four-point Formulas,
rh
f(x)dx
\
(356 q0 + 36O,
-
lOBcp
+ 22p3)
Jo
2h
f(x)dx
(244q,0
+ 360cp1 + l8q
+
Jo
-
r3h
f(x)dx = 2 V3h
(3q,
Jo
2,
+
+ 45q,1 +
Five-point Formulas,
f(x)dx
o
v-Ti-.
lO2Sq,
945
l252cp1
+
-
+ 238q,
- 43(p4),
('2h
Jo
-
/J?
-
('4h
Jo
(195q
315
jo
L
-39'
f(x)dx = 2 V'±.. (355q,0 +
945
f(x)d.x
2 1/4h
(250
+
56q,3
114)I
+ 3O6q,1 + 54q,
+ 4l6p1 + 24cp
+ 31q,4).
+ 84cp -
+
iii
31
S1-po1nt Formu.las,
('h
f(x)dx
I
j
10,395
(1O,932p
+ lS,48'7cp
-
+ 6,O48cp3
('2h
f(x)dx
(3,80'7p
2IB8C4 +
343p)
e
- l,409q,;
+ 6,9l4cp
1S395
1
+
C3h
f(x)dx _V3h
+
-
(2p82
' 36cp
+
3,465
- 4144
+
(2,694q0
+ 4,BS6p
+ l,b54cp
C4h
\
J0
f(x)cix
E'5h
J
f(x)dx
o
2079
+ 56p5)e
6lq
+
(972cp0
+ 3,0243
296p
1q395
+
+ 46O
+ 4Op
l,6BSp
+
+
4, Seven-point Formula,
(16h
'\
j
o
f(x)clx
2V6h
225,225
(46,494
+ 93,096cL1
+
7944Op
-ul6lOcp4
+
36,936
+
4,764p)
5
Eight-point
Forrriala
7h
f(x)dx -
f
(B'7,86?,656(p0
-
28,378,350
540'02'7,726p1
+ 1,5?l,254,44Ocp,
+
+
22'439'416'e
-
1,334,B?4,?B6cp
- 55,542,195cp.r),
f(x) = /è(x) where 9(x) has a zero of first order,
Type B1
1,
236O483l6O(4
Four-point Forniulas,
Çh
= 2
(869e + 2829
66p
139)
+
945
Jo
2h
2h'
(449
945
I
+ 4OBp' + 1869
-
893),
o
¡'3h
2hV3h
(2p0 + 9
=
2
+ 189
+
693)P
Five-point Formulas,
___
¡'h
f(x)clx
jo
-
(1,7159e + 6,9129
10395
+ 994Ç3 -
r
2h
f(x)dx = 2hV2h (4909e
10,395
- 2,öl4cp
17794)1
+ 4,4649i + 2,0829
J
-
112pm +
694)1
'3h
f(x)dx =
hV'
1155
(
(1059a + 7029
o
j
-
2794)!
+
1026
+
50493
V
I'
4h
8ht4h
10,395
f(x)dx
(70
+
3
+
864
1,568
+ 552
+ 4llq,4)i
Six-point Formulas,
f(x)dx
I
=
675,675
(103,436cp0
+ 489'475p1
J
- 243,80O
+ 145'°00Ç3
-Sl7OO(4
2h
i
j
f(x)d-x
=
4h/2h
(15,611
675,675
+
13h
=-
225,225
+ 8:O39p5),
+
146,65O
Z.
-
64,525w
+
-
(18,558
+
314p5)t
146,475
+
+ l8O,9OOp
- 14,8504
4h
8hJ
f(x)dx - 675,675
117,450
+ 1,917(p5
(5,422p0 + 5l,8OO(p
)
+ 44,6OOcp. +
+
3lO75q
-
93'200(pa
vi
(5h
f(x)dx
4,,
(2,o36
27,027
o
+ 11,725
+
21400Ç
+ l24OOcp5
+
34lOO(4
+
8429Q5)I
Seven-point Formula,
C6h
12hvi
f(x)dx
J
+ lO,728p
225,225
0
+
+ 2376Ocp3 +
+
Type C, 1(x)
v'G(x), where 9(x)
+
5,562q).
has two poles of the first
order,
Three-point Formula,
1,
fil
=
1
J
2,
Five-poInt
cl
((-i)
(x)
Forirrila
dx =
6
5,
+ 2tp(0) +
i/l-x
{(-l) + 2(-.5) + 2(.5)
+
Nine-point Formula
il
p(x)dx
\
j-1
Vi-x2
L_ [69-1
630
+ l?6p(-.?ö)
28O(0)
l26(-.5)
+
336p(-25)
-
l26p(.5) + l'76(.75) + 69cp(l)].
+ 336cp(p25)
BIBLI OGRAPIfY
Bennett, Albert A, "The Interpolational Polynomial",
Bulletin of National Research Council, No, 92,
pp, 11-51, 1933,
Encykiopadie Der Mathematischen Wissenschaften, Band II,
3; Heft 2, Leipsiz and Berlin: B.G, Teubner,
Fraser, D,C,, Newton's InterpolatIon Formulas, 2nd Edition,
London: Layton, 1927,
Mime-Thompson, L,M, The Calculus of Finite Differences,
p.
176,
London: MacMillan and Co,, 1933,
Mime, W,E,, Step-by-Step Methods of Integration, Bulletin
of National Researeh Council, No1 92, pp, 71-87,
1933.