License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ITCS.2023.42
URN: urn:nbn:de:0030-drops-175450
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Dey, Papri ; Kannan, Ravi ; Ryder, Nick ; Srivastava, Nikhil

Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization

LIPIcs-ITCS-2023-42.pdf (0.7 MB)


We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An Õ(n^{ω+3}a+n⁴a²+n^ωlog(1/ε)) time algorithm for finding an ε-approximation to the Jordan Normal form of an integer matrix with a-bit entries, where ω is the exponent of matrix multiplication. (2) An Õ(n⁶d⁶a+n⁴d⁴a²+n³d³log(1/ε)) time algorithm for ε-approximately computing the spectral factorization P(x) = Q^*(x)Q(x) of a given monic n× n rational matrix polynomial of degree 2d with rational a-bit coefficients having a-bit common denominators, which satisfies P(x)⪰0 for all real x. The first algorithm is used as a subroutine in the second one.
Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best running time was an unspecified polynomial in n of degree at least twelve [Cai, 1994]. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself.

BibTeX - Entry

  author =	{Dey, Papri and Kannan, Ravi and Ryder, Nick and Srivastava, Nikhil},
  title =	{{Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{42:1--42:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-175450},
  doi =		{10.4230/LIPIcs.ITCS.2023.42},
  annote =	{Keywords: Symbolic algorithms, numerical algorithms, linear algebra}

Keywords: Symbolic algorithms, numerical algorithms, linear algebra
Collection: 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)
Issue Date: 2023
Date of publication: 01.02.2023

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