Stability of a method for multiplying complex matrices with three real matrix multiplications

By use of a simple identity, the product of two complex matrices can be formed with three real matrix multiplications and five real matrix additions, instead of the four real matrix multiplications and two real matrix additions required by the conventional approach. This alternative method reduces the number of arithmetic operations, even for small dimensions, achieving a saving of up to 25 percent. The numerical stability of the method is investigated. The method is found to be less stable than conventional multiplication but stable enough to warrant practical use. Issues involved in the choice of method for complex matrix multiplication are discussed, including the relative efficiency of real and complex arithmetic and the backward stability of block algorithms. Key words, matrix multiplication, complex matrix, Strassen’s method, Winograd’s identity, numerical stability, error analysis, level-3 BLAS AMS(MOS) subject classifications. 65F05, 65G05