Question

Given arrays of the following sizes, determine the optimal order of multiplication. Show the arrays M and P. Az x Az x A3 x AA 3* 4 4*5 5 7 7*3

Answer 1

Suppose there are 2 arrays X and Y. Size of x = a*b Size of y = 6*c Size of resulting array Xxy = a*c Number of operations required for XxY= axbxc We have to find the order in which A,xAz «Az xA4 should be multiplied where minimum number of operations are required. Since matrix multiplication is associative, we can do the multiplication in any order. So, we will find number of operations required for each order 1) A, x ((A2xA3) x Au) For A₂ xAz , no. of ops = 4 x 5 x 7 = 140 Size of AzxAz = 4*7 For (A₂xAz) xA4, no. of ops = 140+ (4x7x 3) = 224 Size of A,xA3)xA4 = 4*3 For A,x ((A,xA3)xA4), no. of Obs-224 + (3x4x 3) = 260.

2) (A, + (A₂xA;)) xA4 For (A, + (A₂xA;)), no. of ops=140+ 3x4x7 = 224 Size of (A, x(A,xAx)) 3*7 For (A, x(A2xAz) x Ay, no. of obs = 224 + 3x 7x3 - 287 3) AixAz) x (Az x Ay) For A,xAz , no. of obs = 3x4x5 = 60 Size of A,xA2 = 3*5 For Azxd4 , no. of ops = 5x7x3 = 105 Size of Az xA4 = 5*3 For A, xA2)x (Az xAw), no. of obs = 60+105+ 3x5x3= 210 4) A,x (Az x (AzxAy)) For (A2 x (AzxAy)), no of obs= 105+ 4x5x3=165 Size of (Az x (A3xAy)) = 4*3 For A,x (A2x (AzxAy))), no. of ops = 165+ 3x4x3= 2011 5) ((A, ₃A₂) xA₃) xA4 For (AixA)) xA3, no. of eps = 60 + 3x5x7 = 165 Size of (AxA2)xA3 = 347 For A,xA2)xA37x Ay, no. of obs=165+ 3x 7x3= 228.

So, optimal order of multiplication is : (A,+A₂) × (A₃xA4) with 210 operations M= A, XA₂ IP= AzxA4