Levinson-type algorithms for polynomial fitting and for Cholesky and Q factors of Hankel and Vandermonde matrices

Abstract

This paper presents Levinson (1947)-type algorithms for (i) polynomial fitting (ii) obtaining a Q decomposition of Vandermonde matrices and a Cholesky factorization of Hankel matrices (iii) obtaining the inverse of Hankel matrices. The algorithm for the least-squares solution of Hankel systems of equations requires 3n2+9n+3 multiply and divide operation (MDO). The algorithm for obtaining an orthogonal representation of an (m×n) Vandermonde matrix X and computing the Cholesky factors F of Hankel matrices requires 5mn+n2 +2n-3m MDO, and the algorithm for generating the inverse of Hankel matrices requires 3(n2+n-2)/2 MDO. Our algorithms have been tested by means of fitting of polynomials of various orders and Fortran versions of all subroutines are provided in the Appendix.