Lecture 11 { Fast matrix multiplication

The main topic of this lecture is fast matrix multiplication. This topic is covered very well in textbooks, so the notes will be more sketchy than usual, and are meant mainly to record the topics covered. 1.1 Multiplying complex numbers Given two complex numbers, a + ıb and c + ıd, we wish to compute their product (ac − bd)+ ı(ad+ bc). Hence we need to compute two values, M1 = (ac − bd) and M2 = (ad+ bc). We assume that multiplication of real numbers is much more expensive then their addition or subtraction, and hence we wish to minimize the number of multiplications. The naive computation of M1 and M2 uses four multiplications. This can be reduced to three as follows. Compute P1 = ac, P2 = bd, and P3 = (a + b)(c + d). Now M1 = P1 − P2 and M2 = P3−P1−P2. We have used three multiplications, two additions (for computing P3), and three subtractions. 1.2 A more general scenario More generally, we may study a scenario in which we are given two types of input variables