25-Adaptive-Quadrature

25-Adaptive-Quadrature - Limitations Algorithm quad...

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Limitations Algorithm quad Adaptive Quadrature Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) Adaptive Quadrature MATH 2070U 1 / 19 Limitations Algorithm quad Adaptive Quadrature 1 Limitations of composite quadrature 2 Algorithm for adaptive quadrature 3 quad in M ATLAB c ± D. Aruliah (UOIT) Adaptive Quadrature MATH 2070U 2 / 19 Limitations Algorithm quad Reminder: quadrature formulas H : = ( b - a ) / M , x k = a + kH ( k = 0: M ), x ` = ( x ` - 1 + x ` ) /2 ( ` = 1: M ) midpoint formula I mp ( f ) : = ( b - a ) f ± a + b 2 ² compsite midpoint formula I c mp ( f ) : = H M k = 1 f ( x k ) trapezoidal formula I t ( f ) : = b - a 2 [ f ( a ) + f ( b )] composite trapezoidal formula I c t ( f ) : = H " f ( a ) + f ( b ) 2 + M - 1 k = 1 f ( x k ) # Simpson formula I s ( f ) : = b - a 6 ³ f ( a ) + 4 f ± a + b 2 ² + f ( b ) ´ composite Simpson formula I c s ( f ) : = H 6 M k = 1 [ f ( x k - 1 ) + 4 f ( x k ) + f ( x k )] c ± D. Aruliah (UOIT) Adaptive Quadrature MATH 2070U 4 / 19 Limitations Algorithm quad Consider function humps in M ATLAB h ( x ) = 1 ( x - 0.3 ) 2 + 0.01 + 1 ( x - 0.9 ) 2 + 0.04 Basic Simpson formula approximation to R 1 0 h ( x ) dx c ± D. Aruliah (UOIT) Adaptive Quadrature MATH 2070U 5 / 19
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Limitations Algorithm quad Composite Simpson formula ( M = 2) for R 1 0 h ( x ) dx Generated using simpplot(2) Try simpplot(3) , simpplot(4) , . . . c ± D. Aruliah (UOIT) Adaptive Quadrature MATH 2070U 6 / 19 Limitations Algorithm quad Adaptive quadrature motivation Composite quadrature formulas approximate I ( f ) = Z b a f ( x ) dx with uniform subintervals I
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25-Adaptive-Quadrature - Limitations Algorithm quad...

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