Evaluation#
- sage.quadratic_forms.quadratic_form__evaluate.QFEvaluateMatrix(Q, M, Q2)[source]#
Evaluate this quadratic form \(Q\) on a matrix \(M\) of elements coercible to the base ring of the quadratic form, which in matrix notation is given by:
\[Q_2 = M^t\cdot Q\cdot M.\]Note
This is a Python wrapper for the fast evaluation routine
QFEvaluateMatrix_cdef()
. This routine is for internal use and is called more conveniently asQ(M)
. The inclusion ofQ2
as an argument is to avoid having to create aQuadraticForm()
here, which for now creates circular imports.INPUT:
Q
–QuadraticForm
over a base ring \(R\)M
– aQ.dim()
\(\times\)Q2.dim()
matrix of elements of \(R\)
OUTPUT: a
QuadraticForm
over \(R\)EXAMPLES:
sage: from sage.quadratic_forms.quadratic_form__evaluate import QFEvaluateMatrix 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: Q2 = QuadraticForm(ZZ, 2) sage: M = Matrix(ZZ, 4, 2, [1,0,0,0, 0,1,0,0]); M [1 0] [0 0] [0 1] [0 0] sage: QFEvaluateMatrix(Q, M, Q2) Quadratic form in 2 variables over Integer Ring with coefficients: [ 0 2 ] [ * 7 ]
>>> from sage.all import * >>> from sage.quadratic_forms.quadratic_form__evaluate import QFEvaluateMatrix >>> Q = QuadraticForm(ZZ, Integer(4), range(Integer(10))); Q Quadratic form in 4 variables over Integer Ring with coefficients: [ 0 1 2 3 ] [ * 4 5 6 ] [ * * 7 8 ] [ * * * 9 ] >>> Q2 = QuadraticForm(ZZ, Integer(2)) >>> M = Matrix(ZZ, Integer(4), Integer(2), [Integer(1),Integer(0),Integer(0),Integer(0), Integer(0),Integer(1),Integer(0),Integer(0)]); M [1 0] [0 0] [0 1] [0 0] >>> QFEvaluateMatrix(Q, M, Q2) Quadratic form in 2 variables over Integer Ring with coefficients: [ 0 2 ] [ * 7 ]
- sage.quadratic_forms.quadratic_form__evaluate.QFEvaluateVector(Q, v)[source]#
Evaluate this quadratic form \(Q\) on a vector or matrix of elements coercible to the base ring of the quadratic form.
If a vector is given, then the output will be the ring element \(Q(v)\), but if a matrix is given, then the output will be the quadratic form \(Q'\) which in matrix notation is given by:
\[Q' = v^t\cdot Q\cdot v.\]Note
This is a Python wrapper for the fast evaluation routine
QFEvaluateVector_cdef()
. This routine is for internal use and is called more conveniently asQ(M)
.INPUT:
Q
–QuadraticForm
over a base ring \(R\)v
– a tuple or list (or column matrix) ofQ.dim()
elements of \(R\)
OUTPUT: an element of \(R\)
EXAMPLES:
sage: from sage.quadratic_forms.quadratic_form__evaluate import QFEvaluateVector 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: QFEvaluateVector(Q, (1,0,0,0)) 0 sage: QFEvaluateVector(Q, (1,0,1,0)) 9
>>> from sage.all import * >>> from sage.quadratic_forms.quadratic_form__evaluate import QFEvaluateVector >>> Q = QuadraticForm(ZZ, Integer(4), range(Integer(10))); Q Quadratic form in 4 variables over Integer Ring with coefficients: [ 0 1 2 3 ] [ * 4 5 6 ] [ * * 7 8 ] [ * * * 9 ] >>> QFEvaluateVector(Q, (Integer(1),Integer(0),Integer(0),Integer(0))) 0 >>> QFEvaluateVector(Q, (Integer(1),Integer(0),Integer(1),Integer(0))) 9