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Question 4 [12 marks] Some applications of mathematics require the use of very large matrices (several thousand rows for example) and this in turn directs attention to efficient ways to manipulate them. This question focuses on the efficiency of matrix multiplication, counting the number of numerical arithmetic operations (addition, subtraction and multiplication) involved. We start with very simplest case of 2x2 matrices. (a) The standard way of multiplying 2x2 matrices uses 8 multiplications and 4 additions. List the 8 products for C and thenc alluvy a (b) In a landmark paper of 19697, Volker Strassen surprised the mathematical community by showing how to multiply 2x2 matrices using only 7 multiplications. For the matrix product in (a) the seven products are: P1 = (a + d)(8 +) P3 = 2(t-u) Ps= (a + b) Py = (b-d) (u + v) P2 = (c+d)s P4 = d(u-s) Pe= (c-a)(s+t) ſa blls t w where W.1,y,z are given by: W = P + P4-Ps + P r = P3 + Ps y = P2 + P4 = P1-P2 + P3 + Po- Illustrate the use of this method replacing a,b,c,d with 2,3,5,7, and s, t, u, u with -7,3,5,-2. Show the values of P1...,P and w,r,y,2. Check the result using standard matrix multiplication. (c) Strassen's method saves one multiplication at the cost of adding a lot more addi- tions/subtractions. So at first sight it does not appear to be at all efficient. However, the efficiency improves for larger matrices when the method is combined with a "divide-and-conquer' strategy. This strategy partitions 2n x 2n matrices into nxn quarters so that the multiplcation of a pair of 2nx 2n matrices can be replaced by a series of multiplications and additions of nxn matrices. Consider this 4x4 example: 1102-31 11 012-31 1 2 3-2 31 1 2 3-2 31 3-1 0-2 3-1 0-2 | 3-1 0 2 | 3-1 0 2 M = 1-3 2011-320 1-1 0 1 0 1-1 0 1 0 1 |-3 2 0 1] -3 2 0 1] 0 2 -3-1 1 0 2 -3-1] [102-31 2 3-2 31 [ 0-3 9 61 I 0-3 9 61 3-1 0-2 3-1 0 2 3 6 0 9 MN=1 3 609 WX | 1-3 2 0-1 0 1 011-9 6 0-3 1-960-35 |-3 2 0 1 0 2-3-1 ] [0-9 3-6 | 0-93-6] Using only 2x2 matrices and the standard method [A BIST][WX (not Strassen) of matrix multiplication, verify that: (i.e. verify that AS + BU =W etc.) (d) What is the total number of arithmetic operations (multiplicati of numbers) used to calculate the product MN in (c)? Is there any saving in using partitioning over direct 4x4 multiplication? Show how you arrive at your answer. (e) Generalise your answer to (d) to give a formula for the number of arithmetic operations used to calculate the product of a pair of nxn matrices using the standard method.