Determining Whether Symbolic Polynomials Are Linear

Suppose that, using Matlab's syms expression, I generate a polynomial using a set of symbolic variables. Here is an example of such a polynomial that I could generate:

syms x y z

f = x*y + z + y*z;

Next, I want some function that will take in such a polynomial as well as a subset of symbolic variables that I designate as free. Then, this function will return whether this polynomial is linear in terms of the given subset of variables, treating all other variables as constants. I would like to find a native Matlab function that accomplishes this without having to code one from scratch myself. Let us suppose that such a function can be denoted as "is_linear". Now, I will proceed to show examples of its use on the above polynomial:

is_linear(f, [y z]) % Returns false

is_linear(f, [y]) % Returns true

I was wondering if anyone knew of a native Matlab function that would accomplish this for me or make the process of creating such a function much simpler. Thank you for reading, and I look forward to seeing responses.

Answer 1

[Torsten]

You will have to check the two properties of a linear function:

f(x+y) = f(x) + f(y)
f(a*x) = a*f(x)

for symbolic variables x,y and a.