lecture32 - CMPSC/MATH 451 Numerical Computations Lecture...

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CMPSC/MATH 451 Numerical Computations Lecture 32 Nov 4, 2011 Prof. Kamesh Madduri
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Numerical Integration and Differentiation Newton-Cotes quadrature Degree of a quadrature rule Clenshaw-Curtis quadrature Gaussian quadrature Slides from textbook follow. 2
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Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Stability of Quadrature Rules Absolute condition number of quadrature rule is sum of magnitudes of weights, n X i =1 | w i | If weights are all nonnegative, then absolute condition number of quadrature rule is b - a , same as that of underlying integral, so rule is stable If any weights are negative, then absolute condition number can be much larger, and rule can be unstable Michael T. Heath Scientific Computing 13 / 61
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Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Newton-Cotes Quadrature Newton-Cotes quadrature rules use equally spaced nodes in interval [ a, b ] Midpoint rule M ( f ) = ( b - a ) f ± a + b 2 ² Trapezoid rule T ( f ) = b - a 2 ( f ( a ) + f ( b )) Simpson’s rule S ( f ) = b - a 6 ± f ( a ) + 4 f ± a + b 2 ² + f ( b ) ² Michael T. Heath Scientific Computing 14 / 61
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Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature Other Integration Problems Error Estimation, continued Difference between midpoint and trapezoid rules provides estimate for error in either of them T ( f ) - M ( f ) = 3 E ( f ) + 5 F ( f ) + ··· so E ( f ) T ( f ) - M ( f ) 3 Weighted combination of midpoint and trapezoid rules eliminates E ( f ) term from error expansion I ( f ) = 2 3 M ( f ) + 1 3 T ( f ) - 2 3 F ( f ) + = S ( f ) - 2 3 F ( f ) + which gives alternate derivation for Simpson’s rule and estimate for its error Michael T. Heath Scientific Computing 18 / 61
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Numerical Integration Numerical Differentiation Richardson Extrapolation Quadrature Rules Adaptive Quadrature
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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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lecture32 - CMPSC/MATH 451 Numerical Computations Lecture...

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