Computing the roots of complex orthogonal and kernel polynomials

Journal Article · Fri Jan 01 04:00:00 EST 1988 · SIAM J. Sci. Stat. Comput.; (United States)

DOI:https://doi.org/10.1137/0909001· OSTI ID:5434032

Saylor, P E; Smolarski, D C

A method is presented to compute the roots of complex orthogonal and kernel polynomials. An important application of complex kernel polynomials is the acceleration of iterative methods for the solution of nonsymmetric linear equations. In the real case, the roots of orthogonal polynomials coincide with the eigenvalues of the Jacobi matrix, a symmetric tridiagonal matrix obtained from the defining three-term recurrence relationship for the orthogonal polynomials. In the real case kernel polynomials are orthogonal. The Stieltjes procedure is an algorithm to compute the roots of orthogonal and kernel polynomials bases on these facts. In the complex case, the Jacobi matrix generalizes to a Hessenberg matrix, the eigenvalues of which are roots of either orthogonal or kernel polynomials. The resulting algorithm generalizes the Stieljes procedure. It may not be defined in the case of kernel polynomials, a consequence of the fact that they are orthogonal with respect to a nonpositive bilinear form. (Another consequence is that kernel polynomials need not be of exact degree.) A second algorithm that is always defined is presented for kernel polynomials. Numerical examples are described.

• Research Organization: Dept. of Computer Science, Univ. of Illinois at Urbana-Champaign, Urbana, IL 61801
• OSTI ID: 5434032
• Journal Information: SIAM J. Sci. Stat. Comput.; (United States), Journal Name: SIAM J. Sci. Stat. Comput.; (United States) Vol. 9:1; ISSN SIJCD
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990220* -- Computers
Computerized Models
& Computer Programs-- (1987-1989)
ALGORITHMS
DATA
EIGENVALUES
FUNCTIONS
INFORMATION
ITERATIVE METHODS
MATHEMATICAL LOGIC
MATRICES
NUMERICAL DATA
NUMERICAL SOLUTION
ORTHOGONAL TRANSFORMATIONS
POLYNOMIALS
TRANSFORMATIONS