![]() ![]() Strassen’s method, when carefully implemented, can be faster than conventional matrix multiplication for reasonable dimensions. Laderman, Pan, and Sha (1993) explain that for these methods “very large overhead constants are hidden in the ` ‘ notation”, and that the methods “improve on Strassen’s (and even the classical) algorithm only for immense numbers. In the methods that achieve exponents lower than 2.775, various intricate techniques are used, based on representing matrix multiplication in terms of bilinear or trilinear forms and their representation as tensors having low rank. ![]() The current record upper bound on the exponent is, proved by Alman and Vassilevska Williams (2021) which improved on the previous record of, proved by Le Gall (2014) The following figure plots the best upper bound for the exponent for matrix multiplication over time. Since there are elements of data, which must each participate in at least one operation, the exponent of in the operation count must be at least. Strassen’s work sparked interest in finding matrix multiplication algorithms of even lower complexity. The number of additions can be shown to be of the same order of magnitude, so the algorithm requires operations. Denote by the number of scalar multiplications required to multiply two matrices. Let us examine the number of multiplications for the recursive Strassen algorithm. Assuming is a power of, we can partition and into four blocks of size, apply Strassen’s formulas for the multiplication, and then apply the same formulas recursively on the half-sized matrix products. However, the formulas do not rely on commutativity so are valid when the and are matrices, in which case for large dimensions the saving of one multiplication greatly outweighs the extra additions. ![]() The evaluation requires multiplications and additions instead of multiplications and additions for the usual formulas.Īt first sight, Strassen’s formulas may appear simply to be a curiosity. In 1969 Volker Strassen showed that when the product can be computed from the formulas For over a century after the development of matrix algebra in the 1850s by Cayley, Sylvester and others, all methods for matrix multiplication were based on this formula and required operations. Each element of is an inner product of a row of and a column of, so if this formula is used then the cost of forming is additions and multiplications, that is, operations. The definition of matrix multiplication says that for matrices and, the product is given by. ![]()
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