orthopolynom (version 1.0-6.1)

polynomial.roots: Create a list of polynomial roots

Description

This function returns a list with \(n\) elements containing the roots of the order $k$ monic orthogonal polynomials for orders \(k = 0,\;1,\; \ldots ,\;n\) using a data frame with the monic polynomial recurrence parameter vectors \(\bf{a}\) and \(\bf{b}\)

Usage

polynomial.roots(m.r)

Value

A list with \(n\) elements each of which is a vector of polynomial roots

1

roots of the order 1 monic polynomial

2

roots of the order 2 monic polynomial

...

n

roots of the order \(n\) monic polynomial

Arguments

m.r

monic recurrence data frame with parameters a and b

Author

Frederick Novomestky

Details

The parameter m.r is a data frame with $n$+1 rows and two names columns. The columns which are names a and b correspond to the above referenced vectors. Function jacobi.matrices is used to create a list of symmetric, tridiagonal Jacobi matrices from these named columns. The eigenvalues of the \(k \times k\) Jacobi matrix are the roots or zeros of the order $k$ monic orthogonal polynomial.

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.

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

See Also

monic.polynomial.recurrences, jacobi.matrices

Examples

###
### generate the recurrences data frame for
### the normalized Chebyshev polynomials of
### orders 0 to 10
###
r <- chebyshev.t.recurrences( 10, normalized=TRUE )
###
### construct the corresponding monic polynomial
### recurrences
###
m.r <- monic.polynomial.recurrences( r )
###
### obtain the polynomial roots from the monic polynomial
### recurrences
p.roots <- polynomial.roots( m.r )