Package sage :: Package linalg :: Module sparse_matrix
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Module sage.linalg.sparse_matrix

Sparse matrices

To create an m x n sparse matrix over a ring R, use the command

    SparseMatrix(base_ring, nrows, ncols, entries=[])

where entries is a list of 3-tuples (i,j,x).  The matrix then has i,j
entry equal to x.

WARNING: The i,j pairs *must* be distinct.  The algorithms assume they
are and if they are not, then you will get nonsense.

Classes
Sparse_matrix_generic A generic sparse matrix.
Sparse_matrix_rational A sparse matrix over the rational numbers.
Sparse_vector_space_generic Sparse vector space.
SparseMatrix  
SparseVector A generic sparse vector.
SparseVectorSpace  

Function Summary
  crt_dict(X, M, M2)
Given a list X of dictionaries and a corresponding list M of Integer moduli (of the same length), use the CRT to make a single dict.
  rational_reconstruction(A, m, denom_optimization)
Lift A mod m using rational reconstruction.
  SparseMatrix_from_rows(X)
INPUT: X -- nonempty list of SparseVector rows OUTPUT: Sparse_matrix with those rows.
  SparseMatrix_using_crt(X)
Given Matrix_modint's X, this function uses the chinese remainder theorem to create a SparseMatrix over the integers.

Function Details

crt_dict(X, M, M2)

Given a list X of dictionaries and a corresponding list M of
Integer moduli (of the same length), use the CRT to make
a single dict.

M2 - should be the partial products of elements of M, so
     M2[i] = M[0] * ... * M[i-1],
     and M2[0] = 1.
     This is an optimization.

rational_reconstruction(A, m, denom_optimization=False)

Lift A mod m using rational reconstruction.

Given a sparse matrix A over the integers and an integer m, this
function uses rational reconstruction element-by-element to try
and find a matrix B over the rational numbers that reduces
to A modulo m.  The entries of B will be uniquely determined by
the condition that the numerator and denominator have absolute
value at most sqrt(m/2).  If no such B exists, this function raises
a ValueError.

INPUT:
    A -- SparseMatrix over the integers
    m -- an integer

OUTPUT:
    B -- SparseMatrix over the rationals

SparseMatrix_from_rows(X)

INPUT:
    X -- nonempty list of SparseVector rows
OUTPUT:
    Sparse_matrix with those rows.

SparseMatrix_using_crt(X)

Given Matrix_modint's X, this function uses the chinese remainder
theorem to create a SparseMatrix over the integers. 
INPUT:
   X -- a list of sparse_matrix_pyx.Matrix_modint's modulo coprime moduli.
OUTPUT:
   A single SparseMatrix with integer entries that reduces to all
   the X[i].

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