The Chemistry Maths Book, Second Edition

580 Chapter 20Numerical methods

There are a number of Gaussian quadrature formulas appropriate to several kinds

of integrand. The simplest is the Gauss–Legendre(or, simply, Gaussian) formula

(20.40)

in which the pointsx

i

are the zeros of the Legendre polynomialP

n

(x). The weightsw

i

are then chosen to make the formula exact iff(x)is a polynomial of degree 2 n 1 − 11.

Tabulations of the weights and points to many significant figures (at least 15) exist for

many values of n(up to at least 100), and the formula can give very high accuracy for

integrals that can be written in the form given by (20.40). Any integral with a finite

range of integration ato bcan be transformed into one with range−1to+1 by a

change of variable to ugiven by.

EXAMPLE 20.15Gauss–Legendre quadrature for

Changing the variable of integration to ugiven byx 1 = 1 0.6(1 1 + 1 u)gives

The Gauss–Legendre parameters forn 1 = 14 are (to 8 figures)

u

1

1 = 1 −u

4

1 = 1 −0.86113631, w

1

1 = 1 w

4

1 = 1 0.34785485

u

2

1 = 1 −u

3

1 = 1 −0.33998104, w

2

1 = 1 w

3

1 = 1 0.65214515

ThenI 1 ≈ 1 0.381532. The exact value is 0.381536, and Gauss–Legendre integration with

4 points gives greater accuracy than Simpson’s rule with 12 intervals.

Other Gaussian quadrature formulas of the more general form

(20.41)

exist for a variety of integrand weight functionsW(x)and also for infinite integrals.

The more important are

Gauss–Chebyshev:

Gauss–Laguerre:

Gauss–Hermite:

Z

−

+

−

=

≈

∑

∞

∞

fxe dx fx

x

i

n

ii

() ( )

2

1

w

Z

0 1

∞

fxe dx fx

x

i

n

ii

() ( )

−

=

≈

∑

w

Z

−

+

=

−

≈

∑

1

1

2

1

1

1

fx

x

dx f x

i

n

ii

() w ( )

Z

a

b

i

n

ii

fxWxdx() () ≈ fx( )

=

∑

1

w

Iuedu

u

i

n

i

=. + ≈.

−

+

−. +

=

∑

036 1 036

1

1

0361

1

2

Z () (

()

w 11

0361

2

• 
  













−. +

ue

i

u

i

)

()

Ixedx

x

=.

−

Z

0

12. 2

xbauab=−++













1

2

()()

Z

−

+

=

≈

∑

1

1

1

fxdx fx

i

n

ii

() w ( )