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8-1-1946
Interpolation by means of finite calculus
Otis White Jr
Atlanta University
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White, Otis Jr, "Interpolation by means of finite calculus" (1946). ETD Collection for AUC Robert W. Woodruff Library. Paper 926.
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INTERPOLATION BY MEANS OF FINITE CALCULUS
A THESIS
SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY
IN PARTIAL FULFILLNENT 0]? THE BEQUIR~NTS FOR
THE DEGREE OF MASTER OF SCIENCE
BY
OTIS WHITE, JR
DEPARTMENT OF MATHEMATICS
ATLANTA, GEORGIA
AUGUST 1946
ii
AC~U~TOWT~EDGE~3NT
The writer is greatly indebted to Mr. Charles H. Pugh
who was his instructor in finite differences, and whose
treatment of interpolation aroused the writer’s interest
in further study. His introduction to the writer of the
lozenge-diagram method for the derivation of interpolation
formulas was extremely heliful in the development of this
work.
The writer is also very grateful to Dr. Joseph A. Pierce
who steered this investigation, and whose many suggestions
the writer was glad to incorporate.
iii
TABLE OF CONTENTS
Chapter
I.
II.
III •
Page
INTRODUCTION
.
.
.
.
.
.
.
V.
.
.
.
3
3
THE ORDINARY DIFFERENCE TABLE
1 • D e fi ni t i on s . • • . . . . • . . . .
2. The Difference Table . a • • • • . . .
3. The Sum of a Difference Column . •
4. The Relation between the OperatorL~and E
5. Difference Q~uotients . . • . . . .
6. Central-Difference Notations . . •
THE DEVELOPMENT OF GREGORY7N~WTON FORMULA
7. The Difference of x~P~ • • • • •
8. The Expansion of a Polynomial in
Factorials . . • . . . .
.
a
9. The Gregory-Newton Formula of
Interpolation . . . . . .
lO.TheRexnalnderTerm. .•. . .
.
IV.
.
.
StJ~fi1~LAI~Y
B IBLIOGRAPHY
•
•
•
•
•
5
7
9
11
•
.
.
•
.
.
13
13
•
.
.
14
•
.
.
i8
20
20
DIVIDED DIFFERENCE FORMULAS . . . a • .
•
•
11. The Divided Difference Table . •
.
a
12. Newton’s Interpolation Formula for
Non-Equidistant Arguments . .
13. The Newton-Gauss Interpolation Formula
as a Special Case of Newton’s Formula
14. Lagrange Formula o±’ Interpolation
THE LOZENGE DIAGRAM
15. Definitions . . . . . . . . . . • .
16. The Extended Lozenge-Diagram . .
17. Derivation of Interpolation Formulas
by Repeated Summation of Parts .
4
23
27
29
•
•
32
32
•
38
.
46
•
34
47
iv
LIST O~ TAPLES
Table
Page
1. The Difference Table
.
.
.
.
2. The Central Difference Table
3.
The Divided Difference Table.
.
.
.
.
.
..
.
.
•
4
11
.
.
.
.
.
.
.
22
CHAPTER I
INTRODUCTION
For a number of years man has been seeldng an objective
method for determining the value of certain mathematical
events when given occurrences with which to work.
The
calDulus of observations or finite differences, as it is us
ually called, gives to man a~ method for actually reading be
tween the lines of general mathematical tables and deriving
by an objective means, the values of particular data. Analo
gous to sampling in mathematical statistics, finite differ
ences seeks to define quantitatively a correlation between
the general and the particular.
The general function may be
derived from a given table of values.
The general term from
which the particular terms of a series come may be obtained
by means of formulas in finite differences.
In this thesis, it shall be the aim of the writer to
point out, bQth illustratively and theoretically some of the
uses of finite differences in relation to tire predicting of
mathematical events.
The writer has therefore found it de
sirable to consider sortie specific topics.
Considerable space
is given to the treatment of the difference table, when the
arguments differ by a constant and when the arguments differ
by a variable quantity.
Although, in certain illustrations trigonometric and
logarithmic examples are used, the writer would like to state
].
2
at ~he out—set that this work is devoted primarily to poly
nomials.
When given data that have been collected by means
of extensive observations and that can be characterized gen—
erai].y by a polynomial, it shall be the aim of this work to
develop some methods for determining that polynomial.
In
this paper the writer proposes to obtain certain formulas
based on successive differences of the function which will en
able one to arrive at the desired polynomial.
The foundation upon which the formulas of this thesis
rest is Newton’s general formula for unequal interval~.
The
Gregory, Gauss, and Sterling formulas were derived by Newton,
hence the writer will attach Newton’s name to each of these
formulas.
Various methods will be used to obtain these form
ulas, including the lozenge-diagr~u method as modified by
D.C. Fraser.
After having obtained interpolative methods and formulas,
the writer will use these methods to do actual interpolation.
In the practical illustration of interpolation that are demon
strated in this thesis the following problems will be solved:
(1) given values of a function for a finite set of arguments,
to determine the value of the function for some intermediate
argument; and (2) given a finite series of quantities sub
ject to a determinate law magnitude, to determine the rational
integral function which will give the law of combination up
on which the series of quantities depend.
CHAP’T’~R II
THE ORDINARY DIFFERENCE TABLE
1.
DefiflitiOns. - c~ne of the most important of all opera
tors that are to be defined in this work is the operatorA.
Consider a function f(x) whose values are given for arguments
x , x
x
..., x of the variable x. These arguments, let
1
2
3
n
us say, differ one from another by a constant w.
The first difference of f(x
1
)
is denoted byi~f(x
1
)
and is
defined by the relations
Af(x )f(x )—f(x ).
1
2
1
In the same manner we define the first difference of f(x ) by
2
j.~f(x )~f(x )-f(x ),
2
3
2
and so on.
2
The second differenc,e of f(x ) is denoted by
f(x ) and
1
1
is defined by the relation
2
~f(x)
-z~f(x )-~f(x
2
1
Proceeding in this manner we can form the nth difference of
f(x ).
1
Thus
n
£Sf(x
1
)=
n-i
n-i
j~ f(x )- L~. f(x ).
