# Evaluation#

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 as Q(M). The inclusion of Q2 as an argument is to avoid having to create a QuadraticForm() here, which for now creates circular imports.

INPUT:

• QQuadraticForm over a base ring $$R$$

• M – a Q.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: 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 ]


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 as Q(M).

INPUT:

• QQuadraticForm over a base ring $$R$$

• v – a tuple or list (or column matrix) of Q.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