Numerical Analysis Lecture 21

Chapter 5 Interpolation

Finite Difference Operators Newton’s Forward Difference 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

For given a table of values, the process of estimating the value of y, for any intermediate value of x, is called interpolation.

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.

Thus Similarly

Shift operator, E

The inverse operator E-1 is defined as Similarly,

Average Operator,

Differential Operator, D

Important Results

Newton’s Forward Difference Interpolation Formula

Let y = f (x) be a function which takes values f(x0), f(x0+ h), f(x0+2h), …, corresponding to various equi-spaced values of x with spacing h, say x0, x0 + h, x0 + 2h, … . Suppose, we wish to evaluate the function f (x) for a value x0 + ph, where p is any real number, then for any real number p, we have the operator E such that

This is known as Newton’s forward difference formula for interpolation, which gives the value of f(x0 + ph) in terms of f(x0) and its leading differences.

This formula is also known as Newton-Gregory forward difference interpolation formula. Here p=(x-x0)/h. An alternate expression is

If we retain (r + 1) terms, we obtain a polynomial of degree r agreeing with yx at x0, x1, …, xr. This formula is mainly used for interpolating the values of y near the beginning of a set of tabular values and for extrapolating values of y, a short distance backward from

Numerical Analysis Lecture 21