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r""" 

Sparse matrices over `\ZZ/n\ZZ` for `n` small 

  

This is a compiled implementation of sparse matrices over 

`\ZZ/n\ZZ` for `n` small. 

  

TODO: - move vectors into a Cython vector class - add _add_ and 

_mul_ methods. 

  

EXAMPLES:: 

  

sage: a = matrix(Integers(37),3,3,range(9),sparse=True); a 

[0 1 2] 

[3 4 5] 

[6 7 8] 

sage: type(a) 

<type 'sage.matrix.matrix_modn_sparse.Matrix_modn_sparse'> 

sage: parent(a) 

Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37 

sage: a^2 

[15 18 21] 

[ 5 17 29] 

[32 16 0] 

sage: a+a 

[ 0 2 4] 

[ 6 8 10] 

[12 14 16] 

sage: b = a.new_matrix(2,3,range(6)); b 

[0 1 2] 

[3 4 5] 

sage: a*b 

Traceback (most recent call last): 

... 

TypeError: unsupported operand parent(s) for *: 'Full MatrixSpace of 3 by 3 sparse matrices over Ring of integers modulo 37' and 'Full MatrixSpace of 2 by 3 sparse matrices over Ring of integers modulo 37' 

sage: b*a 

[15 18 21] 

[ 5 17 29] 

  

:: 

  

sage: TestSuite(a).run() 

sage: TestSuite(b).run() 

  

:: 

  

sage: a.echelonize(); a 

[ 1 0 36] 

[ 0 1 2] 

[ 0 0 0] 

sage: b.echelonize(); b 

[ 1 0 36] 

[ 0 1 2] 

sage: a.pivots() 

(0, 1) 

sage: b.pivots() 

(0, 1) 

sage: a.rank() 

2 

sage: b.rank() 

2 

sage: a[2,2] = 5 

sage: a.rank() 

3 

  

TESTS:: 

  

sage: matrix(Integers(37),0,0,sparse=True).inverse() 

[] 

""" 

  

#***************************************************************************** 

# Copyright (C) 2006 William Stein <wstein@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

  

from __future__ import absolute_import 

  

from collections import Iterator, Sequence 

  

from cysignals.memory cimport check_calloc, sig_malloc, sig_free 

from cysignals.signals cimport sig_on, sig_off 

  

from sage.modules.vector_modn_sparse cimport * 

  

from cpython.sequence cimport * 

  

from sage.libs.gmp.mpz cimport mpz_init_set_si 

cimport sage.matrix.matrix as matrix 

cimport sage.matrix.matrix_sparse as matrix_sparse 

cimport sage.matrix.matrix_dense as matrix_dense 

from sage.rings.finite_rings.integer_mod cimport IntegerMod_int, IntegerMod_abstract 

  

from sage.misc.misc import verbose, get_verbose 

  

import sage.rings.all as rings 

  

from sage.matrix.matrix2 import Matrix as Matrix2 

from sage.arith.all import is_prime 

  

from sage.structure.element import is_Vector 

  

cimport sage.structure.element 

  

from sage.data_structures.binary_search cimport * 

from sage.modules.vector_integer_sparse cimport * 

  

from .matrix_integer_sparse cimport Matrix_integer_sparse 

from sage.misc.decorators import rename_keyword 

  

################ 

# TODO: change this to use extern cdef's methods. 

from sage.rings.fast_arith cimport arith_int 

cdef arith_int ai 

ai = arith_int() 

################ 

  

# The 46341 below is because the mod-n sparse code still uses 

# int's, even on 64-bit computers. Improving this is 

# Trac Ticket #12679. 

MAX_MODULUS = 46341 

  

from sage.libs.linbox.linbox cimport Linbox_modn_sparse 

cdef Linbox_modn_sparse linbox 

linbox = Linbox_modn_sparse() 

  

cdef class Matrix_modn_sparse(matrix_sparse.Matrix_sparse): 

def __cinit__(self, parent, entries, copy, coerce): 

matrix.Matrix.__init__(self, parent) 

