Interpolation & Polynomial Approximation Hermite ...

Interpolation & Polynomial Approximation

Hermite Interpolation II

Numerical Analysis (9th Edition) R L Burden & J D Faires

Beamer Presentation Slides prepared by John Carroll

Dublin City University

c 2011 Brooks/Cole, Cengage Learning

Divided Difference Form

Outline

Example

Algorithm

1 Hermite Polynomials Using Divided Differences 2 Example: Computing H5(1.5) Using Divided Differences 3 The Hermite Interpolation Algorithm

Numerical Analysis (Chapter 3)

Hermite Interpolation II

R L Burden & J D Faires 2 / 22

Divided Difference Form

Outline

Example

Algorithm

1 Hermite Polynomials Using Divided Differences 2 Example: Computing H5(1.5) Using Divided Differences 3 The Hermite Interpolation Algorithm

Numerical Analysis (Chapter 3)

Hermite Interpolation II

R L Burden & J D Faires 3 / 22

Divided Difference Form

Example

Hermite Polynomials & Divided Differences

Algorithm

Introduction

There is an alternative method for generating Hermite approximations that has as its basis the Newton interpolatory divided-difference formula at x0, x1, . . . , xn, that is,

n

Pn(x) = f [x0] + f [x0, x1, . . . , xk ](x - x0) ? ? ? (x - xk-1).

k =1

The alternative method uses the connection between the nth divided difference and the nth derivative of f . See Theorem

Numerical Analysis (Chapter 3)

Hermite Interpolation II

R L Burden & J D Faires 4 / 22

Divided Difference Form

Example

Hermite Polynomials & Divided Differences

Algorithm

Construction

Suppose that the distinct numbers x0, x1, . . . , xn are given together with the values of f and f at these numbers. Define a new sequence z0, z1, . . . , z2n+1 by

z2i = z2i+1 = xi , for each i = 0, 1, . . . , n,

and construct the divided difference table See Original Table in a form that uses z0, z1, . . ., z2n+1. Since z2i = z2i+1 = xi for each i, we cannot define f [z2i , z2i+1] by the divided difference formula. However, we will assume, based on the divided-difference theorem See Theorem that the reasonable substitution in this situation is f [z2i , z2i+1] = f (z2i ) = f (xi ).

Numerical Analysis (Chapter 3)

Hermite Interpolation II

R L Burden & J D Faires 5 / 22

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