Function

Scalar input arguments

We can define function as follows

from gradgen import Function, SX

x = SX.sym("x")
y = SX.sym("y")
f = Function("f", [x, y], [x + y, x * y])

where for f we have specified that it is a function of the input variables x and y and returns two outputs. Mathematically, it is the function $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ with

$$f(x, y) = \begin{bmatrix} x + y \\ xy \end{bmatrix}.$$

The function f can be called with numerical arguments

print(f(2.0, 3.0))  # (5.0, 6.0)

and the result is a tuple of float.

Functions can also be called with symbolic arguments

z = SX.sym("z")
w = SX.sym("w")
print(f(z + 1, w.sin()))  # symbolic outputs

and the result is a tuple of SX.

Vector inputs

We can also define functions that take vector inputs. For example, we can define the function $g:\mathbb{R}^2 \to \mathbb{R}$ given by $g(x) = \Vert x \Vert_2$ as follows

from gradgen import Function, SXVector

x = SXVector.sym("x", 2)
g = Function("g", [x], [x.norm2()])

print(g([3.0, 4.0]))  # returns 5.

Or, we can define the function $h:\mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}$ given by $h(x, a) = a\Vert x \Vert_{1}$ as follows

from gradgen import Function, SXVector, SX

x = SXVector.sym("x", 3)
a = SX.sym("a")
h = Function("h", [x, a], [a * x.norm1()])

x0 = [3.0, 4.0, 1.0]
a0 = -1.
h0 = h(x0, a0)

print(h0) 

Composition of functions

Functions can be composed in a very natural manner. Suppose we have a function $f:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}$ given by

$$f(x, a) = \Vert (a^2 + 1) x \Vert_{\infty},$$

and we have the function $g: \mathbb{R}^2 \to \mathbb{R}^2$ given by

$$g(x) = \frac{x}{\Vert x \Vert_2^2}.$$

We can then define the function $h:\mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}$ given by $h(x, a) = f(g(x), a)$. This can be defined as follows

from gradgen import Function, SXVector, SX

x = SXVector.sym("x", 3)
a = SX.sym("a")

f = Function("f", [x, a], [(a**2 + 1) * x.norm_inf()])
g = Function("g", [x], [x / x.norm2sq()])

h = Function("h", [x, a], [f(g(x), a)]) # h(x, a) = f(g(x), a)