Coverage for local/lib/python2.7/site-packages/sage/matrix/constructor.pyx : 87%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
""" General matrix Constructor """
#***************************************************************************** # Copyright (C) 2005 William Stein <wstein@gmail.com> # Copyright (C) 2016 Jeroen Demeyer <jdemeyer@cage.ugent.be> # # 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
import types from .matrix_space import MatrixSpace from sage.rings.ring import is_Ring from sage.rings.all import ZZ, RDF, CDF from sage.arith.srange import srange from sage.structure.coerce cimport is_numpy_type, py_scalar_parent from sage.structure.element cimport Vector from sage.structure.sequence import Sequence from sage.arith.long cimport pyobject_to_long
class MatrixFactory(object): """ Create a matrix.
This implements the ``matrix`` constructor::
sage: matrix([[1,2],[3,4]]) [1 2] [3 4]
It also contains methods to create special types of matrices, see ``matrix.[tab]`` for more options. For example::
sage: matrix.identity(2) [1 0] [0 1]
INPUT:
The matrix command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix.
The entries of a matrix can be specified as a flat list of elements, a list of lists (i.e., a list of rows), a list of Sage vectors, a callable object, or a dictionary having positions as keys and matrix entries as values (see the examples). If you pass in a callable object, then you must specify the number of rows and columns. You can create a matrix of zeros by passing an empty list or the integer zero for the entries. To construct a multiple of the identity (`cI`), you can specify square dimensions and pass in `c`. Calling matrix() with a Sage object may return something that makes sense. Calling matrix() with a NumPy array will convert the array to a matrix.
The ring, number of rows, and number of columns of the matrix can be specified by setting the ``ring``, ``nrows``, or ``ncols`` keyword parameters or by passing them as the first arguments to the function in specified order. The ring defaults to ``ZZ`` if it is not specified and cannot be determined from the entries. If the number of rows and columns are not specified and cannot be determined, then an empty 0x0 matrix is returned.
INPUT:
- ``ring`` -- the base ring for the entries of the matrix.
- ``nrows`` -- the number of rows in the matrix.
- ``ncols`` -- the number of columns in the matrix.
- ``sparse`` -- create a sparse matrix. This defaults to ``True`` when the entries are given as a dictionary, otherwise defaults to ``False``.
- ``entries`` -- see examples below.
OUTPUT:
a matrix
EXAMPLES::
sage: m = matrix(2); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
::
sage: m = matrix(2,3); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
::
sage: m = matrix(QQ,[[1,2,3],[4,5,6]]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Rational Field
::
sage: m = matrix(QQ, 3, 3, lambda i, j: i+j); m [0 1 2] [1 2 3] [2 3 4] sage: m = matrix(3, lambda i,j: i-j); m [ 0 -1 -2] [ 1 0 -1] [ 2 1 0]
::
sage: matrix(QQ, 2, 3, lambda x, y: x+y) [0 1 2] [1 2 3] sage: matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1)) [ 1 1/2 1/3 1/4 1/5] [ 2 1 2/3 1/2 2/5] [ 3 3/2 1 3/4 3/5] [ 4 2 4/3 1 4/5] [ 5 5/2 5/3 5/4 1]
::
sage: v1=vector((1,2,3)) sage: v2=vector((4,5,6)) sage: m = matrix([v1,v2]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
::
sage: m = matrix(QQ,2,[1,2,3,4,5,6]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Rational Field
::
sage: m = matrix(QQ,2,3,[1,2,3,4,5,6]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Rational Field
::
sage: m = matrix({(0,1): 2, (1,1):2/5}); m; m.parent() [ 0 2] [ 0 2/5] Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
::
sage: m = matrix(QQ,2,3,{(1,1): 2}); m; m.parent() [0 0 0] [0 2 0] Full MatrixSpace of 2 by 3 sparse matrices over Rational Field
::
sage: import numpy sage: n = numpy.array([[1,2],[3,4]],float) sage: m = matrix(n); m; m.parent() [1.0 2.0] [3.0 4.0] Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
::
sage: v = vector(ZZ, [1, 10, 100]) sage: m = matrix(v); m; m.parent() [ 1 10 100] Full MatrixSpace of 1 by 3 dense matrices over Integer Ring sage: m = matrix(GF(7), v); m; m.parent() [1 3 2] Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7
