dim1_gauss_wres returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite.
Weights and abscissae are available for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite quadrature, and for a selection of values of (see Section 5).
(a)Gauss–Legendre Quadrature:
where and are finite and it will be exact for any function of the form
(b)Rational Gauss quadrature, adjusted weights:
and will be exact for any function of the form
(c)Gauss–Laguerre quadrature, adjusted weights:
and will be exact for any function of the form
(d)Gauss–Hermite quadrature, adjusted weights:
and will be exact for any function of the form
(e)Gauss–Laguerre quadrature, normal weights:
and will be exact for any function of the form
(f)Gauss–Hermite quadrature, normal weights:
and will be exact for any function of the form
Note: the Gauss–Legendre abscissae, with ,
, are the zeros of the Legendre polynomials; the Gauss–Laguerre abscissae, with , , are the zeros of the Laguerre polynomials; and the Gauss–Hermite abscissae, with , , are the zeros of the Hermite polynomials.
4References
Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press
Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley
Ralston A (1965) A First Course in Numerical Analysis pp. 87–90 McGraw–Hill
Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall
On entry: , the number of weights and abscissae to be returned.
Constraint:
, , , , , , , , , , , , , , or .
Note: if and is not a member of the above list, the maxmium value of stored below will be used, and all subsequent elements of abscis and weight will be returned as zero.
Optional parameter container, derived from Optional.
6Exceptions and Warnings
Errors or warnings detected by the function:
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of
opt.ifail, depending on how errors
and warnings are being handled (see Error Handling for more details).