2
1
Illustration i~ — Find the first difference of the log f(x~;
By definition, we have
i~1og f(x) ~log f(x~1)- log f(x)
f(x*l)
(1)
~log
—
f(x)
3
4
Adding and subtracting f(x) in the numerator of (i), we obtain
~f(x)
~log f(x)
log 1+
-
f(x).
f(x)
Illustration 2.- Find the first difference of
g(x)
By definition
ff(x)~ f(x ~-1)
f(x)
g(x)
g(x).f(x-I-l)
-
g(x+-l).f(x)
g(x+l).g(x)
Adding and subtracting g(x).f(x) in the numerato~,we have
gf(x)~ g(x) [f(x-i-l) - f(x)j_ {f(x) [g(x±i)
~g(x))
g(x-i-i).g(x)
ff(x)’~ g(x)~\f(x)
or~1
\g(x)1
2.
-
-
f(x)~~g(x)
g(x~l).g(x)
The Difference Table.— If y is given by the function
f(a-~-xw) for valuesof the arguments a, a~-w, a-~-2w, ai—3w,
a-*4w, and a~-5w, then we may
exhibit the following difference
table, based on the results of section 1;
Argument.
Entry.
a
f(a)
a+-w
f(a-t’w)
a-f-2w
f(a-f-2w)
a-f-3w
1~f(a)
E,f(a+w)
&(a)
2.
f(a+3w)
~f(a+2w)
I~f(a-fr2w)
,~f(a+w)
a-fr4w
f(a-t-4w)
z~f(à+3w)
~f(a-f-2w)
a-t-5w
f(a~5w)
E~f(a+3w)
IS.f(a+4w)
TABV~ 1
3
~f(a-l-w)
5
and similar for difference of order higher than the third.
The first entry f(a) is called the leading term and the dif—
2
ference~of f(a), that is to say4f(a),.L~ f(a), . . . are
called the leading differences.
3.
The Sum of a Difference Column.:~... By close observatioT’
of the difference table in section 2 it is almost obvious
that the sum of any difference column is equal to the dif
ference between the first and last term of the preceding dif
ference column.
The truth of this statement may be demon
strated by showing that the sum of any one of the columns in
the above difference table is equal to the difference be
tween the first and last term of the column preceding it.
Let us set the difference of the first and last term of the
first differences equal to the sum of the second differences,
and prove that the right-hand side of our resulting equation
is exactly equal to the left-hand side.
Hence we have
2
2
2
2
L\f(a4-4w)_/\f(a)F~f(a)*Af(a÷w)+ ~f(a÷2w)+ L~f(a÷3w).
Substituting in the above equation the values of each of
the terms on the right, we have
~f(a+4w)_~f(a)~ Af(a÷w)—/~f(a)÷/\f(a+2w)-/\f(a+w)
-f~f(a÷3w)-Af(a÷2w)1- ~f(a+4w)- ~f(a~-3w~.
Collectin~ terms, we obtain
~\f(a)~/~ f(ai-4w)-~f(a).
The above fact affords a numerical check on the accuracy
of the difference table.
It will ~e demonetratea in the following illustrations
that in many cases of tabular functions the differences of a
certain order are all zero; or, to be more accurately, they
are smaller than one unit in the last decimial place retained
in the tables in question.
This fact lies at the bases of
the finite differences’ method 6f inter~olatiofl.
Illustration 1.
.-
The following example is a difference
table~ which represents the log tangent of angles from
260 10’ 0” to 26 11’ 30” inclusive at intervals of iouu/~;i~7
Logtat7-e26° 10’ 0”
-
~
~98083
10”
434 054 05228
531 919 6844
20”
487 246 02072
531 879 8870
30”
540 434 00942
~
40”
593 6i8 09147
531 800 3250 -397646
50”
646 798 05197
5
840 1005
760 56ó4
9.691 699 974 10801
531 720 8069 -397428
10”
753 146 18870
531 681 0641 -397316
20”
906 314 29511
531 641 3325
30”
859 478 42836
26°ll’ 0”
~
9.691 380 858 10301
109
109
110
111
107
112
It will be, seen that in this case the third differences
are practically constant when quantities beyond the four
teenth place ~re neglected: any departure from constancy in
the last place being really due to the neglect of the fif
teenth place of decimals in the original entries.
Illustration 2.- By means of the following difference
table find the sum of the first difference column.
7
Argument.
Entry.
0
2
3
a
2
14
3
29
4
50
5
6
77
6
9
6
6
6
21
27
0
0
0
6
33
110
0
Above we have the difference table for the given entries.
In order to find the sum of the first difference column, we
have
~~y=3÷9÷l5~2l~27÷
33
~l16— 2
=108
4.
The Relation Between the Operators~and E.- The
operator
(1)
when acting upon f(a+xw), is by definition
f(ai-xW)~f(a+~jw)
-
f(a-i--xw).
Suppose we let w represent the interval between
sue—
cessive values of the argument of the function f(a), then
we nay define-E as the operation of increasing the argument
by w, hence
B f(a)~ f(a÷v,)
,
or in general
x
B
f(a)~ f(a-t-xw).
8
To show the relation between
A
and E we may write the
right-hand side of equation (1) in terms of E, thus
xl-l
-
x
~f(a+xw)=E
(El)
or
f(a)-E
EX
f(a)
f()
4f(a+xw)~ (E-1) f(a-t--xw).
It is therefore evident that the operators 1~ andL~are con
nected by the relation ~\E-l or E~z~+l.
The operators~and E obey the ordinary laws of Algebra,
i.e. the distributive, associative, commutative, and law of
exponents holds.
5.
Expression for f(a-i--xw).- From the relation that ex
ists between ~ and E, we now may express the general entry,
f(a+xw) in ternis of items coming from the difference table.
Since, as we have stated, ~ and E behave like algebraic syin
x
bole, we rn~ç,r write t~ f(a-~I.-xw)= (t~+1)
Expanding
(2)
(1\~1)X
f(a).
by the binomial th~Orefl1,we get
f(a-~.-xw)=. f(a)+-x/.~f(a)*
x(x—1)
2
23
n3
Illustration.
—
Given the arguments
n
X
~f(a)-f-... J~f(a).
—3,
and the corresponding entries 16, 17, 4, 1,
general expression, f(a+xw).
Forming the difference table, we have
—2, —1, 0, 1,
-8.
Find the
9
Argument.
Entry.