::
sage: g = graphs.PetersenGraph() sage: m = matrix(g); m; m.parent() [0 1 0 0 1 1 0 0 0 0] [1 0 1 0 0 0 1 0 0 0] [0 1 0 1 0 0 0 1 0 0] [0 0 1 0 1 0 0 0 1 0] [1 0 0 1 0 0 0 0 0 1] [1 0 0 0 0 0 0 1 1 0] [0 1 0 0 0 0 0 0 1 1] [0 0 1 0 0 1 0 0 0 1] [0 0 0 1 0 1 1 0 0 0] [0 0 0 0 1 0 1 1 0 0] Full MatrixSpace of 10 by 10 dense matrices over Integer Ring
::
sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent() Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring
::
sage: R = PolynomialRing(QQ, 9, 'x') sage: A = matrix(R, 3, 3, R.gens()); A [x0 x1 x2] [x3 x4 x5] [x6 x7 x8] sage: det(A) -x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8
TESTS:
There are many ways to create an empty matrix::
sage: m = matrix(); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Integer Ring sage: m = matrix(sparse=True); m; m.parent() [] Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring sage: m = matrix(QQ); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Rational Field sage: m = matrix(ring=QQ); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Rational Field sage: m = matrix(0); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Integer Ring sage: m = matrix(0, 0, ring=QQ); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Rational Field sage: m = matrix([]); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Integer Ring sage: m = matrix(QQ, []); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Rational Field sage: m = matrix(QQ, {}); m; m.parent() [] Full MatrixSpace of 0 by 0 sparse matrices over Rational Field
Only a ring and dimensions::
sage: m = matrix(2); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring sage: m = matrix(QQ,2); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: m = matrix(QQ,2,3); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 dense matrices over Rational Field
A ring, dimensions and a scalar::
sage: m = matrix(2,2,1); m; m.parent() [1 0] [0 1] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring sage: m = matrix(QQ,2,2,5); m; m.parent() [5 0] [0 5] Full MatrixSpace of 2 by 2 dense matrices over Rational Field
For non-square matrices, only zero works::
sage: m = matrix(2,3,0); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 dense matrices over Integer Ring sage: m = matrix(QQ,2,3,0); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 dense matrices over Rational Field sage: matrix(QQ,2,3,1) Traceback (most recent call last): ... TypeError: identity matrix must be square sage: matrix(QQ,2,3,5) Traceback (most recent call last): ... TypeError: nonzero scalar matrix must be square
Matrices specified by a list of entries::
sage: m = matrix([[1,2,3],[4,5,6]]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Integer Ring sage: m = matrix(QQ,2,[[1,2,3],[4,5,6]]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Rational Field sage: m = matrix(QQ,3,[[1,2,3],[4,5,6]]); m; m.parent() Traceback (most recent call last): ... ValueError: number of rows does not match up with specified number sage: m = matrix(QQ,2,3,[[1,2,3],[4,5,6]]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Rational Field sage: m = matrix(QQ,2,4,[[1,2,3],[4,5,6]]); m; m.parent() Traceback (most recent call last): ... ValueError: number of columns does not match up with specified number sage: m = matrix([(1,2,3),(4,5,6)]); m; m.parent() [1 2 3] [4 5 6] Full MatrixSpace of 2 by 3 dense matrices over Integer Ring sage: m = matrix([1,2,3,4,5,6]); m; m.parent() [1 2 3 4 5 6] Full MatrixSpace of 1 by 6 dense matrices over Integer Ring sage: m = matrix((1,2,3,4,5,6)); m; m.parent() [1 2 3 4 5 6] Full MatrixSpace of 1 by 6 dense matrices over Integer Ring sage: m = matrix(QQ,[1,2,3,4,5,6]); m; m.parent() [1 2 3 4 5 6] Full MatrixSpace of 1 by 6 dense matrices over Rational Field sage: m = matrix(QQ,3,2,[1,2,3,4,5,6]); m; m.parent() [1 2] [3 4] [5 6] Full MatrixSpace of 3 by 2 dense matrices over Rational Field sage: m = matrix(QQ,2,4,[1,2,3,4,5,6]); m; m.parent() Traceback (most recent call last): ... ValueError: entries has the wrong length sage: m = matrix(QQ,5,[1,2,3,4,5,6]); m; m.parent() Traceback (most recent call last): ... TypeError: cannot construct an element of Full MatrixSpace of 5 by 1 dense matrices over Rational Field from [1, 2, 3, 4, 5, 6]!
