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lecture32

# 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 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
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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