.3
16
7
—2
-1
-9
4
0
6
-.3
—3
1
-6
0
0
—6
6
-9
1
Substituting in equation (2), we obtain
x((-l)
f(a-t-xw)~l-3x+
x(x-1)(x-.2)
(0)-i-
(—6)
2!
32
~l3X.+0*(x_3x
-
2x)(—1)
5x*l.
Therefore, the general expression which gives each entry when
the correspondent argument is given y=x3f3x2-5x-f-l.
6.
Difference Q~.zotients.
—
Mime-Thompson [2; 23] states
that although not as practical as an interpolation device as
the ordinary differences~ of which were explained in preceding
sections, the difference quotients do present a closer analogy
between finite an infinitesimal calculus.
Norlund’s operator
IS,,
We now introduce
which is defined by the relation
1~f(x)~ f(xt~-f(x)
We ca1l~f(x) the first difference quotient of f(x).
This symbol has the advantage that
lim-:f(x)D f(x)
W40
Where D denotes the operator of differentiation, ifl ~h.j:~ caee~.
10
The operation can be repeated, thus
2
,/lf(x-j-W)- Af(x)
~f(x)~t~ J~f(x)j~
w
wLw J
w
f(x-~-2w)- 2f(xI-W) +f(x)
The operator
ference operator
when :~1, becomes our ordinary dif
LI.
Illustration.
-
Calculate the difference quotients of a
Getting the first difference quotient, we have
x+w.x
a
—a
x
~a
Thus
=
—
a
~ x
xiaW_l~fl,
~sa~a
w
~ W/
1-
(1)
x
~
w
a-i
writing
)
1
W
a(i~bw)
wehave
x
x
Wn
(l-f--bw)
b (i+’bw)
(2)
Since
X
wbx
urn (i*bw) ~e
W40
We have as a limiting case of (2)
n
D
e
bx
n bx
~b e
Thus, in fin~.te calculus (l-g--w)
x
x
wpiays the part of e
.•
1).
7.
~ ...The notation of cen
tral differences is extremely useful when interpolation
formulas invo].Ting the differences on.a line horizontal with
a particular entry are needed.
If we introduce the operator.~~;3qdefiflOd by
2n
U—fl,
2n+l
2n4-l
U
U
—
.
k4-f
The difference table in section 2 becomes:
TABL1~ 2
T}11i~ CT~TRAIi DIFFERENCE PABI~E
Argument.
Entry.
a-2w
u
-2
a-w
j~U~L
i
U
-l
uo
a
a÷w
U
a+2w
u
a+.3w
u
A
,~u-l
~Uj
~
U~,
0
L
1
~
2
~u
~-
~.
~u
1
3
Aui
U
‘~
~u,
~Q~U
4 0
~
u1
1
2
3
The operator~is the central difference operator and
the differences in the above table are known as central
differences.
This table and the table used in section 2
differ only in notations. It will be seen that/.au~u-u
2
U ~ ~ and so on.
If carefully observed,
0
t
,
it will be noted also th~.t the difference~of any horizontal
12
line with u
are labelled with the suffix k +-~.. Therefore
k
the interpolation formulas based on central difference take
the differences used, exactly or nearly so, from a single
horizontal line.
The arithmetic mean of successive differ
ences in the same vertical column is denoted by
A
and is
labelled with the arithmetic mean of the suffixes of the en
tries from which this expression arises.
Thus
~ (z~ u4-
4~\
u)y.& ~ ~o’
~
2 u~+
~i
When these are entered in the difference table the lines
a, a+w, and the line between, will have the following ap
pearance:
a
a+ w
whereyu
u
A2
j-~A u0
u
A
Au1
denotes
.~
(u
u
01
+
u0
u1
u1
.
.
.
.
.
.
.
.
CHA~P~’1~R i~:i
TitJi~ DEVELOP~NT c~.f GREGORY-NEw~PoN FORMULA
8.
‘Phe Differences of x
(p)
.-
The expression x
(p)
is
read x upper p and is defined as
(p)
(i) x
x(x—l)(x—2)...(x—p÷1)
In reality (1) is a polynomial of degree p, expressed
in terms of factorials.
If we suppose the interval of the
arguments in the difference table be unity, we have
a
a(a-l)(a-2)...(a_p~2)(a_p~1), and
(p)
(a-t-l)
~‘(a$1)a(a—l)...(a—p43)(a..p÷2).
By definition
(p.)
(p)
(2) j~,a
(a+1)
or~a
(p)
-
a
(p)
(a4-].) a(a—l)...(a—p÷3)(a_p+2) —a(a—l)..(a—p-g-l).
Factoring, we get
i~a
(p)
a(a—l).~.(a-p 2)(a+l—a4p-l),
(p)
(p-i)
or~a
px
so that
(p)
(3)Ax
(p—i)
~px
Equation (3) is analogous to the formula of differential
p
p-i
calculus d/dx (x )px
13
14
9.
The Expansion of a Polynomial in B~aotoria1s.- Since
this thesis is devoted chiefly to the consideration of data
whoae general function can be’ expressed in term of a rational
integral function one can easily see from section 6 that if
we have a factorial method for expressing a polynomial the
difference can be calculated readily.
Let P f(x) denote a polynomial in x of degree m.
We may
write ~f(x)~r1+(x~.n+m) Pmif(x)~ where r1 is the remainder
and
the quotient when
is of degree rn-i.
is devided by (x-n÷m), so P1
By repeated application of this trans..
formation, we obtain an expression for a polynomial of the
nth degree in terms of factorials:
P.f(x)~ r-~x
n
1
1
(1)
2
x
=r+r x
1 2
.
-
where r., r
r
.
n
,
.
=.r .~.r x
1
2
,
p
(x)
n-1
(i)
(1)
÷r
-i-r
.
3•
3
x
x
(2)
(2)
÷r
.
(1)
-~-
r x
3
P
4
x
.
(2)
-~-
(x)
n—2
3
P
n-3
(x)
.
r x
4
(3)
-~-
...
-~x
(n)
P (x),
o
...,are constants and P (x) is some constant
2
3
We thus obtain p (x) expressed in a factorial form.
n
5 4
3
2
Illustration... Express y~x~3x ÷4x~-2x÷x+l in terms of
15
factorials and find the successive differences of the facto
rial expression of y,
Using detached coefficients when dividing by x-1, x-2,...1
weget
1
1
2
1
3
1
4
1
3
4
2
1
4
8
10
4
4
10
11
2
12
40
6.
