Numerical Methods

HALDIA INSTITUTE OF
TECHNOLOGY ,
HALDIA
NEWTON’S BACKWARD INTERPOLATION FORMULA
Course Name:- Numerical Methods
Course Code:- M(CS)401
Dept:- Computer Science & Engineering
Name:- Md. Halim
Class Roll:- L18/CS/154
University Roll:- 103011018015

Abstract
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In order to reduce the numerical computations associated to the repeated
application of the existing interpolation formula in computing a large
number of interpolated values, a formula has been derived from Newton’s
backward interpolation formula for representing the numerical data on a pair
of variables by a polynomial curve. Application of the formula to numerical
data has been shown in the case of representing the data on the total
population of India corresponding as a function of time. The formula is
suitable in the situation where the values of the argument (i.e. independent
variable) are at equal interval.
Keywords: Interpolation, newton’s backward interpolation formula,
representation of numerical data

Introduction
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Interpolation, which is the process of computing intermediate values of a
function from the set of given values of the function {Hummel (1947), Erdos
& Turan (1938) et al}, plays significant role in numerical research almost in
all branches of science, humanities, commerce and in technical branches. A
number of interpolation formulas namely Newton’s Backward Interpolation
formula, In case of the interpolation by the existing formulae, the value of
the dependent variable corresponding to each value of the independent
variable is to be computed afresh from the used formula putting the value of
the independent variable in it. That is if it is wanted to interpolate the values
of the dependent variable corresponding to a number of values of the
independent variable by a suitable existing interpolation formula,

Newton’s Backward Interpolation Formulae
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By using Binomial thereom:-
f(a-mh) = [ 1-m + m(m-1) 2…]f(a)
2!
f(a)-m. . f(a)+m(m-1). 2 f(a)+…….
But, f(a) = yo
f(a-mb) = yo - m yo +yo +m(m-1) 2 yo
 Remarks:- Newtons’s backward difference interpolation formula, in used to find value of y near
the end of the table.
 Methods:-
We know if degree of function is ‘n’ then n-th differences are constant and (n+1)th order differences vanish.
Conversaly, If n-th order differences of a tabulated function are constant and (n+1)th order differences are zeros
then the tabulated function represents a polynomial of degree n…

Introduction
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 Q. Given the following data, find the value of f(9) using interpolation
formula.
 Sol. :-
To find f(9) i,e. out side the given data the method of extrapolation is
used. i,e. to find the value of y = f(x) at x = 9 we take , a = 8, h = 2, and
a-mb = 9 -2m = 1 .’. 8 – m(2) = 9 => m = -0.5
Here. We use Newton’s Backward Interpolation Formula.
X 0 2 4 6 8
f(x) 2 5 10 17 6

Newton’s Backward Interpolation Formulae
==================================================================================================================================================================================================
f(a-mh) = (yo) + m yo + m(m-1) 2 yo – m(m-1)(m-2) 3 yo +- - - - - - - -
2! 3!
Here,
yo = 26, y = 9, 2 yo = 2, 3 yo= 0
.’. f(9) = 26-(-0.5)(9)+(-0.5)(-0.5 – 1) 2+0
2
=> 26+4.5+0.75 => 26+5.25 => f(9) = 31.25 Ans.
x f(x) f(x) f2(x) f3(x)
0 2 3 2 0
2 5 5 2 0
4 10 7 2
6 17 9
8 26

Conclusion
==================================================================================================================================================================================================
 The formula described by equation (3.2) can be used to represent a
given set of numerical data on a pair of variables, by a polynomial. The
degree of the polynomial is one less than the number of pairs of
observations. The polynomial that represents the given set of numerical
data can be used for interpolation at any position of the independent
variable lying within its two extreme values. The approach of
interpolation, described here, can be suitably applied in inverse
interpolation also. Newton’s backward interpolation formula is valid for
estimating the value of the dependent variable.

Reference
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1. Bathe KJ, Wilson EL. Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1976.
2. Gerald CF, Wheatley PO. Applied Numerical Analysis, fifth ed., Addison-Wesley Pub. Co., MA, 1994.
3. David R Kincard, Ward Chaney E. Numerical analysis, Brooks /Cole, Pacific Grove, CA, 1991.
4. Endre S, David Mayers. An Introduction to Numerical Analysis, Cambridge, UK, 2003.
5. John H Mathews, Kurtis D Fink. Numerical methods using MATLAB, 4ed, Pearson Education, USA, 2004.
6. James B. Scarborough Numerical Mathematical Analysis, 6 Ed, The John Hopkins Press, USA, 1996.
7. Kendall E. Atkinson An Introduction to Numerical Analysis, 2 Ed., New York. 1989.
8. Chapra SC, Canale RP. Numerical Methods for Engineers, third ed., McGraw-Hill, New York, 2002.
9. Conte SD, Carl de Boor. Elementary Numerical Analysis, 3 Ed, McGraw-Hill, New York, USA, 1980.
10. Robert J Schilling, Sandra L Harries. Applied Numerical Methods for Engineers, Brooks /Cole, Pacific Grove, CA,
2000.

Numerical Methods

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