In this paper we consider deterministic computation of the exact determinant of a dense matrix $M$ of integers. We present a new algorithm with worst case complexity $O\left(n^4(\log n+\log||M||)+n^3\log^2||M||\right)$, where $n$ is the dimension of the matrix and $||M||$ is a bound on the entries in $M$, but with average expected complexity $O\left(n^4+n^3(\log n+\log ||M||)^2\right)$, assuming some plausible properties about the distribution of $M$. We will also describe a practical version of the algorithm and include timing data to compare this algorithm with existing ones. Our result does not depend on fast'' integer or matrix techniques.

### Fast Deterministic Computation of Determinants of Dense Matrices

#### Abstract

In this paper we consider deterministic computation of the exact determinant of a dense matrix $M$ of integers. We present a new algorithm with worst case complexity $O\left(n^4(\log n+\log||M||)+n^3\log^2||M||\right)$, where $n$ is the dimension of the matrix and $||M||$ is a bound on the entries in $M$, but with average expected complexity $O\left(n^4+n^3(\log n+\log ||M||)^2\right)$, assuming some plausible properties about the distribution of $M$. We will also describe a practical version of the algorithm and include timing data to compare this algorithm with existing ones. Our result does not depend on fast'' integer or matrix techniques.
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1999
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/509719
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