  

# allocate memory 

cdef Py_ssize_t i, nr, nc 

cdef int p 

  

nr = parent.nrows() 

nc = parent.ncols() 

p = parent.base_ring().order() 

self.p = p 

  

self.rows = <c_vector_modint*>check_calloc(nr, sizeof(c_vector_modint)) 

  

for i from 0 <= i < nr: 

init_c_vector_modint(&self.rows[i], p, nc, 0) 

  

  

def __dealloc__(self): 

cdef Py_ssize_t i 

if self.rows: 

for i in range(self._nrows): 

clear_c_vector_modint(&self.rows[i]) 

sig_free(self.rows) 

  

def __init__(self, parent, entries, copy, coerce): 

""" 

Create a sparse matrix over the integers modulo ``n``. 

  

INPUT: 

  

- ``parent`` -- a matrix space 

  

- ``entries`` -- can be one of the following: 

  

* a Python dictionary whose items have the 

form ``(i, j): x``, where ``0 <= i < nrows``, 

``0 <= j < ncols``, and ``x`` is coercible to 

an element of the integers modulo ``n``. 

The ``i,j`` entry of ``self`` is 

set to ``x``. The ``x``'s can be ``0``. 

* Alternatively, entries can be a list of *all* 

the entries of the sparse matrix, read 

row-by-row from top to bottom (so they would 

be mostly 0). 

  

- ``copy`` -- ignored 

  

- ``coerce`` -- ignored 

""" 

cdef int s, z, p 

cdef Py_ssize_t i, j, k 

  

cdef PyObject** X 

  

if entries is None: 

return 

  

if isinstance(entries, dict): 

# Sparse input format. 

R = self._base_ring 

for ij, x in entries.iteritems(): 

z = R(x) 

if z != 0: 

i, j = ij # nothing better to do since this is user input, which may be bogus. 

if i < 0 or j < 0 or i >= self._nrows or j >= self._ncols: 

raise IndexError("invalid entries list") 

set_entry(&self.rows[i], j, z) 

elif isinstance(entries, (Iterator, Sequence)): 

if not isinstance(entries, (list, tuple)): 

entries = list(entries) 

# Dense input format 

if len(entries) != self._nrows * self._ncols: 

raise TypeError("list of entries must be a dictionary of (i,j):x or a dense list of n * m elements") 

seq = PySequence_Fast(entries,"expected a list") 

X = PySequence_Fast_ITEMS(seq) 

k = 0 

R = self._base_ring 

# Get fast access to the entries list. 

for i from 0 <= i < self._nrows: 

for j from 0 <= j < self._ncols: 

z = R(<object>X[k]) 

if z != 0: 

set_entry(&self.rows[i], j, z) 

k = k + 1 

else: 

# scalar? 

s = int(self._base_ring(entries)) 

if s == 0: 

return 

if self._nrows != self._ncols: 

raise TypeError("matrix must be square to initialize with a scalar.") 

for i from 0 <= i < self._nrows: 

set_entry(&self.rows[i], i, s) 

  

  

cdef set_unsafe(self, Py_ssize_t i, Py_ssize_t j, value): 

set_entry(&self.rows[i], j, (<IntegerMod_int> value).ivalue) 

  

cdef get_unsafe(self, Py_ssize_t i, Py_ssize_t j): 

cdef IntegerMod_int n 

n = IntegerMod_int.__new__(IntegerMod_int) 

IntegerMod_abstract.__init__(n, self._base_ring) 

n.ivalue = get_entry(&self.rows[i], j) 

return n 

  

