User:Fawly/Computational complexity of matrix multiplication: Difference between revisions

Revision as of 21:51, 23 May 2021

Revision as of 21:53, 23 May 2021

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The simplest approach to computing the product of two {{math|''n'' × ''n''}} matrices ''A'' and ''B'' is to compute the arithmetic expressions coming from the definition of matrix multiplication. In pseudocode:

input is A and B, both n by n matrices

initialize C to be an n by n matrix of all zeros

for i from 1 to n:

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for k from 1 to n:

C[i][j] = C[i][j] + A[i][k]*B[k][j]

output C (as A*B)

return C

This [[algorithm]] requires, in the [[worst-case complexity|worst case]], {{tmath|n^3}} multiplications of scalars and {{tmath|(n^3 -1) n^2}} additions for computing the product of two square {{math|''n''×''n''}} matrices. Its [[computational complexity]] is therefore {{tmath|O(n^3)}}, in a [[model of computation]] where field operations (addition and multiplication) take constant time (in practice, this is the case for [[floating point]] numbers, but not necessarily for integers).

=== Strassen's algorithm ===