SageMath - Precomposing a vector-valued function

Question

Given a function from R into R^n, I'd like to define a new function by precomposition, for example as follows

alpha(x) = [e^x,e^(-x)]
beta(x) = alpha(-x+2)

However attempting to do so in this way throws an error "unable to convert (e^(-x + 2), e^(x - 2)) to a symbolic expression"

Now the similar but simpler version of the code

alpha(x) = e^x
beta(x) = alpha(-x+2)

works perfectly, so the issue arrises from the fact that alpha is multivalued.

The following variant of the original code does exactly what I want

alpha(x) = [e^x,e^(-x)]
beta(x) = [alpha[0](-x+2),alpha[1](-x+2)]

but requires me to assume the length of alpha, which is undesirable. And the obvious solution to that problem

alpha(x) = [e^x,e^(-x)]
for i in range(0,len(alpha)):
  beta[i](x) = alpha[i](x)

or any variant thereupon throws the error "can't assign to function call"

My question is as follows: Is there any way to do this precomposition? In particular without assuming the length of alpha. I control how the functions alpha and beta are defined, so if theres another way of defining them (for example using lambda notation or something like that) that lets me do this, that's acceptable too. But note that I would like to do some equivalent of the following at some point in my code ... + beta.derivative(x).dot_product( ...

Answer 1

Defined as in the question, alpha is not a symbolic function returning vectors, but a vector of callable functions.

Below we describe two other ways of defining alpha and beta, either defining alpha as a vector over the symbolic ring, and defining beta by substitution, or defining alpha and beta as Python functions.

Original approach in the question:

sage: alpha(x) = [e^x, e^-x]

sage: alpha
x |--> (e^x, e^(-x))
sage: alpha.parent()
Vector space of dimension 2 over Callable function ring with argument x

Using a vector over the symbolic ring

sage: alpha = vector([e^x, e^-x])

sage: alpha
(e^x, e^(-x))
sage: alpha.parent()
Vector space of dimension 2 over Symbolic Ring

sage: beta = alpha.subs({x: -x + 2})
sage: beta
(e^(-x + 2), e^(x - 2))

Using Python functions

sage: def alpha(x):
....:     return vector([e^x, e^-x])
....:
sage: def beta(x):
....:     return alpha(-x + 2)
....:
sage: beta(x)
(e^(-x + 2), e^(x - 2))

Some related resources.

Query:

• Ask Sage query: vector function

Questions:

• Ask Sage question 8066: Composite function
• Ask Sage question 24943: Define vector valued function of a vector of symbolic variables?
• Ask Sage question 43550: vector constants and vector functions
• Ask Sage question 9375: functions with vector inputs
• Ask Sage question 23758: Defining a function of vector variables
• Ask Sage question 36127: Plotting 2d vector fields – how to delay function evaluation
• Ask Sage question 10704: How to create a vector function (mapping)?
• Ask Sage question 8924: Basic vector functions in Sage
• Ask Sage question 30025: How can I define a function with quaternion argument, and other non-vector input
• Ask Sage question 47710: Vector valued function: unable to convert to symbolic expression
• Ask Sage question 36159: apply functions iteratively (modified re-post)

Tickets:

• Sage Trac ticket 11507: make f(x,y,z)=vector make a vector-valued function
• Sage Trac ticket 11180: Allow vector input to functions taking vectors
• Sage Trac ticket 28640: Manifolds: Vector Valued Forms