computational complexity

, Volume 24, Issue 4, pp 777–821 | Cite as

On the complexity of inverting integer and polynomial matrices

  • Arne Storjohann


An algorithm is presented that probabilistically computes the exact inverse of a nonsingular n × n integer matrix A using \({({n^3(\log||A||+\log \kappa(A)))}^{1+o(1)}}\) bit operations. Here, \({||A||= \max_{ij}|A_{ij}|}\) denotes the largest entry in absolute value, \({\kappa(A) := n ||A^{-1}||\,||A||}\) is the condition number of the input matrix, and the “+o(1)” in the exponent indicates a missing factor \({c_1 {(\log n)}^{c_2}{({\rm loglog} ||A||)}^{c_3}}\) for positive real constants c 1, c 2, c 3. A variation of the algorithm is presented for polynomial matrices that computes the inverse of a nonsingular n × n matrix whose entries are polynomials of degree d over a field using \({{(n^3d)}^{1+o(1)}}\) field operations. Both algorithms are randomized of the Las Vegas type: failure may be reported with probability at most 1/2, and if failure is not reported, then the output is certified to be correct in the same running time bound.


Integer matrix polynomial matrix matrix inverse Smith normal form bit complexity randomized algorithm 

Subject classification

68W30 15A35 


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© Springer Basel 2015

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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