Create list of Jacobi quadrature rules

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Description

This function returns a list with n elements containing the order k quadrature rule data frame for the Jacobi P polynomial for orders k = 1,\;2,\; … ,\;n.

Usage

1
jacobi.p.quadrature.rules(n,alpha,beta,normalized=FALSE)

Arguments

n

integer value for the highest order

alpha

numeric value for the first polynomial parameter

beta

numeric value for the second polynomial parameter

normalized

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 quadrature rule data frame

1

Quadrature rule for the order 1 Jacobi polynomial

2

Quadrature rule for the order 2 Jacobi polynomial

...

n

Quadrature rule for the order n Jacobi polynomial

Author(s)

Frederick Novomestky fnovomes@poly.edu

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

Examples

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###
### construct the list of quadrature rules for
### the Jacobi orthogonal polynomials
### of orders 1 to 5
### alpha = 3 and beta = 2
###
orthogonal.rules <- jacobi.p.quadrature.rules( 5, 3, 2 )
print( orthogonal.rules )
###
### construct the list of quadrature rules for
### the Jacobi orthonormal polynomials
### of orders 1 to 5
### alpha = 3 and beta = 2
###
orthonormal.rules <- jacobi.p.quadrature.rules( 5, 3, 2, TRUE )
print( orthonormal.rules )

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