Gaussian Quadrature We use the following formula with weights w i and - PowerPoint PPT Presentation

Dec 06, 2022 •331 likes •384 views

Gaussian Quadrature We use the following formula with weights w i and nodes x i 1 n f ( x ) dx w i f ( x i ) 1 i =1 The nodes and weights are of the form: n (# of points) x i w i 2 0.5773502691896257 1.0000000000000000

Gaussian Quadrature We use the following formula with weights w i and nodes x i � 1 n � f ( x ) dx ≈ w i f ( x i ) − 1 i =1 The nodes and weights are of the form: n (# of points) x i w i 2 0.5773502691896257 1.0000000000000000 -0.5773502691896257 1.0000000000000000 3 0.7745966692414834 0.5555555555555556 0 0.8888888888888888 -0.7745966692414834 0.5555555555555556 4 0.8611363115940525 0.3478548451374544 0.3399810435848563 0.6521451548625460 -0.3399810435848563 0.6521451548625460 -0.8611363115940525 0.3478548451374544 · · · · · · · · · These can be found in most Numerical analysis texts. September 24, 2019 1 / 4
Matlab implementation We approximate an integral over [ a , b ] as follows: � b n � f dx = w i f (˜ ˜ x i ) a i =1 b − a x i = ( b + a ) + x i ( b − a ) where ˜ w i = w i and ˜ 2 2 The function lgwt computes ˜ w i and ˜ x i for any n on an interval [ a , b ]. [x,w]=lgwt(4,0,pi); We can then specify the function and approximate the integral >> f=sin(x); >> approx_int = sum(w.*f) approx_int = 1.999984228457722 September 24, 2019 2 / 4
Comparison to Composite Trapezoidal rule Gaussian Quadrature is far superior! n Trapeziodal Error Gaussian Error 2 0.429203673205103 0.064180425348863 4 0.103881102062960 1.577154227838662e-05 8 0.025768398054449 4.440892098500626e-15 16 0.006429656227660 4.440892098500626e-16 September 24, 2019 3 / 4
All is not lost for Composite methods! � 2 π e sin( x ) dx 0 n Trapeziodal Error Gaussian Error 2 1.67174121383325e+00 1.52774484383558e+00 4 3.43969188091968e-02 2.02753056344147e-01 8 1.25168893738703e-06 1.222935441335338e-03 16 6.21724893790088e-15 4.724131308364576e-09 September 24, 2019 4 / 4
They perfom quite well for periodic integrals September 24, 2019 4 / 4

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