Evaluation¶
- sage.quadratic_forms.quadratic_form__evaluate.QFEvaluateMatrix(Q, M, Q2)¶
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:
Q2 = M^t * Q * 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:
Q – QuadraticForm over a base ring R
M – a Q.dim() x Q2.dim() matrix of elements of R
OUTPUT:
Q2 – 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 ]
- sage.quadratic_forms.quadratic_form__evaluate.QFEvaluateVector(Q, v)¶
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:
\[Q' = v^t * Q * v.\]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:
Q – QuadraticForm 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