Papers › The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications
The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications
Elias Jarlebring, Gustaf Lorentzon
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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_(2ᵐ)^* 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_(2ᵐ)^* and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by m², which is subsequently proven to be sharp, i.e., (Π_(2ᵐ)^*)=m². 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_(2ᵐ)^*, 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.
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