######################################################################## 

# LEVEL 2 functionality 

# * def _pickle 

# * def _unpickle 

# * cdef _add_ 

# * cdef _mul_ 

# * cpdef _cmp_ 

# * __neg__ 

# * __invert__ 

# * __copy__ 

# * _multiply_classical 

# * _list -- list of underlying elements (need not be a copy) 

# * x _dict -- sparse dictionary of underlying elements (need not be a copy) 

######################################################################## 

# def _pickle(self): 

# def _unpickle(self, data, int version): # use version >= 0 

# cpdef _add_(self, right): 

# cdef _mul_(self, Matrix right): 

# cpdef int _cmp_(self, Matrix right) except -2: 

# def __neg__(self): 

# def __invert__(self): 

# def __copy__(self): 

# def _multiply_classical(left, matrix.Matrix _right): 

# def _list(self): 

  

def _dict(self): 

""" 

Unsafe version of the dict method, mainly for internal use. This 

may return the dict of elements, but as an *unsafe* reference to 

the underlying dict of the object. It might be dangerous if you 

change entries of the returned dict. 

  

EXAMPLES:: 

  

sage: MS = MatrixSpace(GF(13), 50, 50, sparse=True) 

sage: m = MS.random_element(density=0.002) 

sage: m._dict() 

{(4, 44): 7, (5, 25): 4, (26, 9): 9, (43, 43): 6, (44, 38): 1} 

  

TESTS:: 

  

sage: parent(m._dict()[26,9]) 

Finite Field of size 13 

""" 

d = self.fetch('dict') 

if not d is None: 

return d 

  

cdef Py_ssize_t i, j, k 

d = {} 

cdef IntegerMod_int n 

for i from 0 <= i < self._nrows: 

for j from 0 <= j < self.rows[i].num_nonzero: 

n = IntegerMod_int.__new__(IntegerMod_int) 

IntegerMod_abstract.__init__(n, self._base_ring) 

n.ivalue = self.rows[i].entries[j] 

d[(int(i),int(self.rows[i].positions[j]))] = n 

self.cache('dict', d) 

return d 

  

def _pickle(self): 

""" 

TESTS:: 

  

sage: M = Matrix( GF(2), [[1,1,1,1,0,0,0,0,0,0]], sparse=True ) 

sage: loads(dumps(M)) 

[1 1 1 1 0 0 0 0 0 0] 

sage: loads(dumps(M)) == M 

True 

""" 

return self._dict(), 1 

  

def _unpickle(self, data, version): 

if version == 1: 

self.__init__(self.parent(), data, copy=False, coerce=False) 

else: 

raise ValueError("unknown matrix format") 

  

cdef sage.structure.element.Matrix _matrix_times_matrix_(self, sage.structure.element.Matrix _right): 

""" 

This code is implicitly called for multiplying self by another 

sparse matrix. 

  

EXAMPLES: 

sage: a = matrix(GF(43), 3, 3, range(9), sparse=True) 

sage: b = matrix(GF(43), 3, 3, range(10,19), sparse=True) 

sage: a*b 

[ 2 5 8] 

[33 2 14] 

[21 42 20] 

sage: a*a 

[15 18 21] 

[42 11 23] 

[26 4 25] 

sage: c = matrix(GF(43), 3, 8, range(24), sparse=True) 

sage: a*c 

[40 0 3 6 9 12 15 18] 

[26 38 7 19 31 0 12 24] 

[12 33 11 32 10 31 9 30] 

  

Even though sparse and dense matrices are represented 

differently, they still compare as equal if they have the 

same entries: 

sage: a*b == a._matrix_times_matrix_dense(b) 

True 

sage: d = matrix(GF(43), 3, 8, range(24)) 

sage: a*c == a*d 

True 

 

TESTS: 

 

The following shows that :trac:`23669` has been addressed:: 

  

sage: p = next_prime(2**15) 

sage: M = Matrix(GF(p), 1,3, lambda i,j: -1, sparse=True); M 

[32770 32770 32770] 

sage: M*M.transpose() # previously returned [32738] 

[3] 

  

""" 

cdef Matrix_modn_sparse right, ans 

right = _right 

  

cdef c_vector_modint* v 

  

