Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization

Authors Papri Dey, Ravi Kannan, Nick Ryder, Nikhil Srivastava

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Papri Dey
  • Georgia Tech, Atlanta, GA, USA
Ravi Kannan
  • Microsoft Research, Bangalore, India
Nick Ryder
  • OpenAI, San Francisco, CA, USA
Nikhil Srivastava
  • UC Berkeley, CA, USA


We thank the anonymous referees of a previous version of this paper, whose thoughtful comments greatly improved the presentation. We thank Bill Helton, Clément Pernet, Pablo Parrilo, Mario Kummer, Rafael Oliveira, and Rainer Sinn for helpful discussions, as well as the Simons Institute for the Theory of Computing, where a large part of this work was carried out during the "Geometry of Polynomials" program.

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Papri Dey, Ravi Kannan, Nick Ryder, and Nikhil Srivastava. Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

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ACM Subject Classification
  • Mathematics of computing → Numerical analysis
  • Symbolic algorithms
  • numerical algorithms
  • linear algebra


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