Fast algorithms for complex matrix multiplication using surrogates

Novel fast algorithms for multiplying square complex matrices are presented. The algorithms are based on concepts from fast methods of complex multiplication in which a surrogate is used for the square root of minus one. Previous methods imposed the structure of a finite ring or field on the problem. The novel algorithms also use a surrogate, but do not require the imposed structure and its inherent rounding. The number of real matrix multiplications required can be reduced from four to two for even dimension, and to 2+1/N² for odd dimension N. The disadvantage of the algorithms is the imposition of a requirement on the structure of one of the two complex matrices being multiplied. The 2×2 case of the algorithm can be adapted to computing Givens rotations, resulting in a 17% savings in real matrix multiplications