# Build a table that gives the nonzero positions in each column of right 

nonzero_positions_in_columns = [set([]) for _ in range(right._ncols)] 

cdef Py_ssize_t i, j, k 

for i from 0 <= i < right._nrows: 

v = &(right.rows[i]) 

for j from 0 <= j < right.rows[i].num_nonzero: 

nonzero_positions_in_columns[v.positions[j]].add(i) 

  

ans = self.new_matrix(self._nrows, right._ncols) 

  

# Now do the multiplication, getting each row completely before filling it in. 

cdef int x, y, s 

  

for i from 0 <= i < self._nrows: 

v = &self.rows[i] 

for j from 0 <= j < right._ncols: 

s = 0 

c = nonzero_positions_in_columns[j] 

for k from 0 <= k < v.num_nonzero: 

if v.positions[k] in c: 

y = get_entry(&right.rows[v.positions[k]], j) 

x = v.entries[k] * y 

s = (s + x) % self.p 

set_entry(&ans.rows[i], j, s) 

return ans 

  

  

def _matrix_times_matrix_dense(self, sage.structure.element.Matrix _right): 

""" 

Multiply self by the sparse matrix _right, and return the 

result as a dense matrix. 

  

EXAMPLES: 

sage: a = matrix(GF(10007), 2, [1,2,3,4], sparse=True) 

sage: b = matrix(GF(10007), 2, 3, [1..6], sparse=True) 

sage: a * b 

[ 9 12 15] 

[19 26 33] 

sage: c = a._matrix_times_matrix_dense(b); c 

[ 9 12 15] 

[19 26 33] 

sage: type(c) 

<type 'sage.matrix.matrix_modn_dense_double.Matrix_modn_dense_double'> 

  

sage: a = matrix(GF(2), 20, 20, sparse=True) 

sage: a*a == a._matrix_times_matrix_dense(a) 

True 

sage: type(a._matrix_times_matrix_dense(a)) 

<type 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'> 

""" 

cdef Matrix_modn_sparse right 

cdef matrix_dense.Matrix_dense ans 

right = _right 

  

cdef c_vector_modint* v 

  

# Build a table that gives the nonzero positions in each column of right 

nonzero_positions_in_columns = [set([]) for _ in range(right._ncols)] 

cdef Py_ssize_t i, j, k 

for i from 0 <= i < right._nrows: 

v = &(right.rows[i]) 

for j from 0 <= j < right.rows[i].num_nonzero: 

nonzero_positions_in_columns[v.positions[j]].add(i) 

  

ans = self.new_matrix(self._nrows, right._ncols, sparse=False) 

  

# Now do the multiplication, getting each row completely before filling it in. 

cdef int x, y, s 

  

for i from 0 <= i < self._nrows: 

v = &self.rows[i] 

for j from 0 <= j < right._ncols: 

s = 0 

c = nonzero_positions_in_columns[j] 

for k from 0 <= k < v.num_nonzero: 

if v.positions[k] in c: 

y = get_entry(&right.rows[v.positions[k]], j) 

x = v.entries[k] * y 

s = (s + x)%self.p 

ans.set_unsafe_int(i, j, s) 

#ans._matrix[i][j] = s 

return ans 

  

######################################################################## 

# LEVEL 3 functionality (Optional) 

# * cdef _sub_ 

# * __deepcopy__ 

# * __invert__ 

# * Matrix windows -- only if you need strassen for that base 

# * Other functions (list them here): 

# x - echelon form in place 

# x - nonzero_positions 

######################################################################## 

def swap_rows(self, r1, r2): 

self.check_bounds_and_mutability(r1,0) 

self.check_bounds_and_mutability(r2,0) 

self.swap_rows_c(r1, r2) 

  

cdef swap_rows_c(self, Py_ssize_t n1, Py_ssize_t n2): 

""" 

Swap the rows in positions n1 and n2. No bounds checking. 

