24- Numerical Integration

24- Numerical Integration - Definite integrals Quadrature...

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Unformatted text preview: Definite integrals Quadrature formulas Errors Numerical Integration Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Numerical Integration MATH 2070U 1 / 27 Definite integrals Quadrature formulas Errors Numerical Integration 1 Definite integrals 2 Elementary quadrature formulas Midpoint formula Trapezoidal formula Simpson formula 3 Errors & accuracy of quadrature formulas c D. Aruliah (UOIT) Numerical Integration MATH 2070U 2 / 27 Definite integrals Quadrature formulas Errors Definite integrals Recall definite integral I ( f ) : = I ( f ; a , b ) = Z b a f ( x ) dx is (signed) area under graph of y = f ( x ) between x = a and x = b Quadrature or numerical integration : numerical approximation of definite integrals c D. Aruliah (UOIT) Numerical Integration MATH 2070U 4 / 27 Definite integrals Quadrature formulas Errors Evaluation/computation of definite integrals The Fundamental theorem of calculus: I ( f ) = Z b a f ( x ) dx = F ( b )- F ( a ) where F is an antiderivative or primitive of f Definite integral R b a f ( x ) dx is a number Indefinite integral R f ( x ) dx is a function Many integrands do not admit analytic antiderivatives e.g. erf ( x ) : = 2 Z x e- t 2 dt (the error function) e.g. E ( k ) : = Z 2 p 1- k 2 sin 2 d (complete elliptic integral) c D. Aruliah (UOIT) Numerical Integration MATH 2070U 5 / 27 Definite integrals Quadrature formulas Errors Quadrature formulas and composite formulas Partition [ a , b ] into M subintervals I k = [ x k- 1 , x k ] ( k = 1: M ), where x ` : = a + ` H = a + ` ( b- a ) M ( ` = 0: M ) Partitioning [ a , b ] as described yields I ( f ) = Z b a f ( x ) dx = M k = 1 Z I k f ( x ) dx = M k = 1 Z x k x k- 1 f ( x ) dx Composite quadrature formulas refer to basic formulas applied to partitioned interval [ a , b ] c D. Aruliah (UOIT) Numerical Integration MATH 2070U 7 / 27 Definite integrals Quadrature formulas Errors Midpoint formula Midpoint formula Midpoint formula The midpoint formula I mp ( f ) is given by I mp ( f ) : = ( b- a ) f a + b 2 ....
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24- Numerical Integration - Definite integrals Quadrature...

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