Transcription of Numerical integration: Gaussian quadrature rules
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Numerical integration: Gaussian quadrature rules
APMA 0160 (A. Yew) Spring 2011 Numerical integration: Gaussian quadrature rulesMatlab s built-in Numerical integration function[Q,fcount]=quad(f,a,b,tol)is essentially oursimp_compextrcode with some further efficiency-enhancing thatquadrequires scalar functions to be defined with elementwise operations, sof(x) =21+x2should be entered asf=inline( (1+x.^2) , x )orf=@(x) (1+x.^2)The default tolerance forquadis 10 has another efficient integration command calledquadl, with the same input and outputarguments. The method underlyingquadlis a Gaussian quadrature rule .Recall that each Newton Cotes quadrature rule came from integrating the Lagrange polynomial thatinterpolates the integrandfatnequally spacednodes in the interval [a,b]. Thus, in general, we expectthe degree of exactness of the rule to ben 1 (though, as we ve seen, some rules turn out to have ahigher-than-expected degree of exactness ).
Gauss–Legendre rules. They have degree of exactness 2n −1 (and order 2n). Gauss–Legendre rules are open rules, and because the nodes are often positioned at irrational points in the interval, when we code the adaptive composite rules by repeatedly halving the interval, many extra function evaluations may need to be performed.
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