## Abstract

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.

Original language | English |
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Pages (from-to) | 3018-3031 |

Number of pages | 14 |

Journal | IEEE Transactions on Information Theory |

Volume | 70 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2024 |

## Keywords

- Gohberg-Semencul inversion formulas
- Levinson algorithms
- Toeplitz matrices
- integer algorithms