Numerical
Analysis
Lecture 18
Chapter 5
Interpolation
Finite Difference Operators
Newton’s Forward Difference
Interpolation Formula
Newton’s Backward Difference
Interpolation Formula
Lagrange’s Interpolation
Formula
Divided Differences
Interpolation in Two Dimensions
Cubic Spline Interpolation
Introduction
Finite differences play an
important role in numerical
techniques, where
tabulated values of the
functions are available.
For instance, consider a
function
y  f ( x).
As x takes values
x0 , x1 , x2 ,
, xn ,
let the corresponding values
of y be
y0 , y1 , y2 ,
, yn .
That is, for given a table of
values,
( xk , yk ), k  0,1, 2,
, n;
the process of estimating
the value of y, for any
intermediate value of x, is
called interpolation.
The method of computing
the value of y, for a given
value of x, lying outside
the table of values of x is
known as extrapolation.
If the function f (x) is
known, the value of y
corresponding to any x
can be readily
computed to the
desired accuracy.
For interpolation of a
tabulated function, the
concept of finite differences
is important. The knowledge
about various finite
difference operators and
their symbolic relations are
very much needed to
establish various
interpolation formulae.
Finite
Difference
Operators
Finite Difference Operators
Forward Differences
Backward Differences
Central Difference
Forward
Differences
For a given table of values
( xk , yk ), k  0,1, 2,..., n
with equally spaced abscissas
of a function y  f ( x),
we define the forward difference
operator  as follows
To be explicit, we write
y0  y1  y0
y1  y2  y1
yn 1  yn  yn 1
These differences are
called first differences of
the function y and are

y
denoted by the symbol i
Here,  is called the first
difference operator
Similarly, the differences of
the first differences are
called second differences,
defined by
 y0  y1  y0 ,
2
 y1  y2  y1
2
Thus, in general
 yi  yi 1  yi
2
Here  is called the second
difference operator. Thus,
continuing, we can define,
r-th difference of y, as
2
r 1
r 1
 yi   yi 1   yi
r
By defining a difference table
as a convenient device for
displaying various
differences, the above
defined differences can be
written down systematically
by constructing a difference
table for values
( xk , yk ), k  0,1,...,6
Forward Difference Table
This difference table is called
forward difference table or
diagonal difference table.
Here, each difference is
located in its appropriate
column, midway between the
elements of the previous
column.
Please note that the subscript
remains constant along each
diagonal of the table. The first
term in the table, that is y0 is
called the leading term, while
the differences
y0 ,  y0 ,  y0 ,...
2
3
are called leading differences
Example
Construct a forward difference
table for the following values of
x and y:
Solution
Example
Express  y0 and  y0 in
terms of the values of
the function y.
2
3
Solution:
Noting that each higher order
difference is defined in terms
of the lower order difference,
we have
 y0  y1  y0  ( y2  y1 )  ( y1  y0 )
2
 y2  2 y1  y0
and
 3 y0   2 y1   2 y0  (y2  y1 )  ( y1  y0 )
 ( y3  y2 )  ( y2  y1 )  ( y2  y1 )  ( y1  y0 )
 y3  3 y2  3 y1  y0
Hence, we observe that the
coefficients of the values of y,
2
3
in the expansion of  y0 ,  y0 ,
are binomial coefficients.
Thus, in general, we arrive at
the following result: -
 y0  yn  C1 yn1  C2 yn2
n
n
 C3 yn3 
n
n
 (1) y0
n
Example
Show that the value of yn can
be expressed in terms of the
leading value y0 and the
leading differences
y0 ,  y0 ,
2
,  y0 .
n
Solution
The forward difference table will be
y1  y0  y0 or y1  y0  y0 

