lecture34 - CMPSC/MATH 451 Numerical Computations Lecture...

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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 34 Nov 9, 2011 Prof. Kamesh Madduri Class Overview • Progressive Gaussian quadrature • Composite quadrature • Adaptive quadrature • Octave code for quadrature rules (see tutorial) • Slides from textbook follow. 2 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Progressive Gaussian Quadrature Avoiding this additional work is motivation for Kronrod quadrature rules Such rules come in pairs, n-point Gaussian rule G n , and (2 n + 1)-point Kronrod rule K 2 n +1 , whose nodes are optimally chosen subject to constraint that all nodes of G n are reused in K 2 n +1 (2 n + 1)-point Kronrod rule is of degree 3 n + 1 , whereas true (2 n + 1)-point Gaussian rule would be of degree 4 n + 1 In using Gauss-Kronrod pair, value of K 2 n +1 is taken as approximation to integral, and error estimate is given by (200 | G n- K 2 n +1 | ) 1 . 5 Michael T. Heath Scientific Computing 31 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Progressive Gaussian Quadrature, continued Because they efficiently provide high accuracy and reliable error estimate, Gauss-Kronrod rules are among most effective methods for numerical quadrature They form basis for many quadrature routines available in major software libraries Pair ( G 7 , K 15 ) is commonly used standard Patterson quadrature rules further extend this idea by adding 2 n + 2 optimally chosen nodes to 2 n + 1 nodes of Kronrod rule K 2 n +1 , yielding progressive rule of degree...
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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lecture34 - CMPSC/MATH 451 Numerical Computations Lecture...

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