Algorithms for Matrix Multiplication

Strassen’s and Winograd’s algorithms for n × n matrix multiplication are investigated and compared with the normal algorithm. The normal algorithm requires n3 + O(n2) multiplications and about the same number of additions. Winograd’s algorithm almost halves the number of multiplications at the expense of more additions. Strassen’s algorithm reduces the total number of operations to O(n2.82) by recursively multiplying 2n× 2n matrices using seven n× n matrix multiplications. Floating-point error bounds are obtained for both Strassen’s and Winograd’s methods. It is shown that Strassen’s method satisfies a certain numerical stability property (albeit weaker than that satisfied by the normal method); and that scaling is essential for numerical accuracy using Winograd’s method. In practical cases, for moderate n, Winograd’s method appears to be slightly faster than the other two methods, but the gain is, at most, about 20 percent. Strassen’s method should be faster for sufficiently large n, and this will be important in the future as memory sizes increase. An attempt to generalize Strassen’s method is described.