y2  y1  y1 or y2  y1  y1 

y3  y2  y2 or y3  y2  y2 
Similarly,
y1  y0   y0 or y1  y0   y0 


2
2
y2  y1   y1 or y2  y1   y1 

2
2
Similarly, we can also write
 y1   y0   y0 or  y1   y0   y0 


2
2
3
2
2
3
 y2   y1   y1 or  y2   y1   y1 

2
2
3
2
2
3
y2  (y0   y0 )  ( y0   y0 )
2
2
3
 y0  2 y0   y0
2
3
y3  y2  y2
 ( y1  y1 )  (y1   y1 )
2
 y0  3y0  3 y0   y0
2
 (1  ) y0
3
3
Similarly, we can symbolically
write
y  (1   ) y ,
1
0
y2  (1   ) y0 ,
2
y3  (1   ) y0
3
........
yn  (1   ) y0
n
Hence, we obtain
yn  y0  C1y0  C2  y0
n
n
 C3 y0 
n
3
  y0
n
n
yn   Ci  y0
n
i 0
i
2
Numerical
Analysis
Lecture 18
Backward
Differences
For a given table of values
( xk , yk ), k  0,1, 2,..., n
of a function y = f (x) with
equally spaced abscissas, the
first backward differences are
usually expressed in terms of
the backward difference
operator  as
yi  yi  yi 1i  n, (n 1),
y1  y1  y0
OR
y2  y2  y1
yn  yn  yn 1
,1
The differences of these
differences are called second
differences and they are
denoted by 2 y , 2 y , , 2 y .
2
3
n
That is
 y1  y2  y1
2
 y2  y3  y2
2
 yn  yn  yn 1
2
Thus, in general, the second
backward differences are
 yi  yi yi 1, i  n,(n 1),..., 2
2
while the k-th backward
differences are given as
k 1
k 1
 yi   yi  yi 1, i  n,(n 1),..., k
k
These backward differences
can be systematically
arranged for a table of values
( xk , yk ), k  0,1,...,6
Backward Difference Table
From this table, it can
be observed that the
subscript remains
constant along every
backward diagonal.
Example
Show that any value of y
can be expressed in terms
of yn and its backward
differences.
Solution
From yi  yi  yi 1i  n, (n 1), ,1
We get
From
We get
yn1  yn yn
yn2  yn1  yn1
 yi  yi yi 1, i  n,(n 1),..., 2
2
yn1  yn 2 yn
From these equations, we obtain
yn2  yn  2yn   yn
2
Similarly, we can show that
yn3  yn  3yn  3 yn  yn
2
3
Symbolically, these results can
be rewritten as follows:
yn 1  (1  ) yn
yn  2  (1  ) yn
2
yn 3  (1  ) yn
3
.......
yn  r  (1  ) yn
r
ynr  yn  C1yn  C2 yn 
n
n
2
 (1)  yn
r
r
Central
Differences
In some applications, central
difference notation is found to be
more convenient to represent the
successive differences of a
function. Here, we use the symbol
to represent central difference
operator and the subscript of  y
for any difference as the average of
the subscripts

 y1 2  y1  y0 ,
In general
 y3 2  y2  y1 ,
 yi  yi(1 2)  yi(1 2)
Higher order differences are
defined as follows:
 yi   yi(1 2)   yi(1 2)
2
 yi  
n
n1
yi(1 2)  
n1
yi (1 2)
These central differences can
be systematically arranged as
indicated in the Table
Thus, we observe that all
the odd differences have a
fractional suffix and all the
even differences with the
same subscript lie
horizontally.
The following alternative
notation may also be adopted
to introduce finite difference
operators. Let y = f (x) be a
functional relation between x
and y, which is also denoted
by yx.
Suppose, we are given
consecutive values of x
differing by h say x, x + h, x +2h,
x +3h, etc. The corresponding
values of y are yx , yxh , yx2h , yx3h ,
As before, we can form the
differences of these values.
Thus
yx  yxh  yx  f ( x  h)  f ( x)
 yx  yxh  yx
2
Similarly
yx  yx  yxh  f ( x)  f ( x  h)
 yx  yx ( h / 2)  yx ( h / 2)
h

 f x 
2

h

f x 
2

Numerical
Analysis
Lecture 18