Gaussian Quad

8/11/2019 Gaussian Quad

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GaussianGaussian QuadraturesQuadratures Newton-Cotes Formulae use evenly-spaced functional values

Did not use the flexibility we have to select the quadrature points

In fact a quadrature point has several degrees of freedom.Q(f)=i=1m cif(xi)

A formula with m function evaluations requires specification of2m numbers ci andxi

Gaussian Quadratures select both these weights and locations so that a higher order

polynomial can be integrated (alternatively the error is proportionalto a higher derivatives)

Price: functional values must now be evaluated at non-

uniformly distributed points to achieve higher accuracy Weights are no longer simple numbers

Usually derived for an interval such as [-1,1]

Other intervals [a,b] determined by mapping to [-1,1]


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Gaussian Quadrature on [-1, 1]

Two function evaluations:

Choose (c1, c2, x1, x2) such that the method yields exactintegral for f(x) = x0, x1, x2, x3

)x(fc)x(fc)x(fc)x(fcdx)x(f nn2211i1

1

n

1i

i

=

L

)f(xc)f(xc

f(x)dx:2n

2211

1

1

=

x2x1-1 1


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Finding quadrature nodes and weights

One way is through the theory of orthogonal

polynomials.

Here we will do it via brute force Set up equations by requiring that the 2mpoints

guarantee that a polynomial of degree 2m-1 is integrated

exactly. In general process is non-linear

(involves a polynomial function involving the unknown point

and its product with unknown weight) Can be solved by using a multidimensional nonlinear solver

Alternatively can sometimes be done step by step


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Gaussian Quadrature onGaussian Quadrature on [[--1, 1]1, 1]

Exact integral forf = x0, x1, x2, x3

Four equations for four unknowns

)f(xc)f(xcf(x)dx:2n 22111

1

=

=

=

=

3

1x

3

1x

1c

1c

xcxc0dxxxf

xcxc3

2dxxxf

xcxc0xdxxf

cc2dx11f

2

1

2

1

3

22

3

1

1

1 1

33

2

22

2

1

1

11

22

221

1

1 1

2

1

11

)3

1(f)

3

1(fdx)x(fI

1

1


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Error

If we approximate a function with a Gaussian quadrature

formula we cause an error proportional to 2n th

derivative


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Choose (c1, c2, c3, x1, x2, x3) such that the methodyields exact integral forf(x) = x0, x1, x2,x3,x4, x5

)x(fc)x(fc)x(fcdx)x(f:3n 3322111

1

x3x1-1 1x2


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5

33

5

22

5

11

1

1

55

433

422

411

1

1

44

3

33

3

22

3

11

1

1

33

2

33

2

22

2

11

1

1

22

332211

1

1

321

1

1

0

52

0

3

2

0

21

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcxdxxf

cccxdxf

++===

++===

++===

++===

++===

++===

=

=

=

=

=

=

5/3

0

5/3

9/5

9/8

9/5

3

2

1

3

2

1

x

x

x

c

c

c


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Exact integral forf = x0, x1, x2, x3,x4, x5

)5

3(f9

5)0(f9

8)5

3(f9

5dx)x(fI1

1


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Gaussian Quadrature onGaussian Quadrature on [a, b][a, b]

Coordinate transformation from [a,b] to [-1,1]

t2t1a b

==

1

1

1

1

b

adx)x(gdx)

2

ab)(

2

abx

2

ab(fdt)t(f

=

=

=

bt1xat1x

2

abx

2

abt


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Example: Gaussian QuadratureExample: Gaussian Quadrature

Evaluate

Coordinate transformation

Two-point formula

33.34%)(543936.3477376279.3468167657324.9

e)3

44(e)

3

44()

3

1(f)

3

1(fdx)x(fI 3

44

3

441

1

=

=

926477.5216dtteI 40

t2=

+ =+==

=+=++=

1

1

1

1

444

0

2 )()44(

2dxdt;2222

dxxfdxexdtteI

xabxabt

xt


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Example: Gaussian QuadratureExample: Gaussian QuadratureThree-point formula

Four-point formula

4.79%)(106689.4967

)142689.8589(9

5)3926001.218(

9

8)221191545.2(

9

5

e)6.044(95e)4(

98e)6.044(

95

)6.0(f9

5)0(f

9

8)6.0(f

9

5dx)x(fI

6.0446.04

1

1

=

%)37.0(54375.5197

)339981.0(f)339981.0(f652145.0

)861136.0(f)861136.0(f34785.0dx)x(fI1

1

=


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Other rules

Gauss-Lobatto:

requiring end points be included in the formula

Gauss-Radau Require one end point be in the formula

Gaussian Quad

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