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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].

2 (the midpoint of the interval), then the roots ˜x i in [−1,1] are transformed to the nodes x i in [a,b] via x i = h 2 x˜ i +c , and the quadrature formula for approximating R b a f(x)dx will be h 2 times the formula for approximating the equivalent integral over [−1,1].

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