20
50
-
47
and for the factorial form of y, we have
y~x~5~ ~ ~ ~~(2)~ llx*l.
The suocessive differences are given by
y=~5x
2
(4)
+-52x
(3)
+ 156x
5
4 ‘282x 1 100
312xf 282
4
~ 120x 43l2
~ 120
(2)
~100x+i.
(2)
(~)
3
(2)
~y.~6ox 4-
+-141x
16
10.
The Gregory-Newton Formula of Inter?o1ation.~ The~
formula that we shall develop in this section is very valuable
in determining the polynomial f(a-’-xw) when its values are
given for the arguments a, aw, ai-2w,
i.e. its values are
...,
given on equidistant intervals, ~This, formula is also useful
in computing the values of the function between two arguments,
say al-~iw~ and a+iw.
Let us first express f(a+xw) in the factorial form, hence
(1)
f(a~4-xw)=A~AxtAx
01
2
(2)
(3)
-I- Ax
3
(4)
(fl)
-i-Ax .j— •..1-.Ax
4~
n.
Taking the successive differences of (1) by applying the
operation denoted by equation
(3)
7,
section
we obtain
(2)
(3)
f(a-I-xw)~A÷2Ax-i-3Ax+4Ax
12
3
4
(2)
(n—i)
• •+flAflX
(3)
2
L~f(a+xw)
(4)
3
A
(2)
2A -1- 6A xfl2A x
2
3
4
f(a-f-xw)= 6A -J--24A
3
4
•
-~-
4-
(~i-2I
...+n(n-1)~x
...+n(n-l)(n-2)A x
n
•
(n—3)
17
n
L~f(a+~xw)= n~ A
n
The values of the coefficients A
,
A , A ,
0
1.
2
putting x~ 0 in each of the equations (2),
...,
(3),
A~are found by
(4),
...
so
that
2.
A
=
f(a), A
0
L~Sf(a), A
1
Af(a)
~——
2
,
3
2!
3~
Af(a)
A
~
n
ii!
Equation (1) now becomes
x(x—i)
f(a÷xw)~f(a)÷ xZ~f(a).~
2
—~
2’
n
x~.1)(x-2).
. .
(x—n÷’1)
~4
f(a)
4If the abo4e equation is observed closely it is readily
recognized ~as the equation obtained by expanding (~+~l)X
multiplying each term by f(a) in section
5
and
and is known as the
Gregory-Newton interpolation formula.
Illustration.- Using the difference. table in illustration
1, section
3~
find the entry for x ~1.0l, and also find the
general entry f(a+xw).
Writing a~l, w=l, x~.O1 and. substituting in the Gregory~
Newton formula, we get
f(l.0l)~5+9 ~(.0i)+-
6(.ol—l).ol
=5~..09 + .0297
~5. 1197
2!
18
The general entry f(a i-xw) is found by replacing the dif-.
ferencea of f(a) in the Gregory-Newton formula by the success
ive differences of f(o).
Thus
x(x-1)6
f(a+xw)~ 2-t-3x÷-—--------2!
2
-2±3x-i-3x -3x
2
=3x+2
From our, illustration in sections 3 and. 4 it is readily
noticeable that the nth differences inadifference scheme of
which the origin is a polynomial of the nth degree are equal,
whereas the (n+l)th differences are all zero.
At .thiä point
another similarity between finite and infinitesimal calculus
may be pointed out.
The nth derivative of a polynomial of
degree n is a constant whole the(n+l)th derivative is zero.
11.
The Remainder Term.
—
It is stated in Milne-Thomson
[2;61] that the process of interpolation applied to the values
in a given table cannot give an accuracy greater than that of
the values in the table, which are themselves usually approxi
mations, unless, their general function- can.~e expressed in
terms of a polynomial.
In attempting--to---a-t-ta-i-n---t-he--u-tmo-s-t--------
accuracy which the table permits, when the Gregory-Newton
formula of interpolation is used it is common practice to omit
from the interpolation formula t~he first term which ceases to
influence the result obtained.
The question thenarises as to
how far the result so obtained represents the desired approxl
19
For a rational and integral function we have observed in
previous discussions that the nth differences are all constant,
hence the Gregory-Newton formula in section 10 will give us
accurate interpolation.
The problem arises when we have data
whose general function is either trigonometric, logarithmic, or
expnential.
In this eventit is necessary for us to consider
the remainder term for the Gregory-Newton formula.
Let us write the Gregory-Newton formula of section 10 in
the following manner;
(1)
f(a .~-xw)
x(x—1)
f(a)÷ xI~f(a)
.2
-t-----
2~
÷x~x-.1)...Lx-n.~2)
n-it
n-i
~ f(a)4R (x),
fl
where
x(x—1)...(x—n41)
(2)
R(x)=
n
and where f
(~)
n
f
—
n
f (a~e.vw)
(~)a
-
.
The argument~lies somewhere in the interval bounded by
the greatest and least of x, a, a+nw.
Formula(1) is the Gregory-Newton interpolation formula
for forward differences.
The differences employed with this
formula are the same as those of section 10, i.e. they lie on
a line sloping downwards from f(a).
Providing the remainder
term R (x) can be calculated, we can derive f(x) in terms of
n
f(a) regardless of whether we have observed data whose general
function is a polynomial.
The illustrations of section 10
are examples of polynomial interpolation, hence the remainder
term is non-existent.
CHAPTER 1V’
DIVIDED DIF~RENCES FOBMIJLAS
10.
The Divided Difference Table.- Previous to this
chapter we have assumed that the arguments from which our
difference table was computed differed from one another by
a constant quantity;
but quite frequently it is not possible
to complete a difference table with arguments of this nature.
Statisticians, biometricians and others who deal scientifical
ly with m~athematical tables often collect data where the
arguments differ by variable quantities.
For example, when
astronomical observations are disturbed by clouds there are
gaps in the records.
Let us consider a function f(x) whose values are given
for the values a
a
,
equal.
—
a
,
a
1
,
—
2
.
.,
.
a
of the variable x.
The
n
a . . ., a. —a
need not be
1
0
2
1
3-2
nn—l
In place of the ordinary, differences we now introduce
intervals a
0
a
,
a
,
a
-
what are known as divided differences.
The divided differences of the first order for the argu
ments a
,
a
0
,
is denoted by Fa a )and is defined by the re
Lou
1
lation
f(a
0
)
-
f(a
3.
