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""" Miscellaneous matrix functions
"""
#***************************************************************************** # Copyright (C) 2008 William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** from six.moves import range from six import iteritems
from sage.categories.fields import Fields from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.integer_ring import ZZ _Fields = Fields()
def row_iterator(A):
def row_reduced_form(M,transformation=False): """ This function computes a row reduced form of a matrix over a rational function field `k(x)`, for `k` a field.
INPUT:
- `M` - a matrix over `k(x)` or `k[x]` for `k` a field. - `transformation` - A boolean (default: `False`). If this boolean is set to `True` a second matrix is output (see OUTPUT).
OUTPUT:
If `transformation` is `False`, the output is `W`, a row reduced form of `M`.
If `transformation` is `True`, this function will output a pair `(W,N)` consisting of two matrices over `k(x)`:
1. `W` - a row reduced form of `M`. 2. `N` - an invertible matrix over `k(x)` satisfying `NW = M`.
EXAMPLES:
The function expects matrices over the rational function field, but other examples below show how one can provide matrices over the ring of polynomials (whose quotient field is the rational function field).
::
sage: R.<t> = GF(3)['t'] sage: K = FractionField(R) sage: import sage.matrix.matrix_misc sage: sage.matrix.matrix_misc.row_reduced_form(matrix([[(t-1)^2/t],[(t-1)]])) doctest:...: DeprecationWarning: Row reduced form will soon be supported only for matrices of polynomials. See http://trac.sagemath.org/21024 for details. [ 0] [(t + 2)/t]
The last example shows the usage of the transformation parameter.
:: sage: Fq.<a> = GF(2^3) sage: Fx.<x> = Fq[] sage: A = matrix(Fx,[[x^2+a,x^4+a],[x^3,a*x^4]]) sage: from sage.matrix.matrix_misc import row_reduced_form sage: row_reduced_form(A,transformation=True) ( [ x^2 + a x^4 + a] [1 0] [x^3 + a*x^2 + a^2 a^2], [a 1] )
NOTES:
See docstring for row_reduced_form method of matrices for more information. """
# determine whether M has polynomial or rational function coefficients
#Compute the base polynomial ring else:
# calculate least-common denominator of matrix entries and clear # denominators. The result lies in R else: # No need to clear denominators
else:
else:
def prm_mul(p1, p2, mask_free, prec): """ Return the product of ``p1`` and ``p2``, putting free variables in ``mask_free`` to `1`.
This function is mainly use as a subroutine of :func:`permanental_minor_polynomial`.
INPUT:
- `p1,p2` -- polynomials as dictionaries
- `mask_free` -- an integer mask that give the list of free variables (the `i`-th variable is free if the `i`-th bit of ``mask_free`` is `1`)
- `prec` -- if `prec` is not None, truncate the product at precision `prec`
EXAMPLES::
sage: from sage.matrix.matrix_misc import prm_mul sage: t = polygen(ZZ, 't') sage: p1 = {0: 1, 1: t, 4: t} sage: p2 = {0: 1, 1: t, 2: t} sage: prm_mul(p1, p2, 1, None) {0: 2*t + 1, 2: t^2 + t, 4: t^2 + t, 6: t^2} """ return p continue else:
def permanental_minor_polynomial(A, permanent_only=False, var='t', prec=None): r""" Return the polynomial of the sums of permanental minors of ``A``.
INPUT:
- `A` -- a matrix
- `permanent_only` -- if True, return only the permanent of `A`
- `var` -- name of the polynomial variable
- `prec` -- if prec is not None, truncate the polynomial at precision `prec`
The polynomial of the sums of permanental minors is
.. MATH::
\sum_{i=0}^{min(nrows, ncols)} p_i(A) x^i
where `p_i(A)` is the `i`-th permanental minor of `A` (that can also be obtained through the method :meth:`~sage.matrix.matrix2.Matrix.permanental_minor` via ``A.permanental_minor(i)``).
The algorithm implemented by that function has been developed by P. Butera and M. Pernici, see [BP2015]_. Its complexity is `O(2^n m^2 n)` where `m` and `n` are the number of rows and columns of `A`. Moreover, if `A` is a banded matrix with width `w`, that is `A_{ij}=0` for `|i - j| > w` and `w < n/2`, then the complexity of the algorithm is `O(4^w (w+1) n^2)`.
INPUT:
- ``A`` -- matrix
- ``permanent_only`` -- optional boolean. If ``True``, only the permanent is computed (might be faster).
- ``var`` -- a variable name
EXAMPLES::
sage: from sage.matrix.matrix_misc import permanental_minor_polynomial sage: m = matrix([[1,1],[1,2]]) sage: permanental_minor_polynomial(m) 3*t^2 + 5*t + 1 sage: permanental_minor_polynomial(m, permanent_only=True) 3 sage: permanental_minor_polynomial(m, prec=2) 5*t + 1
::
sage: M = MatrixSpace(ZZ,4,4) sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1]) sage: permanental_minor_polynomial(A) 84*t^3 + 114*t^2 + 28*t + 1 sage: [A.permanental_minor(i) for i in range(5)] [1, 28, 114, 84, 0]
An example over `\QQ`::
sage: M = MatrixSpace(QQ,2,2) sage: A = M([1/5,2/7,3/2,4/5]) sage: permanental_minor_polynomial(A, True) 103/175
An example with polynomial coefficients::
sage: R.<a> = PolynomialRing(ZZ) sage: A = MatrixSpace(R,2)([[a,1], [a,a+1]]) sage: permanental_minor_polynomial(A, True) a^2 + 2*a
A usage of the ``var`` argument::
sage: m = matrix(ZZ,4,[0,1,2,3,1,2,3,0,2,3,0,1,3,0,1,2]) sage: permanental_minor_polynomial(m, var='x') 164*x^4 + 384*x^3 + 172*x^2 + 24*x + 1
ALGORITHM:
The permanent `perm(A)` of a `n \times n` matrix `A` is the coefficient of the `x_1 x_2 \ldots x_n` monomial in
.. MATH::
\prod_{i=1}^n \left( \sum_{j=1}^n A_{ij} x_j \right)
Evaluating this product one can neglect `x_i^2`, that is `x_i` can be considered to be nilpotent of order `2`.
