orthopolynom (version 1.0-5)

orthogonal.polynomials: Create orthogonal polynomials

Description

Create list of orthogonal polynomials from the following recurrence relations for \(k = 0,\;1,\; \ldots ,\;n\).

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

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

orthogonal.polynomials(recurrences)

Arguments

recurrences

a data frame containing the parameters of the orthogonal polynomial recurrence relations

Value

A list of \(n + 1\) polynomial objects

1

Order 0 orthogonal polynomial

2

Order 1 orthogonal polynomial

...

n+1

Order \(n\) orthogonal 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, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.

Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

Examples

# NOT RUN {
###
### generate the recurrence relations for T Chebyshev polynomials of orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=FALSE )
print( r )
###
### generate the list of orthogonal polynomials
###
p.list <- orthogonal.polynomials( r )
print( p.list )
# }