2010/3/11 Page 1 Gaussian quadrature Advisor: J. T. Chen
Reporter: Jia-Wei Lee Date: Mar., 11, 2010
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2010/3/11 ( JWLee) Page 2 Trapezoidal rule where is weighting
coefficient
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2010/3/11 ( JWLee) Page 3 Gaussian quadrature f (x) is the
polynomial of degree 2n 1 or less are the zeros of polynomial
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2010/3/11 ( JWLee) Page 4 Legendre polynomial Legendres
equation Legendre polynomial Here are n distinct roots of in the
interval (1,-1)
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2010/3/11 ( JWLee) Page 5 The orthogonality of Legendre
polynomial vectors functions Legendre polynomial and are
orthogonal
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2010/3/11 ( JWLee) Page 6 Gaussian quadrature f (x) is the
polynomial of degree 2n 1 or less are the zeros of Legendre
polynomial Therefore the integral can be obtained by using n
points
2010/3/11 ( JWLee) Page 8 Gaussian quadrature Let where
(Lagrange polynomial)
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2010/3/11 ( JWLee) Page 9 Weighting coefficients and associated
points Number of points Weighting coefficients Associated points
Because the integral interval is from 1 to 1.
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2010/3/11 ( JWLee) Page 10 The general integral interval
(a,b)
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2010/3/11 ( JWLee) Page 11 Numerical examples Ex 1 Exact
solution is a polynomial degree of 7 Number of points
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2010/3/11 ( JWLee) Page 12 Numerical examples Ex 2 Exact
solution Number of points
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2010/3/11 ( JWLee) Page 13 Numerical examples Ex 3 Exact
solution Number of points
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2010/3/11 ( JWLee) Page 14 Fortran program
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2010/3/11 ( JWLee) Page 15 Fortran program
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2010/3/11 ( JWLee) Page 16 The end Thanks for your kind
attentions
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2010/3/11 ( JWLee) Page 17 Gaussian quadrature f (x) is the
polynomial of degree 2n 1 or less are the zeros of Legendre
polynomial Therefore the integral can be obtained by using n
points