Matrices specified by a dict of entries::
sage: m = matrix({(1,1): 2}); m; m.parent() [0 0] [0 2] Full MatrixSpace of 2 by 2 sparse matrices over Integer Ring sage: m = matrix({(1,1): 2}, sparse=False); m; m.parent() [0 0] [0 2] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring sage: m = matrix(QQ,{(1,1): 2}); m; m.parent() [0 0] [0 2] Full MatrixSpace of 2 by 2 sparse matrices over Rational Field sage: m = matrix(QQ,3,{(1,1): 2}); m; m.parent() [0 0 0] [0 2 0] [0 0 0] Full MatrixSpace of 3 by 3 sparse matrices over Rational Field sage: m = matrix(QQ,3,4,{(1,1): 2}); m; m.parent() [0 0 0 0] [0 2 0 0] [0 0 0 0] Full MatrixSpace of 3 by 4 sparse matrices over Rational Field sage: m = matrix(QQ,2,{(1,1): 2}); m; m.parent() [0 0] [0 2] Full MatrixSpace of 2 by 2 sparse matrices over Rational Field sage: m = matrix(QQ,1,{(1,1): 2}); m; m.parent() Traceback (most recent call last): ... IndexError: invalid entries list sage: m = matrix({}); m; m.parent() [] Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring sage: m = matrix(QQ,{}); m; m.parent() [] Full MatrixSpace of 0 by 0 sparse matrices over Rational Field sage: m = matrix(QQ,2,{}); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 sparse matrices over Rational Field sage: m = matrix(QQ,2,3,{}); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 sparse matrices over Rational Field sage: m = matrix(2,{}); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 sparse matrices over Integer Ring sage: m = matrix(2,3,{}); m; m.parent() [0 0 0] [0 0 0] Full MatrixSpace of 2 by 3 sparse matrices over Integer Ring
Matrices with zero rows or columns::
sage: m = matrix(0,2); m; m.parent() [] Full MatrixSpace of 0 by 2 dense matrices over Integer Ring sage: m = matrix(2,0); m; m.parent() [] Full MatrixSpace of 2 by 0 dense matrices over Integer Ring sage: m = matrix(0,[1]); m; m.parent() Traceback (most recent call last): ... ValueError: entries has the wrong length sage: m = matrix(1,0,[]); m; m.parent() [] Full MatrixSpace of 1 by 0 dense matrices over Integer Ring sage: m = matrix(0,1,[]); m; m.parent() [] Full MatrixSpace of 0 by 1 dense matrices over Integer Ring sage: m = matrix(0,[]); m; m.parent() [] Full MatrixSpace of 0 by 0 dense matrices over Integer Ring sage: m = matrix(0,{}); m; m.parent() [] Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring sage: m = matrix(0,{(1,1):2}); m; m.parent() Traceback (most recent call last): ... IndexError: invalid entries list sage: m = matrix(2,0,{(1,1):2}); m; m.parent() Traceback (most recent call last): ... IndexError: invalid entries list
Check conversion from numpy::
sage: import numpy sage: n = numpy.array([[numpy.complex(0,1),numpy.complex(0,2)],[3,4]],complex) sage: m = matrix(n); m; m.parent() [1.0*I 2.0*I] [ 3.0 4.0] Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field sage: n = numpy.array([[1,2],[3,4]],'int32') sage: m = matrix(n); m; m.parent() [1 2] [3 4] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring sage: n = numpy.array([[1,2,3],[4,5,6],[7,8,9]],'float32') sage: m = matrix(n); m; m.parent() [1.0 2.0 3.0] [4.0 5.0 6.0] [7.0 8.0 9.0] Full MatrixSpace of 3 by 3 dense matrices over Real Double Field sage: n = numpy.matrix([[1,2,3],[4,5,6],[7,8,9]],'float64') sage: m = matrix(n); m; m.parent() [1.0 2.0 3.0] [4.0 5.0 6.0] [7.0 8.0 9.0] Full MatrixSpace of 3 by 3 dense matrices over Real Double Field sage: n = numpy.array([[1,2,3],[4,5,6],[7,8,9]],'complex64') sage: