# Generic Asymptotically Fast Strassen Algorithms¶

Sage implements asymptotically fast echelon form and matrix multiplication algorithms.

class sage.matrix.strassen.int_range(indices=None, range=None)

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)]
```

Default:

```sage: 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)]
```

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)]
```

AUTHORS:

intervals()

Return the list of intervals.

OUTPUT:

A 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'>
```
to_list()

Return the (sorted) list of integers represented by this object.

OUTPUT:

A 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]
```
```sage: I = int_range(34, 9)
sage: 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]
```
sage.matrix.strassen.strassen_echelon(A, cutoff)

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
```
```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
```

AUTHORS:

sage.matrix.strassen.strassen_window_multiply(C, A, B, cutoff)

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. 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
```

AUTHORS:

sage.matrix.strassen.test(n, m, R, c=2)

INPUT:

• `n` - integer

• `m` - integer

• `R` - ring

• `c` - integer (optional, 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
```