# Use with ParameterizedFunctions

In the latexalign tutorial I mentioned that one can use latexalign directly on a ParameterizedFunction. Here, I make a somewhat more convoluted and hard-to-read example (you'll soon se why):

using Latexify
using ParameterizedFunctions
copy_to_clipboard(true)

ode = @ode_def positiveFeedback begin
dx = y*y*y/(k_y_x + y) - x - x
dy = x^n_x/(k_x^n_x + x^n_x) - y
end k_y k_x n_x

latexify(ode)
\begin{align} \frac{dx}{dt} =& \frac{y \cdot y \cdot y}{k_{y\_x} + y} - x - x \\ \frac{dy}{dt} =& \frac{x^{n_{x}}}{k_{x}^{n_{x}} + x^{n_{x}}} - y \\ \end{align}

This is pretty nice, but there are a few parts of the equation which could be reduced. Using a keyword argument, you can utilise the SymEngine.jl to reduce the expression before printing.

latexify(ode, field=:symfuncs)
\begin{align} \frac{dx}{dt} =& -2 \cdot x + \frac{y^{3}}{k_{y\_x} + y} \\ \frac{dy}{dt} =& - y + \frac{x^{n_{x}}}{k_{x}^{n_{x}} + x^{n_{x}}} \\ \end{align}

### Side-by-side rendering of multiple system.

A vector of ParameterizedFunctions will be rendered side-by-side:

ode2 = @ode_def negativeFeedback begin
dx = y/(k_y + y) - x
dy = k_x^n_x/(k_x^n_x + x^n_x) - y
end k_y k_x n_x

latexify([ode, ode2])
\begin{align} \frac{dx}{dt} &= \frac{y \cdot y \cdot y}{k_{y\_x} + y} - x - x & \frac{dx}{dt} &= \frac{y}{k_{y} + y} - x & \\ \frac{dy}{dt} &= \frac{x^{n_{x}}}{k_{x}^{n_{x}} + x^{n_{x}}} - y & \frac{dy}{dt} &= \frac{k_{x}^{n_{x}}}{k_{x}^{n_{x}} + x^{n_{x}}} - y & \\ \end{align}

Another thing that I have found useful is to display the parameters of these functions. The parameters are usually in a vector, and if it is somewhat long, then it can be annoying to try to figure out which element belongs to which parameter. There are several ways to solve this. Here are two:

## lets say that we have some parameters
param = [3.4,5.2,1e-2]
latexify(ode.params, param)
\begin{align} k_{y} =& 3.4 \\ k_{x} =& 5.2 \\ n_{x} =& 0.01 \\ \end{align}

or

latexify([ode.params, param]; env=:array, transpose=true)
$$$\begin{equation} \left[ \begin{array}{ccc} k_{y} & k_{x} & n_{x} \\ 3.4 & 5.2 & 0.01 \\ \end{array} \right] \end{equation}$$$

signif.() is your friend if your parameters have more significant numbers than you want to see.

### Get the jacobian, hessian, etc.

ParameterizedFunctions symbolically calculates the jacobian, inverse jacobian, hessian, and all kinds of goodness. Since they are available as arrays of symbolic expressions, which are latexifyable, you can render pretty much all of them.

latexify(ode.symjac)
$$$\begin{equation} \left[ \begin{array}{cc} -2 & \frac{3 \cdot y^{2}}{k_{y\_x} + y} - \frac{y^{3}}{\left( k_{y\_x} + y \right)^{2}} \\ \frac{x^{-1 + n_{x}} \cdot n_{x}}{k_{x}^{n_{x}} + x^{n_{x}}} - \frac{x^{-1 + 2 \cdot n_{x}} \cdot n_{x}}{\left( k_{x}^{n_{x}} + x^{n_{x}} \right)^{2}} & -1 \\ \end{array} \right] \end{equation}$$$