# The Matrix Computation Toolbox

### Nick Higham (view profile)

11 Sep 2002 (Updated )

A collection of M-files for carrying out various numerical linear algebra tasks.

strassenw(A, B, nmin)
```function C = strassenw(A, B, nmin)
%STRASSENW Strassen's fast matrix multiplication algorithm (Winograd variant).
%          C = STRASSENW(A, B, NMIN), where A and B are matrices of dimension
%          a power of 2, computes the product C = A*B.
%          Winograd's variant of Strassen's algorithm is
%          used recursively until dimension <= NMIN is reached,
%          at which point standard multiplication is used.
%          The default is NMIN = 8 (which minimizes the total number of
%          operations).

%          Reference:
%          N. J. Higham, Accuracy and Stability of Numerical Algorithms,
%          Second edition, Society for Industrial and Applied Mathematics,
%          Philadelphia, PA, 2002; chap. 23.

if nargin < 3, nmin = 8; end

n = length(A);
if n ~= 2^( log2(n) )
error('The matrix dimension must be a power of 2.')
end

if n <= nmin
C = A*B;
else
m = n/2; i = 1:m; j = m+1:n;

S1 = A(j,i) + A(j,j);
S2 = S1 - A(i,i);
S3 = A(i,i) - A(j,i);
S4 = A(i,j) - S2;
S5 = B(i,j) - B(i,i);
S6 = B(j,j) - S5;
S7 = B(j,j) - B(i,j);
S8 = S6 - B(j,i);

M1 = strassenw( S2, S6, nmin);
M2 = strassenw( A(i,i), B(i,i), nmin);
M3 = strassenw( A(i,j), B(j,i), nmin);
M4 = strassenw( S3, S7, nmin);
M5 = strassenw( S1, S5, nmin);
M6 = strassenw( S4, B(j,j), nmin);
M7 = strassenw( A(j,j), S8, nmin);

T1 = M1 + M2;
T2 = T1 + M4;

C = [ M2+M3 T1+M5+M6; T2-M7  T2+M5 ];

end
```