43: Gauss forward formula involves differences below
the central line and even differences on the line in A, then
it is useful if
O<p<1
1<p 2
O<p<5
O<p<10
Answers
Answer:
INTERPOLATION
Chapter Objectives
O Introduction
O Newton’s forward interpolation formula
O Newton’s backward interpolation formula
O Central difference interpolation formulae
O Gauss’s forward interpolation formula
O Gauss’s backward interpolation formula
O Stirling’s formula
O Bessel’s formula
O Everett’s formula
O Choice of an interpolation formula
O Interpolation with unequal intervals
O Lagrange’s interpolation formula
O Divided differences
O Newton’s divided difference formula
O Relation between divided and forward differences
O Hermite’s interpolation formula
O Spline interpolation—Cubic spline
O Double interpolation
O Inverse interpolation274 • NUMERICAL METHODS IN ENGINEERING AND SCIENCE
O Iterative method
O Objective type of questions
7.1 Introduction
Suppose we are given the following values of y f(x) for a set of values
of x:
x: x0 x1 x2 xn
y: Y0 y1 y2
yn
.
Then the process of finding the value of y corresponding to any value of
x xi between x0
and xn is called interpolation. Thus interpolation is the
technique of estimating the value of a function for any intermediate value
of the independent variable while the process of computing the value of the
function outside the given range is called extrapolation. The term interpola-
tion however, is taken to include extrapolation.
If the function f(x) is known explicitly, then the value of y correspond-
ing to any value of x can easily be found. Conversely, if the form of f(x) is not
known (as is the case in most of the applications), it is very difficult to de-
termine the exact form of f(x) with the help of tabulated set of values (xi
, yi
).
In such cases, f(x) is replaced by a simpler function (x) which assumes the
same values as those of f(x) at the tabulated set of points. Any other value
may be calculated from (x) which is known as the interpolating function or
smoothing function. If (x) is a polynomial, then it called the interpolating
polynomial and the process is called the polynomial interpolation. Similarly
when (x) is a finite trigonometric series, we have trigonometric interpola-
tion. But we shall confine ourselves to polynomial interpolation only.
The study of interpolation is based on the calculus of finite differences.
We begin by deriving two important interpolation formulae by means of
forward and backward differences of a function. These formulae are often
employed in engineering and scientific investigations.
7.2 Newton’s Forward Interpolation Formula
Let the function y f(x) take the values y0
, y1
, , yn corresponding to
the values x0, x1, , xn of x. Let these values of x be equispaced such that
xi x0 ih (i 0, 1, ). Assuming y(x) to be a polynomial of the nth degree
in x such that 0 01 1 () () , ,, . ( ) n n yx y yx y yx y We can write
O Lagrange’s method