)
f(a.
1
)
-
f(a
0.
Eaa~
Loll
ElO
a—a
01
a—a
10
In the like manner we define the divided difference of the
first order for the arguments a
,
2
2~
a
by
1
21
f(a
f~
1-2
)
)
f(a
-
a~_
2
1j~a
2-
a
1
1
and so on.
The divided difference of the second order for the argu
ments a, a, a
is denoted by
relation
a
aJ and is defined by the
raa]-i~a
11 Li 0
1-2
[a
.L
2
a
1
al
Oj a
—
a
2-
0
The divided difference the (n÷ i)th order may be formed in the
same way, since the order of ~a divided difference is less by
unity than the number of arguments required for definition.
Thus
Ia a
.
1-n n-i
-r
•
.aj
oi~
n
~
n-i
a
0
]
I
a
—
a.
0
The results of the above definitions, using powers of the
letter D to denote the order of the divided difference column,
may be e~~~sëd in the following scheme:
22
TABLE 3
T}33~ .DIVTDED DIFFEBBNCE TABI~
1
Argument.
a
Entry.
•f(a)
o
0
a
f(a)
2.
2.
D
Faa
Lb
f’aal
1211
a
f(a)
2
2
a
3
2
D
D
laaa
‘L210
faaaa
L32l0
,
faaa
L321
faal
[32j
.
f(a)
3
3,
.
.
•
a
a.
faa
1
[nn~11
•
a
)aa
jnn-ln—2
f(a)
n
n
Illustration.- Compute the divided difference table for
the values 1342, 2210, 27~8, 58~0, 6878, 9282 given by the
arguments 11, 13, 14, 18, 19, 21.
Using the divided difference scheme, we have
Argument
Entry
11
1342
13
2210
D
434
14
2758
-
18
5850
19
6878
21
9282
548
773
D
2
D
38
45
i
1
51
1028
1202
58
0
1
4
23
13.
‘sinterpolation ~ormu1a for Non-Equidistant
Arguments /i;2~.. When the intervals of the arguments are un
to
equal, it is necessary
have a formula other than th~ Gregory..
Newton formula ~for determining unknown polynomials or for de
termining entries for arguments between two given arguments.
The basis of all interpolation formulas is the formula to be
developed in this section.
Writing 5~ for a
we have by
0
definition
ía
a a ...a
12
a .~a1 J~a ...a
Li
2
n~
1
_~
-
in4
n
[a
L1
1
n-i
...a
n
a ...a
.1 J~ea ...a
2
n-].~
1
n—2
________
_________
n—lI
3C—a
n-i
ra
~-i
...a
1
a ...a
I J~ca
2
n—2~
1
n—3
-
n-2
n-2
f~a..,a
‘1
n_3.L
...a
(a al ~i~a
1 2~ L i
a7~~~—
1 21
~-a2 it-a2
~
(~a
-
n-2
j~a...a
Li
n—4
~
n-2
1-
n-i
—~
n-2
24
1
L~
f(a )
1
X-a
f(~)
____
1
1
By repeatedly substituting for the second member of the
right of each identity its value as given by the succeeding
identity, we have
f~a ...al
t-12
nJ
Fxa
L1
~ia ...a
L12
n-i
a ...a
2
n
x- a
n
(s-a )(~-a
n
n—i
~ä, a ...a
112
n-2
-
(s-a )(5~-a
)(~-a
n
n-i
n-2
Faa
[12
(~-L~ ) (X-a
) ...(~-a
n
n-i
2
f(a~)
(~—a )... (X—a )(x—a )
2
1
÷
(A)
(x-a )...
n
(~—a ),
1
f(~)~ f(a) ~ (i-a) faa) ~(~-a)(~-a) [aaaJ~
25
4-(~-a )(~-a )(ic-a )f a a a a
1
2
3
~ ...~-(5c—a )...(i~—a
1
7
)fa a ...a
n_1L-12
1-(x—a )(x—a )...(~—a )j~a a •..a 7.
1
2
n
12
n-~
This is Newton’s general interpolation formula with the
remainder term
(i)
R (~)z (i—a )(x-a )...(i~—a )~a. a •..a
1
2
n
12
n
]
The formula is a pure identity and is therefore true without any
restriction on the form of f(~).
For data whose general function is rational and integral
we know that the nth differences are constant and the (n-1)th
differences vanish.
Hence for polynomial interpolation
~ for unequal intervals becomes
(B)
f(i)
f(a
1
)
+
(5c-a )(~~a
(i-a )fa a
1
12
1
•.+(~—a )(~—a )...(~—a
1
2
n-i
)
2
) fa a a
123
~{a a •..a 7
12
nJ
Illustration 1.- From the divided difference table given
8,
in the illustration in e•ction
compute the value of the
polynomial f(x) using Newton’s Interpol~ion:~formu1a.
From the given illustration, we have
f(a
0
)
1342, [a a
01
434 ~ 1a a a
1-012
]
38
and fa a a a
0123
1.
Substituting these values in Newton’s formula, we obtain
26
(38)
f(x) =1342-+(x-l1) 434-,-(x—11)(x-13)
(x—ll)(x—13)(x_14)
-~
13424-434x-4774
3
-
912x
2
38x -t-479x-2002.
Collecting terms, we get the desired po1y~iomial
3
‘c.
Illustration 2.- Calculate f(19) using the following
divided difference table:
x
f(x)
D
11
14646
17
83526
D
2
1)
3
D
4
11480
21
194486
23
279846
31
923526
1626
27740
42680
2490
3778
72
92
80460
Applying Newton’s formula for unequal intervals
f(l9)
=
14646 ~(19-ii) 11480 ~ (19-11) (19-17) 1626
-t(19—11) (19—17) (19—21) (19—23)
146464- 91840 ~-260l6 .128-2304
or
f(19)
14.
Newton
~3
130326.
The Gregory-Newton Formula as a Special aase of
2ormula.
-
The Gregory-Newton formula may be regarded
as the special case of the formula of the last section when
the intervals of the argument are equal.
27
For in Newton’s formula for unequal intervals suppose
that we put
a=a,
0
a=ai-w,
1
a=.a~-2w,
2
...
,~=a-~-xw.
