Hi,
Is it possible to define functions from $\mathbb{R}^m$ to $\mathbb{R}^n$ in Sage? And then, compose them, compute their jacobian (see also this previous post), use them for change of variables, ...
Is there something wrong with the following example for computing Jacobian ?
sage: var('x y')
sage: f(x, y) = (x^2 + y, x - sin(y))
sage: f.derivative()
[ (x, y) |--> 2*x (x, y) |--> 1]
[ (x, y) |--> 1 (x, y) |--> -cos(y)]
sage: f.derivative()(x=0, y=2)
[ 0 1]
[ 1 -cos(2)]
Concerning composition, this is indeed more tricky, because:
sage: f(x, y).parent()
Vector space of dimension 2 over Symbolic Ring
So, you could try:
sage: g(x, y) = (2*y, x)
sage: g(f[0], f[1])
(2*x - 2*sin(y), x^2 + y)
Thanks! For Jacobian, there is something wrong: the string is ugly.
Use: g(*f) to substitute in the components of f into g.
Also, we could print the callable matrices more simply, like we print the callable vectors. Compare: f(x,y)=(3*x,y+x,sin(x)) (we print the (x,y)|--> outside of the vector). We should make a callable symbolic matrix class that prints the domain variables outside of the matrix. It would be the analog to: http://hg.sagemath.org/sage-main/src/0f8fd922eaed351e39f913f1317d319dcceb4c01/sage/modules/vector_callable_symbolic_dense.py?at=default (and shouldn't be that hard to do).
Also, you don't have to do var('x y') because f(x,y)=... automatically declares the variables x and y.
Reformulating some of my comments above as an answer so they don't get lost:
To compose, you can do:
f(x,y)=(x+2*y, x^2)
g(x,y)=(sin(x), cos(x*y))
f(*g)
Note also that I didn't have to declare x and y to be variables; that is automatically done when they are used as inputs in an f(x,y)=... declaration.
As for printing of symbolic matrices, see my comment above. I would love if someone would write the necessary straightforward class to improve the printing.
You mean like http://trac.sagemath.org/sage_trac/ticket/12075?
Asked: 2013-06-26 13:05:03 +0200
Seen: 605 times
Last updated: Jun 29 '13
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