orthonormal.polynomials {orthopolynom}R Documentation

Create orthonormal polynomials

Description

Create list of orthonormal polynomials from the following recurrence relations for $k$ = 0, 1,...$n$.

c_k p_{k+1}( x ) = ( d_k + e_k x ) p_k ( x ) - f_k p_{k-1} ( x )

We require that p_{-1} ( x ) = 0 and p_0 ( x ) = 1. The coefficients are the column vectors {bf{c}}, {bf{d}}, {bf{e}} and {bf{f}}.

Usage

orthonormal.polynomials(recurrences, p.0)

Arguments

recurrences a data frame containing the parameters of the orthonormal polynomial recurrence relations
p.0

{ a polynomial object for the order 0 orthonormal polynomial }

Details

The argument is a data frame with $n$+1 rows and four named columns. The column names are c, d, e and f. These columns correspond to the column vectors described above.

Value

A list of polynomial objects

1 Order 0 orthonormal polynomial
2 Order 1 orthonormal polynomial
n+1 Order $n$ orthonormal polynomial

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Abramowitz and Stegun (1968) and Press, et. al. (1992)

Examples

r <- chebyshev.t.recurrences( 10, normalized=TRUE )
norm <- sqrt( pi )
p.0 <- polynomial( c( 1 / norm ) )
p.list <- orthonormal.polynomials( r, p.0 )

[Package orthopolynom version 1.0-1 Index]