Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases

The definition, in previous studies, of vector Stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series....

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 384; pp. 21 - 42
Main Author: Van Iseghem, Jeannette
Format: Journal Article
Language:English
Published: New York, NY Elsevier Inc 01.06.2004
Elsevier Science
Subjects:
Algebra
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Hermite–Padé approximants
Linear and multilinear algebra, matrix theory
Mathematics
QD algorithm
Sciences and techniques of general use
Stieltjes continued fraction
Stieltjes series
Theoretical computing
Stieltjes continued fraction
QD algorithm
47B36
41A21
Stieltjes series
Hermite–Padé approximants
11J70
30B70
Band matrix
Diagonal matrix
Series expansion
Sparse matrix
Inverse transformation
Stieltjes transformation
Continued fractions
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:The definition, in previous studies, of vector Stieltjes continued fractions in connection with spectral properties of band operators with intermediate zero diagonals, left unsolved the question of a direct definition of their coefficients in terms of the original data, a vector of Stieltjes series. The subject was more undefined in the matrix case. A new version of the QD algorithm for matrix problem, allows to extend to the vector and matrix cases the result of Stieltjes, expansion of a (scalar) function in terms of a Stieltjes continued fraction. Beside this connection, it solves the inverse Miura transform and gives interesting identities between general band matrix and sparse band matrix. Finally, as a consequence, we extend to some dynamical systems a method known for Toda lattices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2003.12.032