orthonormal.polynomials {orthopolynom} | R Documentation |
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}}.
orthonormal.polynomials(recurrences, p.0)
recurrences |
a data frame containing the parameters of the orthonormal polynomial recurrence relations |
p.0 |
{ a polynomial object for the order 0 orthonormal polynomial }
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.
A list of polynomial objects
1 |
Order 0 orthonormal polynomial |
2 |
Order 1 orthonormal polynomial |
n+1 |
Order $n$ orthonormal polynomial |
Frederick Novomestky fnovomes@poly.edu
Abramowitz and Stegun (1968) and Press, et. al. (1992)
r <- chebyshev.t.recurrences( 10, normalized=TRUE ) norm <- sqrt( pi ) p.0 <- polynomial( c( 1 / norm ) ) p.list <- orthonormal.polynomials( r, p.0 )