The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications
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| Format: | Preprint |
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2025
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| author | Jarlebring, Elias Lorentzon, Gustaf |
| author_facet | Jarlebring, Elias Lorentzon, Gustaf |
| contents | We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $Π_{2^{m}}^*$ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $Π_{2^{m}}^*$ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$, which is subsequently proven to be sharp, i.e., $\dim(Π_{2^m}^*)=m^2$. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $Π_{2^{m}}^*$, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications Jarlebring, Elias Lorentzon, Gustaf Numerical Analysis 65F60, 14M99 We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $Π_{2^{m}}^*$ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $Π_{2^{m}}^*$ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$, which is subsequently proven to be sharp, i.e., $\dim(Π_{2^m}^*)=m^2$. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $Π_{2^{m}}^*$, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial. |
| title | The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications |
| topic | Numerical Analysis 65F60, 14M99 |
| url | https://arxiv.org/abs/2504.01500 |