Coverage for local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__variable_substitutions.py : 79%
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""" Variable Substitution, Multiplication, Division, Scaling
""" #***************************************************************************** # Copyright (C) 2007 William Stein and Jonathan Hanke # # 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/ #*****************************************************************************
import copy
def swap_variables(self, r, s, in_place = False): """ Switch the variables `x_r` and `x_s` in the quadratic form (replacing the original form if the in_place flag is True).
INPUT:
`r`, `s` -- integers >= 0
OUTPUT:
a QuadraticForm (by default, otherwise none)
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 4, range(1,11)) sage: Q Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 4 ] [ * 5 6 7 ] [ * * 8 9 ] [ * * * 10 ]
sage: Q.swap_variables(0,2) Quadratic form in 4 variables over Integer Ring with coefficients: [ 8 6 3 9 ] [ * 5 2 7 ] [ * * 1 4 ] [ * * * 10 ]
sage: Q.swap_variables(0,2).swap_variables(0,2) Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 4 ] [ * 5 6 7 ] [ * * 8 9 ] [ * * * 10 ]
"""
else: ## Switch diagonal elements
## Switch off-diagonal elements
def multiply_variable(self, c, i, in_place = False): """ Replace the variables `x_i` by `c*x_i` in the quadratic form (replacing the original form if the in_place flag is True).
Here `c` must be an element of the base_ring defining the quadratic form.
INPUT:
`c` -- an element of Q.base_ring()
`i` -- an integer >= 0
OUTPUT:
a QuadraticForm (by default, otherwise none)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7]) sage: Q.multiply_variable(5,0) Quadratic form in 4 variables over Integer Ring with coefficients: [ 25 0 0 0 ] [ * 9 0 0 ] [ * * 5 0 ] [ * * * 7 ]
"""
else: ## Stretch the diagonal element
## Switch off-diagonal elements
def divide_variable(self, c, i, in_place = False): """ Replace the variables `x_i` by `(x_i)/c` in the quadratic form (replacing the original form if the in_place flag is True).
Here `c` must be an element of the base_ring defining the quadratic form, and the division must be defined in the base ring.
INPUT:
`c` -- an element of Q.base_ring()
`i` -- an integer >= 0
OUTPUT:
a QuadraticForm (by default, otherwise none)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7]) sage: Q.divide_variable(3,1) Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 0 0 0 ] [ * 1 0 0 ] [ * * 5 0 ] [ * * * 7 ]
"""
else: ## Stretch the diagonal element
## Switch off-diagonal elements
def scale_by_factor(self, c, change_value_ring_flag=False): """ Scale the values of the quadratic form by the number `c`, if this is possible while still being defined over its base ring.
If the flag is set to true, then this will alter the value ring to be the field of fractions of the original ring (if necessary).
INPUT:
`c` -- a scalar in the fraction field of the value ring of the form.
OUTPUT:
A quadratic form of the same dimension
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [3,9,18,27]) sage: Q.scale_by_factor(3) Quadratic form in 4 variables over Integer Ring with coefficients: [ 9 0 0 0 ] [ * 27 0 0 ] [ * * 54 0 ] [ * * * 81 ]
sage: Q.scale_by_factor(1/3) Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 0 0 0 ] [ * 3 0 0 ] [ * * 6 0 ] [ * * * 9 ]
""" ## Try to scale the coefficients while staying in the ring of values.
## Check if we can preserve the value ring and return result. -- USE THE BASE_RING FOR NOW... # This is a hack: we would like to use QuadraticForm here, but # it doesn't work by scoping reasons. except Exception: if not change_value_ring_flag: raise TypeError("Oops! We could not rescale the lattice in this way and preserve its defining ring.") else: raise UntestedCode("This code is not tested by current doctests!") F = R.fraction_field() list2 = [F(x) for x in new_coeff_list] Q = copy.deepcopy(self) Q.__init__(self.dim(), F, list2, R) ## DEFINE THIS! IT WANTS TO SET THE EQUIVALENCE RING TO R, BUT WITH COEFFS IN F. #Q.set_equivalence_ring(R) return Q
def extract_variables(self, var_indices): """ Extract the variables (in order) whose indices are listed in var_indices, to give a new quadratic form.
INPUT:
var_indices -- a list of integers >= 0
OUTPUT:
a QuadraticForm
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 4, range(10)); Q Quadratic form in 4 variables over Integer Ring with coefficients: [ 0 1 2 3 ] [ * 4 5 6 ] [ * * 7 8 ] [ * * * 9 ] sage: Q.extract_variables([1,3]) Quadratic form in 2 variables over Integer Ring with coefficients: [ 4 6 ] [ * 9 ]
"""
def elementary_substitution(self, c, i, j, in_place = False): ## CHECK THIS!!! """ Perform the substitution `x_i --> x_i + c*x_j` (replacing the original form if the in_place flag is True).
INPUT:
`c` -- an element of Q.base_ring()
`i`, `j` -- integers >= 0
OUTPUT:
a QuadraticForm (by default, otherwise none)
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 4, range(1,11)) sage: Q Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 4 ] [ * 5 6 7 ] [ * * 8 9 ] [ * * * 10 ]
sage: Q.elementary_substitution(c=1, i=0, j=3) Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 6 ] [ * 5 6 9 ] [ * * 8 12 ] [ * * * 15 ]
::
sage: R = QuadraticForm(ZZ, 4, range(1,11)) sage: R Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 4 ] [ * 5 6 7 ] [ * * 8 9 ] [ * * * 10 ]
::
sage: M = Matrix(ZZ, 4, 4, [1,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1]) sage: M [1 0 0 1] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: R(M) Quadratic form in 4 variables over Integer Ring with coefficients: [ 1 2 3 6 ] [ * 5 6 9 ] [ * * 8 12 ] [ * * * 15 ]
"""
else: ## Adjust the a_{k,j} coefficients else:
def add_symmetric(self, c, i, j, in_place = False): """ Performs the substitution `x_j --> x_j + c*x_i`, which has the effect (on associated matrices) of symmetrically adding `c * j`-th row/column to the `i`-th row/column.
NOTE: This is meant for compatibility with previous code, which implemented a matrix model for this class. It is used in the local_normal_form() method.
INPUT:
`c` -- an element of Q.base_ring()
`i`, `j` -- integers >= 0
OUTPUT:
a QuadraticForm (by default, otherwise none)
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 3, range(1,7)); Q Quadratic form in 3 variables over Integer Ring with coefficients: [ 1 2 3 ] [ * 4 5 ] [ * * 6 ] sage: Q.add_symmetric(-1, 1, 0) Quadratic form in 3 variables over Integer Ring with coefficients: [ 1 0 3 ] [ * 3 2 ] [ * * 6 ] sage: Q.add_symmetric(-3/2, 2, 0) ## ERROR: -3/2 isn't in the base ring ZZ Traceback (most recent call last): ... RuntimeError: Oops! This coefficient can't be coerced to an element of the base ring for the quadratic form.
::
sage: Q = QuadraticForm(QQ, 3, range(1,7)); Q Quadratic form in 3 variables over Rational Field with coefficients: [ 1 2 3 ] [ * 4 5 ] [ * * 6 ] sage: Q.add_symmetric(-3/2, 2, 0) Quadratic form in 3 variables over Rational Field with coefficients: [ 1 2 0 ] [ * 4 2 ] [ * * 15/4 ]
"""
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