CSC_349_HANDOUT#23

CSC_349_HANDOUT#23 - large as n becomes large is to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 COMPUTER SCIENCE 349A Handout Number 23 THE RUNGE PHENOMENON The following example is the classical example to illustrate the oscillatory nature and thus the unsuitability of high order interpolating polynomials. It is due to Runge in 1901. Consider the problem of interpolating f ( x ) = 1 1 + 25 x 2 on the interval [ 1, 1] at n + 1 equally-spaced points x i by the interpolating polynomial P n ( x ). In the figure below, the graphs of ) ( and ) ( ), ( 20 5 x P x P x f are shown. f(x) degree 5 degree 20 x
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Runge proved that as n → ∞ , P n ( x ) diverges from ) ( x f for all values of x such that 0.726 x < 1 (except for the points of interpolation x i ). The interpolating polynomials do approximate ) ( x f well for 726 . 0 < x . One way to see that the difference between ) ( and ) ( x P x f n becomes arbitrarily
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: large as n becomes large is to consider the error term for polynomial interpolation = + + = n i i n n x x n f x P x f ) 1 ( ) ( )! 1 ( ) ( ) ( ) ( . As n , it can be shown that ) ( ) ( x P x f n (at some points x in [ 1, 1] ). Note: if the points of interpolation are not constrained to be equally spaced, then it is possible to choose the points of interpolation x i so that ) ( ) ( lim x f x P n n = . However, there is no known rule that indicates how to choose appropriate points of interpolation x i to guarantee such convergence of the interpolating polynomials for arbitrary continuous functions ) ( x f ....
View Full Document

Page1 / 2

CSC_349_HANDOUT#23 - large as n becomes large is to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online