assn7 - AMATH 341 CM 271 CS 371 Assignment 7 Due Monday...

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AMATH 341 / CM 271 / CS 371 Assignment 7 Due : Monday March 29, 2010 Instructor: K. D. Papoulia 1. Consider a general quadrature scheme for the interval [ a,b ] that uses distinct quadrature points x 1 ,...,x n and weights w 1 ,...,w n where the weights are initially unknown. Suppose we require this rule to be exact for polynomials of degree up to n - 1. Then an exact formula for the weight w k is given by the integral over [ a,b ] of the k th polynomial in the Lagrange interpolating basis for the points x 1 ,...,x n . (This polynomial appears, e.g., inside the parentheses in the summation at the bottom of p. 93 of the Moler text, which is p. 1 of the ‘Interpolation’ chapter on-line). Explain why this is so. [Hint: what is the outcome of the quadrature rule if it is applied to this Lagrange basis polynomial?] Solution. The k th polynomial in the Lagrange interpolating basis is l k ( x ) = ( x - x 1 ) ··· ( x - x k - 1 )( x - x k +1 ) ··· ( x - x n ) ( x k - x 1 ) ··· ( x k - x k - 1 )( x k - x k +1 ) ··· ( x k - x n ) . This polynomial has the property, apparent by inspection, that l k ( x m ) = 1 if m = k else l k ( x m ) = 0. Also apparent is that its degree is n - 1. Since the quadrature rule is supposed to be exact for polynomials of n - 1 or less, then it is exact for l k , i.e., Z b a l k ( x ) dx = n X i =1 w i l k ( x i ) . However, the right-hand side simplifies to w k because of the property of the Lagrange basis polynomials, so we conclude that w k must be the value of the exact integral of the Lagrange basis polynomial over [ a,b ]. 2. Consider integrating a function over the interval [0 , ) using the Matlab function quad . This function takes as its first argument a function handle (the integrand), its second and third arguments the limits of integration, and its fourth the accuracy requirement. [Special note: the function handle sent to
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This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.

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assn7 - AMATH 341 CM 271 CS 371 Assignment 7 Due Monday...

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