""" 

cdef c_vector_modint tmp 

tmp = self.rows[n1] 

self.rows[n1] = self.rows[n2] 

self.rows[n2] = tmp 

  

def _echelon_in_place_classical(self): 

""" 

Replace self by its reduction to reduced row echelon form. 

  

ALGORITHM: We use Gauss elimination, in a slightly intelligent way, 

in that we clear each column using a row with the minimum number of 

nonzero entries. 

  

TODO: Implement switching to a dense method when the matrix gets 

dense. 

""" 

x = self.fetch('in_echelon_form') 

if not x is None and x: return # already known to be in echelon form 

self.check_mutability() 

  

cdef Py_ssize_t i, r, c, min, min_row, start_row 

cdef int a0, a_inverse, b, do_verb 

cdef c_vector_modint tmp 

start_row = 0 

pivots = [] 

fifth = self._ncols / 10 + 1 

tm = verbose(caller_name = 'sparse_matrix_pyx matrix_modint echelon') 

do_verb = (get_verbose() >= 2) 

  

for c from 0 <= c < self._ncols: 

if do_verb and (c % fifth == 0 and c>0): 

tm = verbose('on column %s of %s'%(c, self._ncols), 

level = 2, 

caller_name = 'matrix_modn_sparse echelon') 

#end if 

min = self._ncols + 1 

min_row = -1 

for r from start_row <= r < self._nrows: 

if self.rows[r].num_nonzero > 0 and self.rows[r].num_nonzero < min: 

# Since there is at least one nonzero entry, the first entry 

# of the positions list is defined. It is the first position 

# of a nonzero entry, and it equals c precisely if row r 

# is a row we could use to clear column c. 

if self.rows[r].positions[0] == c: 

min_row = r 

min = self.rows[r].num_nonzero 

#endif 

#endif 

#endfor 

if min_row != -1: 

r = min_row 

# print("min number of entries in a pivoting row = ", min) 

pivots.append(c) 

# Since we can use row r to clear column c, the 

# entry in position c in row r must be the first nonzero entry. 

a = self.rows[r].entries[0] 

if a != 1: 

a_inverse = ai.c_inverse_mod_int(a, self.p) 

scale_c_vector_modint(&self.rows[r], a_inverse) 

self.swap_rows_c(r, start_row) 

sig_on() 

for i from 0 <= i < self._nrows: 

if i != start_row: 

b = get_entry(&self.rows[i], c) 

if b != 0: 

add_c_vector_modint_init(&tmp, &self.rows[i], 

&self.rows[start_row], self.p - b) 

clear_c_vector_modint(&self.rows[i]) 

self.rows[i] = tmp 

sig_off() 

start_row = start_row + 1 

  

  

self.cache('pivots',tuple(pivots)) 

self.cache('in_echelon_form',True) 

  

def _nonzero_positions_by_row(self, copy=True): 

""" 

Returns the list of pairs (i,j) such that self[i,j] != 0. 

  

It is safe to change the resulting list (unless you give the option copy=False). 

  

EXAMPLES:: 

sage: M = Matrix(GF(7), [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,1,0]], sparse=True); M 

[0 0 0 1 0 0 0 0] 

[0 1 0 0 0 0 1 0] 

sage: M.nonzero_positions() 

[(0, 3), (1, 1), (1, 6)] 

  

""" 

x = self.fetch('nonzero_positions') 

if not x is None: 

if copy: 

return list(x) 

return x 

nzp = [] 

cdef Py_ssize_t i, j 

for i from 0 <= i < self._nrows: 

for j from 0 <= j < self.rows[i].num_nonzero: 

nzp.append((i,self.rows[i].positions[j])) 

self.cache('nonzero_positions', nzp) 

if copy: 

return list(nzp) 

return nzp 

  

def density(self): 

""" 

Return the density of self, i.e., the ratio of the number of 

nonzero entries of self to the total size of self. 

