The triangular decomposition of Hankel matrices

Author:
James L. Phillips

Journal:
Math. Comp. **25** (1971), 559-602

MSC:
Primary 65F30

DOI:
https://doi.org/10.1090/S0025-5718-1971-0295553-0

MathSciNet review:
0295553

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Abstract: An algorithm for determining the triangular decomposition of a Hankel matrix *H* using operations is derived. The derivation is based on the Lanczos algorithm and the relation between orthogonalization of vectors and the triangular decomposition of moment matrices. The algorithm can be used to compute the three-term recurrence relation for orthogonal polynomials from a moment matrix.

**[1]**Walter Gautschi,*Construction of Gauss-Christoffel quadrature formulas*, Math. Comp.**22**(1968), 251–270. MR**0228171**, https://doi.org/10.1090/S0025-5718-1968-0228171-0**[2]**Alston S. Householder,*The theory of matrices in numerical analysis*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0175290****[3]**Cornelius Lanczos,*An iteration method for the solution of the eigenvalue problem of linear differential and integral operators*, J. Research Nat. Bur. Standards**45**(1950), 255–282. MR**0042791****[4]**William F. Trench,*An algorithm for the inversion of finite Hankel matrices*, J. Soc. Indust. Appl. Math.**13**(1965), 1102–1107. MR**0189232**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0295553-0

Keywords:
Hankel matrix,
triangular decomposition of matrices,
orthogonalization,
orthogonal polynomials,
three-term recurrence relations,
moment matrix,
Lanczos algorithm

Article copyright:
© Copyright 1971
American Mathematical Society