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 | 1 | initial version |
First an answer for the special case, then a more general answer.
Trigonometric functions
In the special case of the example in the question, this helps:
sage: f(x) = 17
sage: g(x) = cos(x)^2
sage: h(x) = cos(2*x)
sage: h(x).simplify_trig()
2*cos(x)^2 - 1
Linear independence of functions
Returning to the general question, one way to check for possible linear dependence relations, or for proofs of independence, would be to project to a finite-dimensional subspace and check there.
Form a vectors from the value of each function at a chosen collection of evaluation points.
If these vectors are linearly independent, then the functions are.
If there is a linear relation between these vectors, then it's worth checking if the linear relation in fact holds for the functions themselves.
Let us illustrate with the three functions above.
Choose a collection of evaluation points:
sage: xx = (0, pi/6, pi/4, pi/3, pi/2)
Form a vector for each function:
sage: u = vector(AA, [f(x) for x in xx])
sage: v = vector(AA, [g(x) for x in xx])
sage: w = vector(AA, [h(x) for x in xx])
To look for dependence relations, compute the left kernel of the matrix with these vectors as rows.
sage: m = matrix([u, v, w])
sage: K = m.kernel()
sage: K
Vector space of degree 3 and dimension 1 over Algebraic Real Field
Basis matrix:
[ 1 -34 17]
This suggests that maybe f - 34*g + 17*h is zero.
Extract the coefficients:
sage: a, b, c = K.basis()[0]
sage: a, b, c
(1, -34, 17)
Check if the linear relation in fact holds for the functions:
sage: bool(a*f(x) + b*g(x) + c*h(x) == 0)
True
| | 2 | No.2 Revision |
First an answer for the special case, then a more general answer.
Trigonometric functions
In the special case of the example in the question, this helps:
sage: f(x) = 17
sage: g(x) = cos(x)^2
sage: h(x) = cos(2*x)
sage: h(x).simplify_trig()
2*cos(x)^2 - 1
Linear independence of functions
Returning to the general question, one way to check for possible linear dependence relations, or for proofs of independence, would be to project to a finite-dimensional subspace and check there.
Form a vectors from the value of each function at a chosen collection of evaluation points.
If these vectors are linearly independent, then the functions are.
If there is a linear relation between these vectors, then it's worth checking if the linear relation in fact holds for the functions themselves.
Let us illustrate with the three functions above.
Choose a collection of evaluation points:
sage: xx = (0, pi/6, pi/4, pi/3, pi/2)
Form a vector for each function:
sage: u = vector(AA, [f(x) vector([f(x) for x in xx])
sage: v = vector(AA, [g(x) vector([g(x) for x in xx])
sage: w = vector(AA, [h(x) vector([h(x) for x in xx])
To look for dependence relations, compute the left kernel of the matrix with these vectors as rows.
sage: m = matrix([u, v, w])
sage: K = m.kernel()
sage: K
Vector space of degree 3 and dimension 1 over Algebraic Real Field
Basis matrix:
[ 1 -34 17]
This suggests that maybe f - 34*g + 17*h is zero.
Extract the coefficients:
sage: a, b, c = K.basis()[0]
sage: a, b, c
(1, -34, 17)
Check if the linear relation in fact holds for the functions:
sage: bool(a*f(x) + b*g(x) + c*h(x) == 0)
True