By constructing a table of divided differences, we see
that
1
ía a
LOll
,jf(a),
w~’
~—
1
ía a 1~—~f(a w)
t121 w
1
a1=_—~
Lol 2~
2~w
2
J~a
,~
‘—~
In the same way we find
3
1
ra a a
L 0 1 3J
3’w
afld so on.
if we now replace x by a +xw, the formula for unequal in
tervals of the argument becomes
2
x(x-l)
x(x-1)(x—2) 3.
3’
2’
which i~ the Gregory-Newton formula.
15.
The Newton-Gauss Interpolation Formula as a Special
Case of Newton’s Formula.- The differences used in NewtonGauss Formula are as nearly as possible taken from the hori
zontal line through u
in the central difference table of
0
chapter II, section
6.
u
in the central difference table
0
corresponds to f(a) in the ordinary difference table,
pose f(ai-xw) is given by the values
of its arguments.
...,
Sup
a-w, a,wa*w, ai-2w,...
28
If in Newton’s Formula we take a ~a, a ~a+ w, a ~-a-w,
1
2
3
aj.2w, a =a-2w and so on, and denote a i-xw by ~, we obtain
a
5
4
(1)
wJi-(x-a)(x-a_w)fa
f(a-r-xw)= f(a)÷ (x-a) fa ag-
a÷wa-wJ
~—(x-a) ~x—a-w) (x—a ~w)La a+ wa—wa ~-2w3
+(x-a) (x-a-w) (x-aw) [a a÷wa-wat2wa-2w~
4-.
The divided differences contained in this equation may be
written in the ordinary notation of differences as follows:
2.
fa a÷wjz—~f(a),
1
2
f(a-w),
2!w
13
a ÷ wa-wa
1-
2w3
f (a..w),
-~
and so on.
Hence equation (1) takes the form
x(x-1)
2
(x*l)x(x-1)
—
2
A
f(a-w)
3
~ f(a-w)
3!
(x
+
1)x(x—1)(x—2)
4
~ f(a-2w)
4!
(x+2)(x÷1)x(x-1)(x-2)
5
A
f(a—2w)
29
This formula is the Newton—Gauss Interpolation Formula.
16.
Lagrange’s Formula of Tnterpo1ation~- If we are
given n values of a function which are not consecutive and
equidistant, we are able to find the value of the function
for any argument by means of the formula of Lagrange
Let f(a), f(b), f(c),
1;38
f(a) be the given values
...,
corresponding to the arguments a, b, o,
...,
n respectively
and let it be required to determine an appropriate general
expression for f(x), where f(x) is some polynomial, i.e. f(x)
is an integral rational function.
Let us assume
(1)
2
n-i
f(x)=.A÷Bx÷Cx~ ...~Ex
and let us determine A,B, C,
...,
by the linear system of
equations formed by making ~a, b, c,
...,
n in succession.
We may express equation (1) in this equivalent form
(2)
f(x)
A(x—b)(x—c)...(x—n)
4-B(x—a) (x—c)..
‘-
C(x.-a) (x—b).
1~~
•
•
•
•
.
.
(x—n)
. .
tx—n)
.
.
S
S
to n terms, each of the n terms in the right-hand member
lacking one of the factors x-a, x-b,
multiplied by an arbitrary constant.
...,
x-n, and each being
Our above assumption is
correct because equation (2) is equivalent to equation (1) in
that it is rational and integral, and contains n undetermined
coefficients.
Making x~ a, we have
f(a)
A(a—b)(a—c)...(a—n)
30
hence
f(a)
(a—b) (a—c).
.
(a—n)
.
In like manner making x-b, we have
f(b)
13=
(b-a)(b-c).
and so on.
.
(b-n)
.
Hence finally,
f(x)
(3)
f(a)
(x-a)(x-b).
.
.
(x—n)
(x—a)(a-b).
(a-n)
.
.
f( b)
~(x-b)(b-a’).
.
.
(b-n)~
f(n)
+
V
(x—n)(n—a)(n—c,.
the required expression.
If ~e may multiply both sides of equation
by (x-a)
(x-b)
.
(4’)
f(x)
f(a)
.
.
(3)
(x-n), we obtain
(x—b)(x—c).
.
.
(x—n)
(a—b)(a—c).
.
.
(a—n)
(x—a)(x—b).
.
.
(x—n)
(b-a)(b-c).
.
.
(b-n)
÷f(b)
(x-a)(x-b)(x-c)
.
.
(n—a)(n—b)(n—c)
.
.
-i-f(n)
V
Equation
(3)
and (4) are known as the interpolation
formulas of Lagrange.
3].
Illustration.— Assume f(x) to be some rational and in
tegral function of zc, find the value of f(a) by means of the
Lagrange interpolation formula from the values
x
5
7
11
13
17
f(x)
150
392
1452
2366
5202
(3),
Substituting in the Lagrange ~‘ormu1a
f(9)
we obtain
150
(9—5)(5—7)(5—n)(5—13)(5—17)
(9—5)(9—7)(9—11)(9-l3)(9—17Y
392
~ (9-7) (7-5) (7-n) (7-13) (7..j~
1452
~(9—ll) (11—5)
(n~7) (11—13) (1i.a7)
?36 6
-I.
(9-l3)(13~5)(13—7)(13~11) (13—17)
5203
~(9—17)(i7—5)(17—7)(l7—i1) (17—13)
f(9)
150
=
-512
or
392
-
4608
1452
—
960
2366
~1-
576
f(9)~ -512( .0302—14205—2.5207
~—512(—1.
6965)
868. 6080.
—
1536
5202
—
23040
j- 1.5403-. 2258)
CHAPTER V
TEE LOZENGE DIAGRAM
17.
Definitions.- The lozenge_dia~ram/~43]is a method
which enables us to find a large number of interpolation
formulas.
Let
(P ) denote the quantity
g
and letEridenote the entry f(atrw).
Let us now show that (p)
g
may be expressed as (P÷i) — (P)
gt-].
g~l
By definition
(P+j) - (p
g~i
g+~
(P 4- 1)!
)~.__________
(gt-l).~[pi-l_(g+].)~
Pt
(g~l.)~p-(g i-i)j!
Multiplying the numerator and denominator of the nega
tive fraction on the right by (P-g), we get
(P41)P~
(p i-i)
g4i.
