I have a function which depends on four parameters and a target value, how can I discover the value for the four parameters that hits my target value?

Question

So I have an equation:

$$F(s,t,u,v)=A$$

Where $A$ is some given value. Is there an iterative method to discover the four parameters that will obtain my given $A$?

If it helps, my function $F$ is a quintic bezier where most of the parameters are determined. I have isolated just these four that are required to fit the value $A$.

$$Q(t)=R(1-t)^5+5S(1-t)^4t+10T(1-t)^3t^2+10U(1-t)^2t^3+5V(1-t)t^4+Wt^5$$

$R,S,T,U,V,W$ are vectors where $R$ and $W$ are known, I have isolated only a single element in each of $S,T,U,V$ that vary as parameters.

Any thoughts?

Answer 1

In computer arithmetic, there's a good chance that there are no values of $s$, $t$, $u$, $v$, such that $F(s,t,u,v) = A$. So, a better approach might be to try to find values of $s$, $t$, $u$, $v$ that minimize $\Vert F(s,t,u,v) - A \Vert^2$. This is a fairly standard minimization problem, and you can apply any of the usual iterative numerical methods. It will be a big help if you have a good place to start the iteration -- somewhere that's close to a solution.

Sources of minimization software are the Numerical Recipes book, the Plato web site, Mathematica, Maple, Sage, this Wikipedia page, etc.