ACM Transactions on Mathematical Software· 2026Q1
The 2M Multiplication Algorithm for Complex Matrices
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- Q1SCImago
- 2026year
Short summary
A new 2M algorithm performs complex matrix multiplication using only 2 real matrix multiplications (GEMMs) of the same size, a reduction from the standard 4M and the previous 3M.
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Key points
- Introduces the 2M algorithm for complex matrix multiplication, requiring only 2 real GEMMs.
- Reduces computational cost compared to the standard 4M and the 3M algorithms.
- Achieves practical, high-performance results for floating-point matrices when combined with the Ozaki-II scheme.
- Derives new algorithms for symmetric rank-k updates (SYRK/HERK) as corollaries.
AI-generated from the title and abstract; the full text is not read.
Abstract
Complex matrix multiplication is typically computed using 4 real matrix multiplications (GEMMs) of the same size. The well-known 3M multiplication algorithm reduces this cost to 3 real GEMMs, together with quadratic time pre- and post-processing steps. In this paper, we reduce 3M to 2M for matrices with integer real and imaginary parts, performing complex GEMM with only 2 real GEMMs of the same size, along with quadratic time pre- and post-processing. For floating-point matrices, 2M multiplication combines naturally with the Ozaki-II scheme, yielding a practical, high-performance algorithm for computing a complex floating-point GEMM in roughly twice the time of a real GEMM of the same size. As corollaries, we derive new algorithms for symmetric rank- \(k\) updates ( SYRK / HERK ) that internally use full rectangular GEMMs.
The authors' abstract, as published at the source. ACM Transactions on Mathematical Software, 2026 · DOI ↗
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Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science