Integer Levinson Algorithm for the Inversion of any Nonsingular Hermitian Toeplitz Matrix
This paper presents an integer preserving (IP) version of the Levinson algorithm to solve a normal set of equations for a Hermitian Toeplitz matrix with any singularity profile. The IP property means that for a matrix with integer entries, the algorithm can be completed over the integer solely by us...
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| Veröffentlicht in: | IEEE transactions on information theory 2024-04, Vol.70 (4), p.1-1 |
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| creator | Bistritz, Yuval Dekel, Idan |
| description | This paper presents an integer preserving (IP) version of the Levinson algorithm to solve a normal set of equations for a Hermitian Toeplitz matrix with any singularity profile. The IP property means that for a matrix with integer entries, the algorithm can be completed over the integer solely by using a ring of integer operations. The IP algorithm provides remedies for unpredictable numerical outcomes when a corresponding floating-point (FP) Levinson algorithm either overlooks zero principal minors (PMs) or applies a singularity skipping routine to a PM that is considered erroneously to be zero. The error-free computational edge of integer arithmetic is also applicable to a non-integer Toeplitz matrix by first scaling it up to an acceptably accurate integer matrix. The proposed algorithm can also be used to obtain the inverse of a nonsingular Hermitian Toeplitz matrix (with any singularity profile) by one of two proposed IP Gohberg-Semencul type inversion formulas. |
| doi_str_mv | 10.1109/TIT.2024.3364574 |
| format | Article |
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The IP property means that for a matrix with integer entries, the algorithm can be completed over the integer solely by using a ring of integer operations. The IP algorithm provides remedies for unpredictable numerical outcomes when a corresponding floating-point (FP) Levinson algorithm either overlooks zero principal minors (PMs) or applies a singularity skipping routine to a PM that is considered erroneously to be zero. The error-free computational edge of integer arithmetic is also applicable to a non-integer Toeplitz matrix by first scaling it up to an acceptably accurate integer matrix. 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The proposed algorithm can also be used to obtain the inverse of a nonsingular Hermitian Toeplitz matrix (with any singularity profile) by one of two proposed IP Gohberg-Semencul type inversion formulas.</description><subject>Algorithms</subject><subject>Correlation</subject><subject>Electronic mail</subject><subject>Floating point arithmetic</subject><subject>Gohberg-Semencul inversion formulas</subject><subject>Integer algorithms</subject><subject>Integers</subject><subject>Levinson algorithms</subject><subject>Mathematical models</subject><subject>Matrix decomposition</subject><subject>Numerical stability</subject><subject>Prediction algorithms</subject><subject>Singularities</subject><subject>Symmetric matrices</subject><subject>Toeplitz matrices</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNpNkE1PAjEURRujiYjuXbho4nqwnzN0SYjKJKibceGqaacdKIEW20LEX-8QWLh6uXnnvpccAO4xGmGMxFNTNyOCCBtRWjJesQswwJxXhSg5uwQDhPC4EIyNr8FNSqs-Mo7JAHzVPtuFjXBu986n4OFkvQjR5eUGdiHCvLSw9nsbk-t3oYPKH-B78Mn5xW6tIpzZuHHZKQ-bYLdrl3_hm8rR_dyCq06tk707zyH4fHluprNi_vFaTyfzoiWM54IZrQ1t9ZjalquS44pQo0hFlBLElKJDhotOYW0UVYJrrRWvbEU106ZiyNAheDzd3cbwvbMpy1XYRd-_lESUAnFCCe8pdKLaGFKKtpPb6DYqHiRG8ihQ9gLlUaA8C-wrD6eKs9b-wxlFWJT0D1t8bgk</recordid><startdate>20240401</startdate><enddate>20240401</enddate><creator>Bistritz, Yuval</creator><creator>Dekel, Idan</creator><general>IEEE</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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The IP property means that for a matrix with integer entries, the algorithm can be completed over the integer solely by using a ring of integer operations. The IP algorithm provides remedies for unpredictable numerical outcomes when a corresponding floating-point (FP) Levinson algorithm either overlooks zero principal minors (PMs) or applies a singularity skipping routine to a PM that is considered erroneously to be zero. The error-free computational edge of integer arithmetic is also applicable to a non-integer Toeplitz matrix by first scaling it up to an acceptably accurate integer matrix. 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| subjects | Algorithms Correlation Electronic mail Floating point arithmetic Gohberg-Semencul inversion formulas Integer algorithms Integers Levinson algorithms Mathematical models Matrix decomposition Numerical stability Prediction algorithms Singularities Symmetric matrices Toeplitz matrices |
| title | Integer Levinson Algorithm for the Inversion of any Nonsingular Hermitian Toeplitz Matrix |
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