To formalize this procedure, consider the algebra `R = K[\eta_1, \eta_2, \ldots, \eta_n]` where the `\eta_i` are commuting, nilpotent of order `2` (i.e. `\eta_i^2 = 0`). Formally it is the quotient ring of the polynomial ring in `\eta_1, \eta_2, \ldots, \eta_n` quotiented by the ideal generated by the `\eta_i^2`.
We will mostly consider the ring `R[t]` of polynomials over `R`. We denote a generic element of `R[t]` by `p(\eta_1, \ldots, \eta_n)` or `p(\eta_{i_1}, \ldots, \eta_{i_k})` if we want to emphasize that some monomials in the `\eta_i` are missing.
Introduce an "integration" operation `\langle p \rangle` over `R` and `R[t]` consisting in the sum of the coefficients of the non-vanishing monomials in `\eta_i` (i.e. the result of setting all variables `\eta_i` to `1`). Let us emphasize that this is *not* a morphism of algebras as `\langle \eta_1 \rangle^2 = 1` while `\langle \eta_1^2 \rangle = 0`!
Let us consider an example of computation. Let `p_1 = 1 + t \eta_1 + t \eta_2` and `p_2 = 1 + t \eta_1 + t \eta_3`. Then
.. MATH::
p_1 p_2 = 1 + 2t \eta_1 + t (\eta_2 + \eta_3) + t^2 (\eta_1 \eta_2 + \eta_1 \eta_3 + \eta_2 \eta_3)
and
.. MATH::
\langle p_1 p_2 \rangle = 1 + 4t + 3t^2
In this formalism, the permanent is just
.. MATH::
perm(A) = \langle \prod_{i=1}^n \sum_{j=1}^n A_{ij} \eta_j \rangle
A useful property of `\langle . \rangle` which makes this algorithm efficient for band matrices is the following: let `p_1(\eta_1, \ldots, \eta_n)` and `p_2(\eta_j, \ldots, \eta_n)` be polynomials in `R[t]` where `j \ge 1`. Then one has
.. MATH::
\langle p_1(\eta_1, \ldots, \eta_n) p_2 \rangle = \langle p_1(1, \ldots, 1, \eta_j, \ldots, \eta_n) p_2 \rangle
where `\eta_1,..,\eta_{j-1}` are replaced by `1` in `p_1`. Informally, we can "integrate" these variables *before* performing the product. More generally, if a monomial `\eta_i` is missing in one of the terms of a product of two terms, then it can be integrated in the other term.
Now let us consider an `m \times n` matrix with `m \leq n`. The *sum of permanental `k`-minors of `A`* is
.. MATH::
perm(A, k) = \sum_{r,c} perm(A_{r,c})
where the sum is over the `k`-subsets `r` of rows and `k`-subsets `c` of columns and `A_{r,c}` is the submatrix obtained from `A` by keeping only the rows `r` and columns `c`. Of course `perm(A, \min(m,n)) = perm(A)` and note that `perm(A,1)` is just the sum of all entries of the matrix.
The generating function of these sums of permanental minors is
.. MATH::
g(t) = \left\langle \prod_{i=1}^m \left(1 + t \sum_{j=1}^n A_{ij} \eta_j\right) \right\rangle
In fact the `t^k` coefficient of `g(t)` corresponds to choosing `k` rows of `A`; `\eta_i` is associated to the i-th column; nilpotency avoids having twice the same column in a product of `A`'s.
For more details, see the article [BP2015]_.
From a technical point of view, the product in `K[\eta_1, \ldots, \eta_n][t]` is implemented as a subroutine in :func:`prm_mul`. The indices of the rows and columns actually start at `0`, so the variables are `\eta_0, \ldots, \eta_{n-1}`. Polynomials are represented in dictionary form: to a variable `\eta_i` is associated the key `2^i` (or in Python ``1 << i``). The keys associated to products are obtained by considering the development in base `2`: to the monomial `\eta_{i_1} \ldots \eta_{i_k}` is associated the key `2^{i_1} + \ldots + 2^{i_k}`. So the product `\eta_1 \eta_2` corresponds to the key `6 = (110)_2` while `\eta_0 \eta_3` has key `9 = (1001)_2`. In particular all operations on monomials are implemented via bitwise operations on the keys. """ raise ValueError('the argument `prec` must be a positive integer')
# build the polynomial p1 = 1 + t sum A_{ij} eta_j else:
# make the product with the preceding polynomials, taking care of # variables that can be integrated else:
return K.zero()
raise RuntimeError("Something is wrong! Certainly a problem in the" " algorithm... please contact sage-devel@googlegroups.com")
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