Generic Asymptotically Fast Strassen Algorithms¶
This implements asymptotically fast echelon form and matrix multiplication algorithms.
- class sage.matrix.strassen.int_range(indices=None, range=None)[source]¶
- Bases: - object- Represent a list of integers as a list of integer intervals. - Note - Repetitions are not considered. - Useful class for dealing with pivots in the Strassen echelon, could have much more general application - INPUT: - It can be one of the following: - indices– integer; start of the unique interval
- range– integer; length of the unique interval
 - OR - indices– list of integers, the integers to wrap into intervals
 - OR - indices–- None(default), shortcut for an empty list
 - OUTPUT: - An instance of - int_range, i.e. a list of pairs- (start, length).- EXAMPLES: - From a pair of integers: - sage: from sage.matrix.strassen import int_range sage: int_range(2, 4) [(2, 4)] - >>> from sage.all import * >>> from sage.matrix.strassen import int_range >>> int_range(Integer(2), Integer(4)) [(2, 4)] - Default: - sage: int_range() [] - >>> from sage.all import * >>> int_range() [] - From a list of integers: - sage: int_range([1,2,3,4]) [(1, 4)] sage: int_range([1,2,3,4,6,7,8]) [(1, 4), (6, 3)] sage: int_range([1,2,3,4,100,101,102]) [(1, 4), (100, 3)] sage: int_range([1,1000,2,101,3,4,100,102]) [(1, 4), (100, 3), (1000, 1)] - >>> from sage.all import * >>> int_range([Integer(1),Integer(2),Integer(3),Integer(4)]) [(1, 4)] >>> int_range([Integer(1),Integer(2),Integer(3),Integer(4),Integer(6),Integer(7),Integer(8)]) [(1, 4), (6, 3)] >>> int_range([Integer(1),Integer(2),Integer(3),Integer(4),Integer(100),Integer(101),Integer(102)]) [(1, 4), (100, 3)] >>> int_range([Integer(1),Integer(1000),Integer(2),Integer(101),Integer(3),Integer(4),Integer(100),Integer(102)]) [(1, 4), (100, 3), (1000, 1)] - Repetitions are not considered: - sage: int_range([1,2,3]) [(1, 3)] sage: int_range([1,1,1,1,2,2,2,3]) [(1, 3)] - >>> from sage.all import * >>> int_range([Integer(1),Integer(2),Integer(3)]) [(1, 3)] >>> int_range([Integer(1),Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(2),Integer(3)]) [(1, 3)] - AUTHORS: - Robert Bradshaw 
 - intervals()[source]¶
- Return the list of intervals. - OUTPUT: list of pairs of integers - EXAMPLES: - sage: from sage.matrix.strassen import int_range sage: I = int_range([4,5,6,20,21,22,23]) sage: I.intervals() [(4, 3), (20, 4)] sage: type(I.intervals()) <... 'list'> - >>> from sage.all import * >>> from sage.matrix.strassen import int_range >>> I = int_range([Integer(4),Integer(5),Integer(6),Integer(20),Integer(21),Integer(22),Integer(23)]) >>> I.intervals() [(4, 3), (20, 4)] >>> type(I.intervals()) <... 'list'> 
 - to_list()[source]¶
- Return the (sorted) list of integers represented by this object. - OUTPUT: list of integers - EXAMPLES: - sage: from sage.matrix.strassen import int_range sage: I = int_range([6,20,21,4,5,22,23]) sage: I.to_list() [4, 5, 6, 20, 21, 22, 23] - >>> from sage.all import * >>> from sage.matrix.strassen import int_range >>> I = int_range([Integer(6),Integer(20),Integer(21),Integer(4),Integer(5),Integer(22),Integer(23)]) >>> I.to_list() [4, 5, 6, 20, 21, 22, 23] - sage: I = int_range(34, 9) sage: I.to_list() [34, 35, 36, 37, 38, 39, 40, 41, 42] - >>> from sage.all import * >>> I = int_range(Integer(34), Integer(9)) >>> I.to_list() [34, 35, 36, 37, 38, 39, 40, 41, 42] - Repetitions are not considered: - sage: I = int_range([1,1,1,1,2,2,2,3]) sage: I.to_list() [1, 2, 3] - >>> from sage.all import * >>> I = int_range([Integer(1),Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(2),Integer(3)]) >>> I.to_list() [1, 2, 3] 
 
- sage.matrix.strassen.strassen_echelon(A, cutoff)[source]¶
- 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 elimination and multiplication must be at least 1
 - OUTPUT: the list of pivot columns - EXAMPLES: - 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 - >>> from sage.all import * >>> A = matrix(QQ, Integer(7), [Integer(5), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), -Integer(1), Integer(0), Integer(0), Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), -Integer(1), Integer(3), Integer(1), Integer(0), -Integer(1), Integer(0), Integer(0), -Integer(1), Integer(0), Integer(1), Integer(2), -Integer(1), Integer(1), Integer(0), -Integer(1), Integer(0), Integer(1), Integer(3), -Integer(1), Integer(1), Integer(0), Integer(0), -Integer(2), Integer(0), Integer(2), Integer(0), Integer(1), Integer(0), Integer(0), -Integer(1), Integer(0), Integer(1), Integer(0), Integer(1)]) >>> B = A.__copy__(); B._echelon_strassen(Integer(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] >>> C = A.__copy__(); C._echelon_strassen(Integer(2)); C == B True >>> C = A.__copy__(); C._echelon_strassen(Integer(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 - >>> from sage.all import * >>> n = Integer(32); A = matrix(Integers(Integer(389)),n,range(n**Integer(2))) >>> B = A.__copy__(); B._echelon_in_place_classical() >>> C = A.__copy__(); C._echelon_strassen(Integer(2)) >>> B == C True - AUTHORS: - Robert Bradshaw 
 
- sage.matrix.strassen.strassen_window_multiply(C, A, B, cutoff)[source]¶
- Multiply 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. - EXAMPLES: The following matrix dimensions are chosen especially to exercise the eight possible parity combinations that could occur 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) #indirect doctest True - >>> from sage.all import * >>> A = MatrixSpace(Integers(Integer(2)**Integer(65)), Integer(64), Integer(83)).random_element() >>> B = MatrixSpace(Integers(Integer(2)**Integer(65)), Integer(83), Integer(101)).random_element() >>> A._multiply_classical(B) == A._multiply_strassen(B, Integer(3)) #indirect doctest True - AUTHORS: - David Harvey 
- Simon King (2011-07): Improve memory efficiency; Issue #11610 
 
- sage.matrix.strassen.test(n, m, R, c=2)[source]¶
- Test code for the Strassen algorithm. - INPUT: - n– integer
- m– integer
- R– ring
- c– integer (default: 2)
 - EXAMPLES: - sage: from sage.matrix.strassen import test sage: for n in range(5): ....: print("{} {}".format(n, test(2*n,n,Frac(QQ['x']),2))) 0 True 1 True 2 True 3 True 4 True - >>> from sage.all import * >>> from sage.matrix.strassen import test >>> for n in range(Integer(5)): ... print("{} {}".format(n, test(Integer(2)*n,n,Frac(QQ['x']),Integer(2)))) 0 True 1 True 2 True 3 True 4 True