orthopolynom (version 1.0-2)

orthonormal.polynomials: Create orthonormal polynomials

Description

Create list of orthonormal polynomials from the following recurrence relations for $k$ = 0, 1,...$n$. $$c_k p_{k+1}\left( x \right) = \left( d_k + e_k x \right) p_k \left( x \right) - f_k p_{k-1} \left( x \right)$$ We require that $p_{-1} \left( x \right) = 0$ and $p_0 \left( x \right) = 1$. 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

Value

• A list of polynomial objects
• 1
  Order 0 orthonormal polynomial
• 2
  Order 1 orthonormal polynomial
• ...
• n+1
  Order $n$ 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.

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 )