Acceleration of complex matrix multiplication using arbitrary precision floating-point arithmetic

TL;DR

This paper extends optimization techniques for complex matrix multiplication to arbitrary precision, demonstrating potential speed improvements through benchmarks and applying these methods to LU decomposition.

Contribution

It introduces the extension of matrix multiplication optimization methods to arbitrary precision complex numbers and verifies their effectiveness via benchmarks.

Findings

Speed-up observed in complex matrix multiplication with optimization methods

Ozaki scheme improves LU decomposition performance

Benchmark results confirm increased computational efficiency

Abstract

Efficient multiple precision linear numerical computation libraries such as MPLAPACK are critical in dealing with ill-conditioned problems. Specifically, there are optimization methods for matrix multiplication, such as the Strassen algorithm and the Ozaki scheme, which can be used to speed up computation. For complex matrix multiplication, the 3M method can also be used, which requires only three multiplications of real matrices, instead of the 4M method, which requires four multiplications of real matrices. In this study, we extend these optimization methods to arbitrary precision complex matrix multiplication and verify the possible increase in computation speed through benchmark tests. The optimization methods are also applied to complex LU decomposition using matrix multiplication to demonstrate that the Ozaki scheme can be used to achieve higher computation speeds.