Gaussian%20quadrature

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Gaussian quadrature

• Advisor J. T. Chen
• Reporter Jia-Wei Lee
• Date Mar., 11, 2010

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Trapezoidal rule
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Gaussian quadrature
f (x) is the polynomial of degree 2n-1 or less
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Legendre polynomial
Legendres equation
Legendre polynomial
Here are n distinct roots of in the
interval (1,-1)
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The orthogonality of Legendre polynomial
vectors
functions
Legendre polynomial
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Gaussian quadrature
f (x) is the polynomial of degree 2n-1 or less
Therefore the integral can be obtained by using n
points
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Gaussian quadrature
Ex (n3)
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Gaussian quadrature
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Weighting coefficients and associated points




Number of points
Associated points
Weighting coefficients
Because the integral interval is from -1 to 1.
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The general integral interval (a,b)
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Numerical examples
Ex 1
Exact solution





Number of points
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Numerical examples
Ex 2
Exact solution
Number of points





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Numerical examples
Ex 3
Exact solution





Number of points
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Fortran program
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Fortran program
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The end
Thanks for your kind attentions
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Gaussian quadrature
f (x) is the polynomial of degree 2n-1 or less
Therefore the integral can be obtained by using n
points