—
P~(P~g)
(p)
gt-1
(g.e~l)’(P—g)’
(Pi ].-)P’
(g÷l)~(P~.g)~
P’(g4 1)
—
g~(g
~,
l)(p—g) ~
32
(g~l)~(P—g—l)~(P—g)
P’(p—g)
(g ~l)~(p-g)~
33
or
(1)
(p +1)
—
(p~
(p~
g
By definition,we know that
(2)
where
¶L~ L-r3~
L_r]
/~ ~-r~
-
f(a-rw)
Multiplying the left-hand side of equation (1) by the
right-hand side of equation (2), and multiplying the righthand side of equation (1) by the left-hand side of equation
(2), we see that
g
g
(P)g £~&r~~ ~(P)g L\T-rj
(~
g1-l
÷ l)gkl
AL-~1
-(p~~
g÷l
~ J..r]
or
(3)
g
(P~ ~
L_r4~.l1+
g~].
(p~~1
j~
g
g~l
[~rJz(P)g ~L_~(Ptl)1~&r1
Suppose we arrange these ternis in the form
of
a ‘!Lozenge” or
an oblique parallelogram so that the terms on the left-hand
side equation
(3)
lie along the two upper sides of the
lozenge and the terms of the right-hand side along the lower.
We obtain the following lozenge diagram as which a line
directed from left to right joining two quantities denotes
the addition. of those quantities.
34
g
~&rJ
(p÷l)
g
.~.
l-.~
g÷.l
Z\
f~r]
g4.l
FIGUI~E I
Equation (3) may be e~qressed by the statement that: in
travelling from the left-hand vertex to the right-hand vertex
of the lozenge in the diagram, the stun of the elements which
lie along the upper route is equal to the sum of the elements
which lie along the lower route [4;44J
18.
Extended ~ozd~nge~Diagram.- It is evident that the
concluding stat~ent of the last section may be extended.
examples let us consider the lozenges corresponding to
Px
px-l
P~x
g=l
g~l
g~2
r~l
r~o
r~l
For
3~
so that the upper vertices of the lozenge, which are of the
form
(P)g z~L~-rj form a sort of difference table:
(x)AE-1J
(x)~[-rj
(x-l)
~foJ
3
(x)~(~—lJ
1
2
(x-2)
~
[oJ
t~r’~
By extending figure 1 of section 13, we shall now develop
a lozenge-diagram which is a modification of the “lozenge”
developed by D.C. Fraserf4;4~/.
As an explanation to this modification of Fraser’s
lozenge-diagram we are letting the powers of K denote the value
of g in any particular column, hence the order of any dif
ference operator found in the column K2 is the second, and
any P found under the same column has a subscript g equal to
two.
The values of P are constant along any diagonal descend
ing from left to right of the diagram, while along a diagonal
ascending from left to right these values increase by unity at
each vertex.
36
We now obtain the following lozenge-diagram:
2
[r]
K
K
K
3
4
K
[~3~
(x 4-3
(~
AE~]
(x ÷1
(x)
(x~l
(x)
(x-l)
~1)}
A[1]
2)
[3~
(xl)
2)~
FIGUBE 2.
Applying the rule of equation
(3)
section 13, we may
form the following sums from the above lozenge, each being
equal:
37
~ {~1]~(xfl)
+(x~1)~[-2~~ (xt2)
[~i]
(x~i)~
&
A2f-2j
A
{-2}
x~2
1)~f.2]~(x1:2)
AL]
By the application of equation (3),we also have
~l]-)~.(x*1)/\L-l]*{O]+(x) L~[—i], hence we may form three
1
other expressions beginning with the term {oJ instead of
j~-1]
and equivalent to those already given, namely,
[o]~(x)~
fO]f(x)~2{~lJ~(x~l)
~3~~2J
and two similar expressions.
By close observation of figure 2 it is readily notice
able that the sum of the elements from
foj
-
along the down-
ward sloping line of zero differences gives the Gregory—
Newton formula for{x]
If we form the identityfOJ÷xLfO]~~l}4(x_l)/~fO], it
is evident that the value of [x] is unaltered if a route is
selected starting from
f ii
instead of fromfO]
.
In general,
the sum of the elements along any route proceeding from any
entry r whatever to the line of zero differences is equal to
[x]
.
From tnis fact it is obvious that many interpolation
formulas may be found by using the lozenge-diagram.
38
19.
Derivation of Interpolation Formulas by Repeated
Summation of Part.- All of the interpolation formulas may be
derived from the lozenge-diagram method by using the formula
(i)
~[xj=A(v{x]
V
where v
q(a -~--xw) and
)
-
xf(a 4-xw).
x
Letting x
(2)
0 in the above formula, we have
v~{0J~(vfOj) _foJL\v
Applying the definition of the first difference to the
right-hand side of equation (2)
v~[0]=v
or
Lu
~v
A[o]=Lll
[11
-
-
~
{oj
-
{oJ
v~fOj v
foj ~
(o].
-
Hence,
(3)
[l~~o~t~jfo].
Letting x=1 in equation (1) and applying the definition
of the first difference to the right—hand side as before, we
obtain
-
or
(4)
[2~[l~tA[l]
‘9
From the lozenge-diagram we may form the following
identity:
fi] ~
(x-l)~
or
(x-l) ~[i]~
{oJ
(x-l)
~
-
x(x-l)
Taking x
(5)
(x-i)MoJ ÷(x)
2
[~J~(x-i)
A fo7
~[o}
[0].
2~ we have
~[i]~Lo}±A
f01.
Substituting for
and j~[l~from equation
(3)
and
(5),
(4) becomes
(6)
{2]:{O1÷2~[o}t
~[o]
Proceeding in this manner we may write an expression for [xl
in terms of [0) and the differences of
Afl +
(A) [x):[O)~ xA[O3~ (x)
+
•
•
•
~ (x)
~
[o}~
(x)
)~o]
.
Thus
~
.
The above equation may be recognized readily as the GregoryNewton Formula.
