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,; … ,;n.
boolean value. if TRUE rules are for orthonormal
polynomials, otherwise they are for orthgonal polynomials
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.
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
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
###
### 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 )