32.8 Generic Asymptotically Fast Strassen Algorithms

Module: sage.matrix.strassen

Generic Asymptotically Fast Strassen Algorithms

SAGE implements asymptotically fast echelon form and matrix multiplication algorithms.

Module-level Functions

strassen_echelon( )

Compute echelon form, in place. Internal function, call with M.echelonize(algorithm="strassen") Based on work of Robert Bradshaw and David Harvey at MSRI workshop in 2006.

Input:

A
- matrix window
cutoff
- size at which algorithm reverts to naive gaussian elemination and multiplication must be at least 1.

Output: The list of pivot columns

sage: A = matrix(QQ, 7, [5, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 3, 1, 0, -1, 0, 0, -1, 0, 1, 2, -1, 1, 0, -1, 0, 1, 3, -1, 1, 0, 0, -2, 0, 2, 0, 1, 0, 0, -1, 0, 1, 0, 1])
sage: B = A.copy(); B._echelon_strassen(1); B
[ 1  0  0  0  0  0  0]
[ 0  1  0 -1  0  1  0]
[ 0  0  1  0  0  0  0]
[ 0  0  0  0  1  0  0]
[ 0  0  0  0  0  0  1]
[ 0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0]
sage: C = A.copy(); C._echelon_strassen(2); C == B
True
sage: C = A.copy(); C._echelon_strassen(4); C == B
True

sage: n = 32; A = matrix(Integers(389),n,range(n^2))
sage: B = A.copy(); B._echelon_in_place_classical()
sage: C = A.copy(); C._echelon_strassen(2)
sage: B == C
True

TESTS:

sage: A = matrix(Integers(7), 4, 4, [1,2,0,3,0,0,1,0,0,1,0,0,0,0,0,1])
sage: B = A.copy(); B._echelon_in_place_classical()
sage: C = A.copy(); C._echelon_strassen(2)
sage: B == C
True

sage: A = matrix(Integers(7), 4, 4, [1,0,5,0,2,0,3,6,5,1,2,6,4,6,1,1])
sage: B = A.copy(); B._echelon_in_place_classical()
sage: C = A.copy(); C._echelon_strassen(2)
sage: B == C
True

Author: Robert Bradshaw

strassen_window_multiply( )

Multiplies the submatrices specified by A and B, places result in C. Assumes that A and B have compatible dimensions to be multiplied, and that C is the correct size to receive the product, and that they are all defined over the same ring.

Uses strassen multiplication at high levels and then uses MatrixWindow methods at low levels. The following matrix dimensions are chosen especially to exercise the eight possible parity combinations that ocould ccur while subdividing the matrix in the strassen recursion. The base case in both cases will be a (4x5) matrix times a (5x6) matrix.

sage: A = MatrixSpace(Integers(2^65), 64, 83).random_element()
sage: B = MatrixSpace(Integers(2^65), 83, 101).random_element()
sage: A._multiply_classical(B) == A._multiply_strassen(B, 3)
True

Author: David Harvey

Class: int_range

class int_range
Useful class for dealing with pivots in the strassen echelon, could have much more general application Author: Robert Bradshaw

Functions: intervals,$ \,$ to_list

Special Functions: __add__,$ \,$ __init__,$ \,$ __iter__,$ \,$ __len__,$ \,$ __mul__,$ \,$ __repr__,$ \,$ __sub__

__add__( )

__init__( )

__iter__( )

__len__( )

__mul__( )

__repr__( )

__sub__( )

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