  

  

EXAMPLES:: 

  

sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8],sparse=True) 

sage: A.density() 

2/3 

  

Notice that the density parameter does not ensure the density 

of a matrix; it is only an upper bound. 

  

:: 

  

sage: A = random_matrix(GF(127),200,200,density=0.3, sparse=True) 

sage: A.density() 

2073/8000 

""" 

cdef Py_ssize_t i, nonzero_entries 

  

nonzero_entries = 0 

for i from 0 <= i < self._nrows: 

nonzero_entries += self.rows[i].num_nonzero 

  

return rings.ZZ(nonzero_entries)/rings.ZZ(self._nrows*self._ncols) 

  

def transpose(self): 

""" 

Return the transpose of self. 

  

EXAMPLES:: 

  

sage: A = matrix(GF(127),3,3,[0,1,0,2,0,0,3,0,0],sparse=True) 

sage: A 

[0 1 0] 

[2 0 0] 

[3 0 0] 

sage: A.transpose() 

[0 2 3] 

[1 0 0] 

[0 0 0] 

  

``.T`` is a convenient shortcut for the transpose:: 

  

sage: A.T 

[0 2 3] 

[1 0 0] 

[0 0 0] 

""" 

cdef int i, j 

cdef c_vector_modint row 

cdef Matrix_modn_sparse B 

  

B = self.new_matrix(nrows = self.ncols(), ncols = self.nrows()) 

for i from 0 <= i < self._nrows: 

row = self.rows[i] 

for j from 0 <= j < row.num_nonzero: 

set_entry(&B.rows[row.positions[j]], i, row.entries[j]) 

if self._subdivisions is not None: 

row_divs, col_divs = self.subdivisions() 

B.subdivide(col_divs, row_divs) 

return B 

  

def matrix_from_rows(self, rows): 

""" 

Return the matrix constructed from self using rows with indices in 

the rows list. 

  

INPUT: 

  

  

- ``rows`` - list or tuple of row indices 

  

  

EXAMPLES:: 

  

sage: M = MatrixSpace(GF(127),3,3,sparse=True) 

sage: A = M(range(9)); A 

[0 1 2] 

[3 4 5] 

[6 7 8] 

sage: A.matrix_from_rows([2,1]) 

[6 7 8] 

[3 4 5] 

""" 

cdef int i,k 

cdef Matrix_modn_sparse A 

cdef c_vector_modint row 

  

if not isinstance(rows, (list, tuple)): 

rows = list(rows) 

  

A = self.new_matrix(nrows = len(rows)) 

  

k = 0 

for ii in rows: 

i = ii 

if i < 0 or i >= self.nrows(): 

raise IndexError("row %s out of range" % i) 

  

row = self.rows[i] 

for j from 0 <= j < row.num_nonzero: 

set_entry(&A.rows[k], row.positions[j], row.entries[j]) 

k += 1 

return A 

  

  

def matrix_from_columns(self, cols): 

""" 

Return the matrix constructed from self using columns with indices 

in the columns list. 

  

EXAMPLES:: 

  

sage: M = MatrixSpace(GF(127),3,3,sparse=True) 

sage: A = M(range(9)); A 

[0 1 2] 

[3 4 5] 

[6 7 8] 

sage: A.matrix_from_columns([2,1]) 

[2 1] 

[5 4] 

[8 7] 

""" 

cdef int i,j 

cdef Matrix_modn_sparse A 

cdef c_vector_modint row 

  

if not isinstance(cols, (list, tuple)): 

cols = list(cols) 

  

A = self.new_matrix(ncols = len(cols)) 

  

cols = dict(zip([int(e) for e in cols],range(len(cols)))) 

  

for i from 0 <= i < self.nrows(): 

row = self.rows[i] 

for j from 0 <= j < row.num_nonzero: 

if int(row.positions[j]) in cols: 

set_entry(&A.rows[i], cols[int(row.positions[j])], row.entries[j]) 

return A 

  

cdef _init_linbox(self): 

sig_on() 

linbox.set(self.p, self._nrows, self._ncols, self.rows) 

sig_off() 

  

@rename_keyword(deprecation=6094, method="algorithm") 

def _rank_linbox(self, algorithm): 

""" 

See self.rank(). 

