arXiv subjects

Gustaf Lorentzon

Publications and source records attributed to Gustaf Lorentzon.

2 recordsLinked to original sources

arXiv2026

Recursive expansion of the matrix step function using polynomials of degree eight

We consider the problem of efficiently computing the matrix step function of a large dense symmetric matrix. To this end, we introduce a recursive polynomial expansion method in which a composite polynomial of high degree is built recursively from component polynomials of degree eight. The component polynomial used in each iteration is designed to achieve strong amplification of the spectral gap across the step while favorably positioning the updated gap for subsequent iterations. A key ingredient is a novel evaluation scheme for arbitrary matrix polynomials of degree exactly eight requiring only three matrix-matrix multiplications and three matrices in memory. This scheme makes available a substantially larger class of component polynomials than previously possible within a three-multiplication budget, thereby expanding the class of composite polynomials that can be generated. Together with our polynomial selection strategy, this leads to a significant and consistent reduction in the number of matrix-matrix multiplications required to compute the matrix step function compared to existing recursive expansion methods.

math.NA↗

arXiv2025

The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications

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 $\Pi_{2^{m}}^*$ 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 $\Pi_{2^{m}}^*$ 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., $\dim(\Pi_{2^m}^*)=m^2$. 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 $\Pi_{2^{m}}^*$, 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.

math.NA↗