m = matrix(n); m; m.parent() [1.0 2.0 3.0] [4.0 5.0 6.0] [7.0 8.0 9.0] Full MatrixSpace of 3 by 3 dense matrices over Complex Double Field sage: n = numpy.matrix([[1,2,3],[4,5,6],[7,8,9]],'complex128') sage: m = matrix(n); m; m.parent() [1.0 2.0 3.0] [4.0 5.0 6.0] [7.0 8.0 9.0] Full MatrixSpace of 3 by 3 dense matrices over Complex Double Field sage: a = matrix([[1,2],[3,4]]) sage: b = matrix(a.numpy()); b [1 2] [3 4] sage: a == b True sage: c = matrix(a.numpy('float32')); c [1.0 2.0] [3.0 4.0] sage: matrix(numpy.array([[5]])) [5] sage: matrix(numpy.matrix([[5]])) [5]
A ring and a numpy array::
sage: n = numpy.array([[1,2,3],[4,5,6],[7,8,9]],'float32') sage: m = matrix(ZZ, n); m; m.parent() [1 2 3] [4 5 6] [7 8 9] Full MatrixSpace of 3 by 3 dense matrices over Integer Ring sage: n = matrix(QQ, 2, 2, [1, 1/2, 1/3, 1/4]).numpy(); n array([[ 1. , 0.5 ], [ 0.33333333, 0.25 ]]) sage: matrix(QQ, n) [ 1 1/2] [1/3 1/4]
The dimensions of a matrix may be given as numpy types::
sage: matrix(numpy.int32(2), ncols=numpy.int32(3)) [0 0 0] [0 0 0]
The dimensions of a matrix must have an integral type::
sage: matrix(RR, 2.0, 2.0) Traceback (most recent call last): ... TypeError: invalid matrix constructor: type matrix? for help
More tests::
sage: v = vector(ZZ, [1, 10, 100]) sage: m = matrix(ZZ['x'], v); m; m.parent() [ 1 10 100] Full MatrixSpace of 1 by 3 dense matrices over Univariate Polynomial Ring in x over Integer Ring sage: matrix(ZZ, 10, 10, range(100)).parent() Full MatrixSpace of 10 by 10 dense matrices over Integer Ring sage: m = matrix(GF(7), [[1/3,2/3,1/2], [3/4,4/5,7]]); m; m.parent() [5 3 4] [6 5 0] Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 7 sage: m = matrix([[1,2,3], [RDF(2), CDF(1,2), 3]]); m; m.parent() [ 1.0 2.0 3.0] [ 2.0 1.0 + 2.0*I 3.0] Full MatrixSpace of 2 by 3 dense matrices over Complex Double Field sage: m = matrix(3,3,1/2); m; m.parent() [1/2 0 0] [ 0 1/2 0] [ 0 0 1/2] Full MatrixSpace of 3 by 3 dense matrices over Rational Field sage: matrix([[1],[2,3]]) Traceback (most recent call last): ... ValueError: list of rows is not valid (rows are wrong types or lengths) sage: matrix([[1],2]) Traceback (most recent call last): ... ValueError: list of rows is not valid (rows are wrong types or lengths) sage: matrix(vector(RR,[1,2,3])).parent() Full MatrixSpace of 1 by 3 dense matrices over Real Field with 53 bits of precision
Check :trac:`10158`::
sage: matrix(ZZ, [[0] for i in range(10^5)]).is_zero() True
Test conversion using a ``_matrix_`` method::
sage: A = gap(MatrixSpace(QQ, 2, 2)(range(4))) sage: matrix(QQ, A) [0 1] [2 3] sage: matrix(A, ring=QQ) [0 1] [2 3]
A redundant ``ring`` argument::
sage: matrix(ZZ, 3, 3, ring=ZZ) Traceback (most recent call last): ... TypeError: invalid matrix constructor: type matrix? for help
TESTS:
Some calls using an iterator (note that xrange is no longer available in Python 3)::
sage: from six.moves import range sage: matrix(QQ, 3, 6, range(18), sparse=true) [ 0 1 2 3 4 5] [ 6 7 8 9 10 11] [12 13 14 15 16 17] sage: matrix(4, 4, range(16)) [ 0 1 2 3] [ 4 5 6 7] [ 8 9 10 11] [12 13 14 15]
AUTHORS:
- William Stein: Initial implementation
- Jason Grout (2008-03): almost a complete rewrite, with bits and pieces from the original implementation
- Jeroen Demeyer (2016-02-05): major clean up, see :trac:`20015` and :trac:`20016` """ def __call__(self, *Args, ring=None, nrows=None, ncols=None, sparse=None):