""" 

if is_prime(self.p): 

x = self.fetch('rank') 

if not x is None: 

return x 

self._init_linbox() 

sig_on() 

# the returend pivots list is currently wrong 

#r, pivots = linbox.rank(1) 

r = linbox.rank(algorithm) 

r = rings.Integer(r) 

sig_off() 

self.cache('rank', r) 

return r 

else: 

raise TypeError("only GF(p) supported via LinBox") 

  

def rank(self, gauss=False): 

""" 

Compute the rank of self. 

  

INPUT: 

  

  

- ``gauss`` - if True LinBox' Gaussian elimination is 

used. If False 'Symbolic Reordering' as implemented in LinBox is 

used. If 'native' the native Sage implementation is used. (default: 

False) 

  

  

EXAMPLES:: 

  

sage: A = random_matrix(GF(127),200,200,density=0.01,sparse=True) 

sage: r1 = A.rank(gauss=False) 

sage: r2 = A.rank(gauss=True) 

sage: r3 = A.rank(gauss='native') 

sage: r1 == r2 == r3 

True 

sage: r1 

155 

  

ALGORITHM: Uses LinBox or native implementation. 

  

REFERENCES: 

  

- Jean-Guillaume Dumas and Gilles Villars. 'Computing the Rank 

of Large Sparse Matrices over Finite 

Fields'. Proc. CASC'2002, The Fifth International Workshop 

on Computer Algebra in Scientific Computing, Big Yalta, 

Crimea, Ukraine, 22-27 sept. 2002, Springer-Verlag, 

http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/POSTSCRIPT/rankjgd.ps 

  

.. NOTE:: 

  

For very sparse matrices Gaussian elimination is faster 

because it barly has anything to do. If the fill in needs to 

be considered, 'Symbolic Reordering' is usually much faster. 

""" 

  

if self._nrows == 0 or self._ncols == 0: 

return 0 

x = self.fetch('rank') 

if not x is None: return x 

  

if is_prime(self.p): 

if gauss is False: 

return self._rank_linbox(0) 

elif gauss is True: 

return self._rank_linbox(1) 

elif gauss == "native": 

return Matrix2.rank(self) 

else: 

raise TypeError("parameter 'gauss' not understood") 

else: 

return Matrix2.rank(self) 

  

def _solve_right_nonsingular_square(self, B, algorithm=None, check_rank = True): 

""" 

If self is a matrix `A`, then this function returns a 

vector or matrix `X` such that `A X = B`. If 

`B` is a vector then `X` is a vector and if 

`B` is a matrix, then `X` is a matrix. 

  

.. NOTE:: 

  

In Sage one can also write ``A B`` for 

``A.solve_right(B)``, i.e., Sage implements the "the 

MATLAB/Octave backslash operator". 

  

INPUT: 

  

  

- ``B`` - a matrix or vector 

  

- ``algorithm`` - one of the following: 

  

- ``'LinBox:BlasElimination'`` - dense elimination 

  

- ``'LinBox:Blackbox'`` - LinBox chooses a Blackbox 

algorithm 

  

- ``'LinBox:Wiedemann'`` - Wiedemann's algorithm 

  

- ``'generic'`` - use generic implementation 

(inversion) 

  

- ``None`` - LinBox chooses an algorithm (default) 

  

- ``check_rank`` - check rank before attempting to 

solve (default: True) 

  

  

OUTPUT: a matrix or vector 

  

EXAMPLES:: 

  

sage: A = matrix(GF(127), 3, [1,2,3,-1,2,5,2,3,1], sparse=True) 

sage: b = vector(GF(127),[1,2,3]) 

sage: x = A \ b; x 

(73, 76, 10) 

sage: A * x 

(1, 2, 3) 

""" 

cdef Matrix_modn_sparse A = self 

cdef Matrix_modn_sparse b 

cdef Matrix_modn_sparse X 

cdef c_vector_modint *x 

  

if self.base_ring() != B.base_ring(): 

B = B.change_ring(self.base_ring()) 

  

if algorithm == "generic" or not is_prime(self.p): 

return Matrix2.solve_right(self, B) 

  

if check_rank and self.rank() < self.nrows(): 

raise ValueError("self must be of full rank.") 

