gaussquad (version 1.0-3)

ultraspherical.quadrature.rules: Create list of ultraspherical quadrature rules

Description

This function returns a list with \(n\) elements containing the order \(k\) quadrature rule data frame for the ultraspherical polynomial for orders \(k = 1,\;2,\; \ldots ,\;n\).

Usage

ultraspherical.quadrature.rules(n,alpha,normalized=FALSE)

Arguments

n

integer value for the highest order

alpha

numeric value for the polynomial parameter

normalized

boolean value. if TRUE rules are for orthonormal polynomials, otherwise they are for orthgonal polynomials

Value

A list with \(n\) elements each of which is a data frame

1

Quadrature rule data frame for the order 1 ultraspherical polynomial

2

Quadrature rule data frame for the order 2 ultraspherical polynomial

...

n

Quadrature rule data frame for the order \(n\) ultraspherical polynomial

Details

An order \(k\) quadrature data frame is a named data frame that contains the roots and abscissa values of the corresponding order \(k\) orthogonal polynomial. The column with name x contains the roots or zeros and the column with name w contains the weights.

References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Stroud, A. H., and D. Secrest, 1966. Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ.

See Also

quadrature.rules, ultraspherical.quadrature

Examples

# NOT RUN {
###
### generate a list of quadrature rules for
### the orthogonal ultraspherical polynomial
### of orders 1 to 5
### the polynomial parameter value alpha is 1.0
###
orthogonal.rules <- ultraspherical.quadrature.rules( 5, 1 )
print( orthogonal.rules )
###
### generate a list of quadrature rules for
### the orthonormal ultraspherical polynomial
### of orders 1 to 5
### the polynomial parameter value alpha is 1.0
###
orthonormal.rules <- ultraspherical.quadrature.rules( 5, 1, TRUE )
print( orthonormal.rules )
# }