Question

1. [7] Strassen's Algorithm: a. [3] Given two 2-by-2 matrices A and B: $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$ $C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}$ (i) [1] Let C = A x B. Find an expression for each value of $c_{ij}$ in C? (ii) [2] Verify that Strassen's formula for computing the C values produces the same results as in (i). You may use the formulas for m given in the notes. b. [4] Use Strassen's algorithm to compute the product of two 4x4 matrix given below. Exit the recursion when $n = 2$. Compute the product of 2-by- 2 matrices by brute force (standard) algorithm. $A = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 4 & 1 & 1 & 0 \\ 0 & 1 & 3 & 0 \\ 5 & 0 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 2 & 1 & 0 & 4 \\ 2 & 0 & 1 & 1 \\ 1 & 3 & 5 & 0 \end{bmatrix}$

          1. [7] Strassen's Algorithm:
a. [3] Given two 2-by-2 matrices A and B:
$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$
$C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}$
(i) [1] Let C = A x B. Find an expression for each value of $c_{ij}$ in C?
(ii) [2] Verify that Strassen's formula for computing the C values
produces the same results as in (i). You may use the formulas for m
given in the notes.
b. [4] Use Strassen's algorithm to compute the product of two 4x4 matrix
given below. Exit the recursion when $n = 2$. Compute the product of 2-by-
2 matrices by brute force (standard) algorithm.
$A = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 4 & 1 & 1 & 0 \\ 0 & 1 & 3 & 0 \\ 5 & 0 & 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 2 & 1 & 0 & 4 \\ 2 & 0 & 1 & 1 \\ 1 & 3 & 5 & 0 \end{bmatrix}$

1. [7] Strassen's Algorithm:
a. [3] Given two 2-by-2 matrices A and B:
A =
< b m a t r i x > and B =
< b m a t r i x >
C =
< b m a t r i x >
(i) [1] Let C = A x B. Find an expression for each value of cij in C?
(ii) [2] Verify that Strassen's formula for computing the C values
produces the same results as in (i). You may use the formulas for m
given in the notes.
b. [4] Use Strassen's algorithm to compute the product of two 4x4 matrix
given below. Exit the recursion when n = 2. Compute the product of 2-by-
2 matrices by brute force (standard) algorithm.
A =
< b m a t r i x >
, B =
< b m a t r i x >

Added by Gema R.

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