# ring argument
# object with _matrix_ method pass else: else: else: return makematrix(*args, sparse=sparse)
# nrows argument pass else:
# ncols argument pass else:
# Now we've taken care of initial ring, nrows, and ncols arguments. # We've also taken care of the Sage object case.
# Now the rest of the arguments are a list of rows, a flat list of # entries, a callable, a dict, a numpy array, or a single value. # If no entries are specified, pass back a zero matrix raise TypeError("when passing in a callable, the dimensions of the matrix must be specified") nrows = ncols
# no entries are specified, pass back the zero matrix # Ensure we have a list of lists, each inner list having the same number of elements # We have a list of rows or vectors
else: # We have a flat list; figure out nrows and ncols
# We have a dictionary: default to sparse # no entries are specified, pass back the zero matrix else: # note that ncols can still be None if nrows is set -- # it will be assigned nrows down below.
# See the construction after the numpy case below. else: # Convert to a numpy array if it was a matrix.
raise TypeError('numpy matrix must be either c_contiguous or f_contiguous')
elif str_dtype.count('object') == 1: # Get the raw nested list from the numpy array # and feed it back into matrix m = matrix([list(row) for row in list(arg)]) else: raise TypeError("cannot convert NumPy matrix to Sage matrix")
# assume that we should just pass the thing into the # MatrixSpace constructor and hope for the best # This is not documented, but it is doctested else: else: # len(args) >= 2
Matrix = matrix = MatrixFactory()
def prepare(w): """ Given a list w of numbers, find a common ring that they all canonically map to, and return the list of images of the elements of w in that ring along with the ring.
This is for internal use by the matrix function.
INPUT:
- ``w`` - list
OUTPUT:
list, ring
EXAMPLES::
sage: sage.matrix.constructor.prepare([-2, Mod(1,7)]) ([5, 1], Ring of integers modulo 7)
Notice that the elements must all canonically coerce to a common ring (since Sequence is called)::
sage: sage.matrix.constructor.prepare([2/1, Mod(1,7)]) Traceback (most recent call last): ... TypeError: unable to find a common ring for all elements
TESTS:
Check that :trac:`19920` is fixed::
sage: import numpy sage: matrix([[numpy.int8(1)]]) [1] """ return Sequence([], ZZ), ZZ
def prepare_dict(w): """ Given a dictionary w of numbers, find a common ring that they all canonically map to, and return the dictionary of images of the elements of w in that ring along with the ring.
This is for internal use by the matrix function.
INPUT:
- ``w`` -- dict
OUTPUT:
dict, ring
EXAMPLES::
sage: sage.matrix.constructor.prepare_dict({(0,1):2, (4,10):Mod(1,7)}) ({(0, 1): 2, (4, 10): 1}, Ring of integers modulo 7) """
def nrows_from_dict(d): """ Given a dictionary that defines a sparse matrix, return the number of rows that matrix should have.
This is for internal use by the matrix function.
INPUT:
- ``d`` - dict
OUTPUT:
integer
EXAMPLES::
sage: sage.matrix.constructor.nrows_from_dict({}) 0
Here the answer is 301 not 300, since there is a 0-th row.
::
sage: sage.matrix.constructor.nrows_from_dict({(300,4):10}) 301 """
def ncols_from_dict(d): """ Given a dictionary that defines a sparse matrix, return the number of columns that matrix should have.
This is for internal use by the matrix function.
INPUT:
- ``d`` - dict
OUTPUT:
integer
EXAMPLES::
sage: sage.matrix.constructor.ncols_from_dict({}) 0
Here the answer is 301 not 300, since there is a 0-th row.
::
sage: sage.matrix.constructor.ncols_from_dict({(4,300):10}) 301 """
from .special import * |