[weight,abscis,ifail] = d01bbf(d01xxx,a,b<,n,itype,ifail>)
This routine returns the weights and abscissae for use in the
Gaussian quadrature of a function f(x). The quadrature takes the
form
n
--
S= > w f(x )
-- i i
i=1
where w are the weights and x are the abscissae.
i i
Weights and abscissae are available for Gauss-Legendre, Gauss-
Rational, Gauss-Laguerre and Gauss-Hermite quadrature, and for a
selection of values of n.
(a) Gauss-Legendre Quadrature:
b
/
S~= |f(x)dx
/
a
where a and b are finite and it will be exact for any
function of the form
2n-1
-- i
f(x)= > c x
-- i
i=0
(b) Gauss-Rational quadrature:
infty a
/ /
S~= | f(x)dx (a+b>0) or S~= | f(x)dx (a+b<0)
/ /
a -infty
and will be exact for any function of the form
2n-1
-- i
> c (x+b)
2n+1 c -- 2n+1-i
-- i i=0
f(x)= > ------= ------------------
-- i 2n+1
i=2 (x+b) (x+b)
(c) Gauss-Laguerre quadrature, adjusted weights option:
infty a
/ /
S~= | f(x)dx (b>0<) or S~= | f(x)dx (b<0)
/ /
a -infty
and will be exact for any function of the form
2n-1
-bx -- i
f(x)=e > c x
-- i
i=0
(d) Gauss-Hermite quadrature, adjusted weights option:
+infty
/
S~= | f(x)dx
/
-infty
and will be exact for any function of the form
2 2n-1
-b(x-a) -- i
f(x)=e > c x (b>0)
-- i
i=0
(e) Gauss-Laguerre quadrature, normal weights option:
infty a
/ -bx / -bx
S~= | e f(x)dx (b>0) or S~= | e f(x)dx
/ /
a -infty
(b<0)
and will be exact for any function of the form
2n-1
-- i
f(x)= > c x
-- i
i=0
(f) Gauss-Hermite quadrature, normal weights option:
+infty 2
/ -b(x-a)
S~= | e f(x)dx
/
-infty
and will be exact for any function of the form:
2n-1
-- i
f(x)= > c x
-- i
i=0
Note: that the Gauss-Legendre abscissae, with a=-1, b=+1, are the
zeros of the Legendre polynomials; the Gauss-Laguerre abscissae,
with a=0, b=1, are the zeros of the Laguerre polynomials; and the
Gauss-Hermite abscissae, with a=0, b=1, are the zeros of the
Hermite polynomials.
d01bbf
Required Input Arguments:
d01xxx function (NAG-Supplied)
Use:
'd01baz' for Gauss-Legendre weights and ascissae
'd01bay' for Gauss-Rational weights and ascissae
'd01bax' for Gauss-Laguere weights and ascissae
'd01baw' for Gauss-Hermite weights and ascissae
a real
b real
Optional Input Arguments: <Default>
n integer 64
itype integer 0
ifail integer -1
Output Arguments:
weight (n) real
abscis (n) real
ifail integer