monic.polynomial.recurrences: Create data frame of monic recurrences

Description

This function returns a data frame with parameters required to construct monic orthogonal polynomials based on the standard recurrence relation for the non-monic polynomials. The recurrence relation for monic orthogonal polynomials is as follows. $$q_{k + 1} \left( x \right) = \left( {x - a_k } \right)\;q_k \left( x \right) - b_k \;q_{k - 1} \left( x \right)$$ We require that $q_{-1} \left( x \right) = 0$ and $q_0 \left( x \right) = 1$. The recurrence for non-monic orthogonal polynomials is given by $$c_k \;p_{k + 1} \left( x \right) = \left( {d_k + e_k \;x} \right)\;p_k \left( x \right) - f_k \;p_{k - 1} \left( x \right)$$ We require that $p_{-1} \left( x \right) = 0$ and $p_0 \left( x \right) = 1$. The monic polynomial recurrence parameters, a and b, are related to the non-monic polynomial parameter vectors c, d, e and f in the following manner. $$a_k = - \frac{{d_k }}{{e_k }}$$ $$b_k = \frac{{c_{k - 1} \;f_k }}{{e_{k - 1} \;e_k }}$$ with $b_0 = 0$.

Usage

monic.polynomial.recurrences(recurrences)

Arguments

recurrences

the data frame of recurrence parameter vectors c, d, e and f

Value

A data frame with $n + 1$ rows and two named columns, a and b.

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

Run this code

# NOT RUN {
###
### construct a list of the recurrences for the T Chebyshev polynomials of
### orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
###
### construct the monic polynomial recurrences from the above list
###
m.r <- monic.polynomial.recurrences( r )
# }