CoppersmithWinograd Matrix Multiplication

Suppose n is even and define

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n/2 f(x) = x2i1T21, VER" i=1 (a) Show that the inner product can be re-expressed as n/2 x+y= (x211 + y2i) (tai + y2i1) f(x) f(y), VX,Y ER". i=1 (b) Now consider the matrix multiplication C = AB for A, B e Rxn. Give an algorithm for computing C that uses only m. + O(n2) scalar multiplications. (Notice that a standard matrix multiplication needs n3 scalar multiplications. Together with other techniques, we can obtain Coppersmith-Winograd algorithm whose complexity is only O(n2.375).) n/2 f(x) = x2i1T21, VER" i=1 (a) Show that the inner product can be re-expressed as n/2 x+y= (x211 + y2i) (tai + y2i1) f(x) f(y), VX,Y ER". i=1 (b) Now consider the matrix multiplication C = AB for A, B e Rxn. Give an algorithm for computing C that uses only m. + O(n2) scalar multiplications. (Notice that a standard matrix multiplication needs n3 scalar multiplications. Together with other techniques, we can obtain Coppersmith-Winograd algorithm whose complexity is only O(n2.375).)