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2010/3/11 計算機在工程上的應用 Page 1 Gaussian quadrature Advisor: J. T. Chen Reporter: Jia-Wei Lee Date: Mar., 11, 2010
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 2 Trapezoidal rule where is weighting coefficient
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2010/3/11 計算機在工程上的應用 ( 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 計算機在工程上的應用 ( JWLee) Page 4 Legendre polynomial Legendre’s equation : Legendre polynomial : Here are n distinct roots of in the interval (1,-1)
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 5 The orthogonality of Legendre polynomial vectors functions Legendre polynomial and are orthogonal
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2010/3/11 計算機在工程上的應用 ( 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
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 7 Gaussian quadrature Ex: (n=3)
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 8 Gaussian quadrature Let : where (Lagrange polynomial)
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2010/3/11 計算機在工程上的應用 ( 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 計算機在工程上的應用 ( JWLee) Page 10 The general integral interval (a,b)
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2010/3/11 計算機在工程上的應用 ( 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 計算機在工程上的應用 ( JWLee) Page 12 Numerical examples Ex : 2 Exact solution : Number of points
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 13 Numerical examples Ex : 3 Exact solution : Number of points
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 14 Fortran program
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 15 Fortran program
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2010/3/11 計算機在工程上的應用 ( JWLee) Page 16 The end Thanks for your kind attentions
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2010/3/11 計算機在工程上的應用 ( 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
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