gaussian quadratures

Gaussian Gaussian 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=1

m ci f(xi)A formula with m function evaluations requires specification of

2m numbers ci and xi• Gaussian Quadratures

– select both these weights and locations so that a higher order polynomial can be integrated (alternatively the error is proportional to 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]

Gaussian Quadrature on [-1, 1]

• Two function evaluations:– Choose (c1, c2, x1, x2) such that the method yields “exact

integral” for f(x) = x0, x1, x2, x3

)x(fc)x(fc)x(fc)x(fcdx)x(f nn2211i

1

1

n

1ii +++=≈∫ ∑−

=

L

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

2211

1

1

+=

= ∫−

x2x1-1 1

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 2m points

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

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

Exact integral for f = x0, x1, x2, x3

– Four equations for four unknowns

)f(xc)f(xcf(x)dx :2n 2211

1

1+== ∫−

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

−=

==

⇒

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+==⇒=

+==⇒=

+==⇒=

+==⇒=

∫

∫∫∫

−

−

−

−

31x

31x

1c1c

xcxc0dxx xf

xcxc32dxx xf

xcxc0xdx xf

cc2dx1 1f

2

1

2

1

322

31

1

1 133

222

21

1

1 122

221

1

1 1

2

1

1 1

)3

1(f)3

1(fdx)x(fI1

1+−== ∫−

Error

• If we approximate a function with a Gaussian quadrature formula we cause an error proportional to 2n thderivative


• Choose (c1, c2, c3, x1, x2, x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5

)x(fc)x(fc)x(fcdx)x(f :3n 332211

1

1++== ∫−

x3x1-1 1x2


533

522

511

1

1

55

433

422

411

1

1

44

333

322

311

1

1

33

233

222

211

1

1

22

332211

1

1

321

1

1

0

52

0

32

0

21

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcdxxxf

xcxcxcxdxxf

cccxdxf

++==⇒=

++==⇒=

++==⇒=

++==⇒=

++==⇒=

++==⇒=

∫

∫

∫

∫

∫

∫

−

−

−

−

−

−

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

=−=

===

⇒

5/3

05/3

9/59/89/5

3

2

1

3

2

1

x

xx

ccc


Exact integral for f = x0, x1, x2, x3, x4, x5

)53(f

95)0(f

98)

53(f

95dx)x(fI

1

1++−== ∫−

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

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

t2t1a b

∫∫∫ −−=

−++

−=

1

1

1

1

b

adx)x(gdx)

2ab)(

2abx

2ab(fdt)t(f

⎩⎨⎧

=⇒==⇒−=

++

−=

bt 1xat1x2

abx2

abt

Example: Gaussian QuadratureExample: Gaussian QuadratureEvaluate

Coordinate transformation

Two-point formula

33.34%)( 543936.3477376279.3468167657324.9

e)3

44(e)3

44()3

1(f)31(fdx)x(fI 3

443

441

1

==+=

++−=+−

==+−

−∫ε

926477.5216dtteI4

0

t2 == ∫

∫∫∫ −−

+ =+==

=+=+

+−

=

1

1

1

1

444

0

2 )()44(

2dxdt ;2222

dxxfdxexdtteI

xabxabt

xt

Example: Gaussian QuadratureExample: Gaussian QuadratureThree-point formula

Four-point formula4.79%)( 106689.4967

)142689.8589(95)3926001.218(

98)221191545.2(

95

e)6.044(95e)4(

98e)6.044(

95

)6.0(f95)0(f

98)6.0(f

95dx)x(fI

6.0446.04

1

1

==

++=

+++−=

++−==

+−

−∫

ε

[ ][ ]

%)37.0( 54375.5197 )339981.0(f)339981.0(f652145.0 )861136.0(f)861136.0(f34785.0dx)x(fI

1

1

==+−+

+−== ∫−

ε

Other rules

• Gauss-Lobatto:– requiring end points be included in the formula

• Gauss-Radau– Require one end point be in the formula