  

if self.nrows() != B.nrows(): 

raise ValueError("input matrices must have the same number of rows.") 

  

if not self.is_square(): 

raise NotImplementedError("input matrix must be square") 

  

self._init_linbox() 

  

matrix = True 

if is_Vector(B): 

matrix = False 

b = self.matrix_space(1, self.ncols(),sparse=True)(B.list()) 

else: 

if not B.is_sparse(): 

B = B.sparse_matrix() 

if isinstance(B, Matrix_modn_sparse): 

b = B 

else: 

raise TypeError("B must be a matrix or vector over the same base as self") 

  

X = self.new_matrix(b.ncols(), A.ncols()) 

  

if algorithm is None: 

algorithm = 0 

elif algorithm == "LinBox:BlasElimination": 

algorithm = 1 

elif algorithm == "LinBox:Blackbox": 

algorithm = 2 

elif algorithm == "LinBox:Wiedemann": 

algorithm = 3 

else: 

raise TypeError("parameter 'algorithm' not understood") 

  

b = b.transpose() # to make walking the rows easier 

for i in range(X.nrows()): 

sig_on() 

x = &X.rows[i] 

linbox.solve(&x, &b.rows[i], algorithm) 

sig_off() 

  

if not matrix: 

# Convert back to a vector 

return (X.base_ring() ** X.ncols())(X.list()) 

else: 

return X.transpose() 

  

def lift(self): 

""" 

Return lift of this matrix to a sparse matrix over the integers. 

  

EXAMPLES: 

sage: a = matrix(GF(7),2,3,[1..6], sparse=True) 

sage: a.lift() 

[1 2 3] 

[4 5 6] 

sage: a.lift().parent() 

Full MatrixSpace of 2 by 3 sparse matrices over Integer Ring 

  

Subdivisions are preserved when lifting:: 

  

sage: a.subdivide([], [1,1]); a 

[1||2 3] 

[4||5 6] 

sage: a.lift() 

[1||2 3] 

[4||5 6] 

""" 

cdef Py_ssize_t i, j 

cdef Matrix_integer_sparse L 

L = Matrix_integer_sparse.__new__(Matrix_integer_sparse, 

self.parent().change_ring(rings.ZZ), 

0, 0, 0) 

  

cdef mpz_vector* L_row 

cdef c_vector_modint* A_row 

for i from 0 <= i < self._nrows: 

L_row = &(L._matrix[i]) 

A_row = &(self.rows[i]) 

sig_free(L_row.entries) 

L_row.entries = <mpz_t*> sig_malloc(sizeof(mpz_t)*A_row.num_nonzero) 

L_row.num_nonzero = A_row.num_nonzero 

if L_row.entries == NULL: 

raise MemoryError("error allocating space for sparse vector during sparse lift") 

sig_free(L_row.positions) 

L_row.positions = <Py_ssize_t*> sig_malloc(sizeof(Py_ssize_t)*A_row.num_nonzero) 

if L_row.positions == NULL: 

sig_free(L_row.entries) 

L_row.entries = NULL 

raise MemoryError("error allocating space for sparse vector during sparse lift") 

for j from 0 <= j < A_row.num_nonzero: 

L_row.positions[j] = A_row.positions[j] 

mpz_init_set_si(L_row.entries[j], A_row.entries[j]) 

L.subdivide(self.subdivisions()) 

return L