Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f ( T ) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t i i are arranged in inc...

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Bibliographic Details
Published in:Numerical algorithms Vol. 90; no. 3; pp. 905 - 930
Main Authors: Kubelík, P., Kurbatov, V. G., Kurbatova, I. V.
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2022
Springer Nature B.V
Subjects:
Accuracy
Algebra
Algorithms
Analytic functions
Computer Science
Decomposition
Eigenvalues
Mathematical analysis
Numeric Computing
Numerical Analysis
Original Paper
Polynomials
Theory of Computation
Divided differences
Multiprecision arithmetic
Schur–Parlett algorithm
Polynomial interpolation
MSC 97N50
MSC 65F60
Matrix function
Reordering
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f ( T ) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t i i are arranged in increasing order. To avoid calculations using the differences t i i − t j j with close (including equal) t i i and t j j , it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f ( T ) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.
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content type line 14
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-021-01214-6