Coverage for local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__evaluate.pyx : 83%

"Evaluation" 

def 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: 

.. MATH:: 

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 

""" 

return QFEvaluateVector_cdef(Q, v) 

cdef QFEvaluateVector_cdef(Q, v): 

""" 

Routine to quickly evaluate a quadratic form Q on a vector v. See 

the Python wrapper function QFEvaluate() above for details. 

""" 

## If we are passed a matrix A, return the quadratic form Q(A(x)) 

## (In matrix notation: A^t * Q * A) 

n = Q.dim() 

tmp_val = Q.base_ring()(0) 

for i from 0 <= i < n: 

for j from i <= j < n: 

tmp_val += Q[i,j] * v[i] * v[j] 

## Return the value (over R) 

return Q.base_ring()._coerce_(tmp_val) 

def 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 ] 

""" 

return QFEvaluateMatrix_cdef(Q, M, Q2) 

cdef QFEvaluateMatrix_cdef(Q, M, Q2): 

""" 

Routine to quickly evaluate a quadratic form Q on a matrix M. See 

the Python wrapper function QFEvaluateMatrix() above for details. 

""" 

## Create the new quadratic form 

n = Q.dim() 

m = Q2.dim() 

## TO DO: Check the dimensions of M are compatible with those of Q and Q2 

## Evaluate Q(M) into Q2 

for k from 0 <= k < m: 

for l from k <= l < m: 

tmp_sum = Q2.base_ring()(0) 

for i from 0 <= i < n: 

for j from i <= j < n: 

if (k == l): 

tmp_sum += Q[i,j] * (M[i,k] * M[j,l]) 

else: 

tmp_sum += Q[i,j] * (M[i,k] * M[j,l] + M[i,l] * M[j,k]) 

Q2[k,l] = tmp_sum 

return Q2