Numerical Analysis Lecture 24

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

Newton’s Forward Difference Interpolation Formula

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

An alternate expression is

NEWTON’S BACKWARD DIFFERENCE INTERPOLATION FORMULA

The formula is,

Alternatively, this formula can also be written as Here

LAGRANGE’S INTERPOLATION FORMULA

Newton’s interpolation formulae can be used only when the values of the independent variable x are equally spaced. Also the differences of y must ultimately become small.

If the values of the independent variable are not given at equidistant intervals, then we have the Lagrange formula

Here the polynomial is of the form or in the form

Here, the coefficients ak are so chosen as to satisfy this equation by the (n + 1) pairs (xi, yi). Thus we get Therefore,

Similarly and

Substituting the values of a0, a1, …, an we get The Lagrange’s formula for interpolation

This formula can be used whether the values x0, x2, …, xn are equally spaced or not. Alternatively, this can also be written in compact form as

Thus introducing Kronecker delta notation Where, We can easily observe that, and Thus introducing Kronecker delta notation

Further, if we introduce the notation That is is a product of (n + 1) factors. Clearly, its derivative contains a sum of (n + 1) terms in each of which one of the factors of will be absent.

We also define, which is same as except that the factor (x–xk) is absent. Then But, when x = xk, all terms in the above sum vanishes except Pk(xk)

Hence,

Finally, the Lagrange’s interpolation polynomial of degree n can be written as

DIVIDED DIFFERENCES

Let us assume that the function y = f (x) is known for several values of x, (xi, yi), for i=0,1,..n. The divided differences of orders 0, 1, 2, …, n are now defined recursively as:

is the zero-th order divided difference

The first order divided difference is defined as

Similarly, the higher order divided differences are defined in terms of lower order divided differences by the relations of the form

Generally

Standard format of the Divided Differences

We can easily verify that the divided difference is a symmetric function of its arguments. That is,

Now,

Therefore

This is symmetric form, hence suggests the general result as

NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA

Let y = f (x) be a function which takes values y0, y1, …, yn corresponding to x = xi, i = 0, 1,…, n. We choose an interpolating polynomial, interpolating at x = xi, i = 0, 1, …, n in the following form

Here, the coefficients ak are so chosen as to satisfy above equation by the (n + 1) pairs (xi, yi). Thus, we have

The coefficients a0, a1, …, an can be easily obtained from the above system of equations, as they form a lower triangular matrix.

The first equation gives The second equation gives

Third equation yields which can be rewritten as that is

Thus, in terms of second order divided differences, we have Similarly, we can show that

Newton’s divided difference interpolation formula can be written as

Newton’s divided differences can also be expressed in terms of forward, backward and central differences. They can be easily derived

Assuming equi-spaced values of abscissa, we have

By induction, we can in general arrive at the result

Similarly

In general, we have

Also, in terms of central differences, we have

In general, we have the following pattern

Numerical Analysis Lecture 24