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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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Presentation on theme: "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."— Presentation transcript:

1 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

2 2010/3/11 計算機在工程上的應用 ( JWLee) Page 2 Trapezoidal rule where is weighting coefficient

3 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

4 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)

5 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

6 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

7 2010/3/11 計算機在工程上的應用 ( JWLee) Page 7 Gaussian quadrature Ex: (n=3)

8 2010/3/11 計算機在工程上的應用 ( JWLee) Page 8 Gaussian quadrature Let ： where (Lagrange polynomial)

9 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.

10 2010/3/11 計算機在工程上的應用 ( JWLee) Page 10 The general integral interval (a,b)

11 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

12 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

13 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

14 2010/3/11 計算機在工程上的應用 ( JWLee) Page 14 Fortran program

15 2010/3/11 計算機在工程上的應用 ( JWLee) Page 15 Fortran program

16 2010/3/11 計算機在工程上的應用 ( JWLee) Page 16 The end Thanks for your kind attentions

17 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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