How do I numerically solve a custom function?

Question

Whenever I use functions like FindRoot or NDSolve, it sends x through the function and deals with the result. That would be fine if I was sending a simple math function through, but I have something more like a short program. Is there any way to make it solve it by putting specific values through the function and looking at the results?

Answer 1

I will give you a simple but realistic example. Imagine you are given two $d$-dimensional vectors $X$ and $B$. Now you are asked to find a matrix say $A$ such that $AX=B.$ How can we use Mathematica to solve this problem?

Prepare the two vectors X and B

d = 6;
SeedRandom[123];
X = RandomInteger[{1, 100}, d];
B = RandomReal[{200, 400}, d];

Now is time to define a function that takes a numerical $d\times d$-dimensional array $A$ as input and computes the norm $|A X-B|.$

Obj[A_?(ArrayQ[#, _, NumericQ] &)] := (A. X - B) // Norm;

At this point one would try to minimize the above function in order to get the matrix $A$ such that $AX=B$ holds approximately. We use the FindMinimum function with a derivative free method option.

FindMinimum[Obj[A], {A, RandomReal[{1, 25}, {d, d}]},
Method -> "PrincipalAxis", AccuracyGoal -> 12, PrecisionGoal -> 60,
MaxIterations -> 1000] // Short

{1.39237*10^-13,{A->{{-66.9631,20.7114,14.3336,11.7857,9.99711,14.8347},{-<<19>>,<<4>>,<<19>>},<<3>>,{<<1>>}}}}

Similar things can be done for functions like FindRoot or NDSolve. Hope this helps.

BR