Again from the lozenge-diagram, let us consider the iden
t i ty
(X)~[0]4(X)~~[-l~ (x)
~f~i] +
(x~1)
~
40
or
x(x-1) ~2~j
3
x(x~l)(x-2)
L]x(x_l)
A2
[~1
3
3!~
(x l)x(x-l)
±—
When x ~2, we have
Substituting in equation
(7)
LsLà],
A2E~ ~
(6)
L2MO]÷ 2LEoJ+
Applying the difference operator
(8)
~f2j=A~]+ 2 ~
for
(7)
we obtain
becomes,
[oJ+~-i]
±
Now if we let x~2 in formula (1), then
AE2]~L3J-
{2]
or
(9)
Substituting for [2] and AL27 from
(9),
equation
(7)
and
(8)
in
we get
(‘°)
From the lozenge-diagram, we know that
(x+ i)~~~-i]+(~÷i)
g~2j
=
and when we let x.~3, we have
4
4
5
(x÷1)
A~2Jf(xt2)5AL-21
4]-
~f_i]
Replacing
[31=L0]÷ 3 ~[o]~
in equation (10) by its equal, then
3 ~ [-1~k 4
~
f-il ~
Proceeding in the manner used to obtain
an expression for [xJ
(B)
{xJ~foJ~
+
~
[3]~
we niay now write
Thus
.
x~~+ (x)
(x÷1)
~j-2J÷A f~2J.
~
~
f-~J~ (xt2)
f-i]
~~2Jt.
which is tne Newton-Gauss formula.
Rewriting (B) in terms of the central—difference nota
tions as given in chapter I, section
(B)’
rxJz~i+ (x)
~~i/2]+(x)
+(x~i)ATOJ+
~
6,
we have
(x~l)
A
fl/2]
.
If we now take the mean of these values of
[X~,
we ob
tain the formula whose central differences lie along the
horizontal line corresponding to
-
{xj
~
[O~(x)
[oJ
[-l/~j~-~/2]
1
2
LA_Li
+
~
4
(x~2)
Lo
-i-
(x.i-i)
~+.
42
By definition, we know that
~L~1/2J+A
~l/2J~~Co]
Hence,
(C)
~=fO1+(x)~
x(x2~ 12)
~foj+
~3
~
Loi*
) ~
~
which is the Newto~i—Stir1ing formula.
In order to obtain another of the important interpolation
formulas let us eliminate the odd differences from formula (B)
By definition, we have the relations
A[o]~ E~]
g
-
{oJ, ~f~i]~-~ [-1] fA ~i’
E-2J=A
L-’]
-
~
~
Formula (B) becomes
[xJ~oJ+x[{1]
(x~ 1)
(x+2)
-
[oJJ(x)~1]
{o]
-
A E-’]
~ (x +1)
{~~L-~- A{~2]~+
.
.
.
~ [-2]
43
Using the relation given in equation (1), section 13
this equation may be written
2
(D)
L\
txj~ (l-x) {OjtxL1]~(x÷i)
2
3
÷(x~2)A {-‘]- (x+i)
A 1-’]
Lo1~(x)
3
~f-2]+.
or in central-difference notations
(D’)
[xJ= (i-x)
foj~
EQ÷
(x+1)
~x
~4fo]-
(x)~~oj- (x~l)
2
,‘≤\[ij
3
4
r~\ ti]÷.
+(xi-2)
5
which is the Laplace-Everett formula.
It is evident that many more interpolation formulas may
he derived by the use of the lozenge-diagram method.
At this
point let us illustrate the use of the last three interpolat—
ion formulas that involve the central—difference notations.
Instead of using data resulting from a rational integral
function in our illustration as we have done previously let
us consider data resulting from the log sin of four angles.
Since the djf~erences of data corciingfromn any function other
than a rational integral function merely approach zero the
accuracy of logarithmic and trigonometric interpolation is
sometimes affected at the last figure beyond the decimal
point.
44
Illustration: Find the value log sin 0 16 8.5 using the
following difference table and formulas ~B’), (C), and (D’).
log sin
7.670 999 750 0
0016?7t1
4488799
8”
7.671 448 629 9
911
7.671 897 046 4
-4634
4484165
-4627
7
4479538
7.672 345 000 2
10’
By definition
Lo]
{x]=f(a+xw),
hence x=1/2, and
7.6714486299.
Substituting in the Newton-Gauss formula (B) we have
f(0° i6’8”.
5)
=
7671448629.9
l/8(—463.4)
7671448629.9
÷ 57.92
=
...
-
+
-
-i-
1/2(448416.5)
3/48 (.7)
224208.25
.043
7671672896.023
log si~~16’8”.5=7.6716728960
Substituting in the Newton-Stirling formulas (c) for
x
1/2,
we
have
/448879.9
f(0° l6’8”.5)
=
7671448629.9
÷ 1/2.1/2 (
\+448416. 5
+(l/4.l/2) (—463.4)
7671448629.9 ÷224323.85
7671672895.83
..
log sin
00
i6’8”.5= 7.6716728960
•
-
57.92
45
Substituting in the Laplace-Everett formula for
x z1/2, we have
f(O i6’8~’.5) =1/2(761448629.9)
-
~/48(-463.4)
÷1/2(761897046.4)
-
3/48(-462.7)
~38O948523. 2
+
+ 28.91
7671672896. 02
log sin 0 l6’8”.5~7.671672896o
28.96
-i-
380724314.95
cHAP’r1~T~ VI
smi~LARY
Interpolation in this thesis has been treated purely
from the standpoint of finite calculus.
Therefore, in
chapter II, some of the theory of finite differences was
introduced.
~he operator~was treated exactly as any other
algebraic symbol, in that it obeys all the..~laws of algebra.
From this introduction of a portion of the theory of finite
differences, the first interpolation formula was developed..
In chapter IV this first formula, as were all the other
formulas, was shown to be a special ease of Newton’s formula
for unequal intervals.
By the use of illustrative difference
tables and various fo~rmulas the problems of interpolation,
as set forth in the introduction, were solved..
In chapter V
the, lozenge-diagram method for derivation of interpolation
formulas, as developed by D.C. Fraser, was modified and used
to develop the Newton interpolation formulas.
46
B IBLIOGRAPHY
[i7
George Boole, Calculus of Finite Differences, ~ew York,
Strechert, 1926.
[2J L.M. Milne_Thompson,CalculLlus of Finite Differences,
London, Macmil1iai~i,
[3J
Samuel Barnard, Higher Algebra, London, Macrnillian, 1936.
[4] E.T. Whittaker, The Calculus of Observations, London,
Blackie, 1937:.
[5]
G.E. Porter, on The Calculus of Finite Differences,
Atlanta, A.tT. Thesis,1942
[6] The Encyclopedia Britannica, Chicago, Britannica
E
47