1. I have a system of linear equations representing a simple statics example (like the struts on a bridge or a stack of blocks or something along those lines). I can form the equation: Aj = b, where A is an nxn matrix of coefficients, and j and b are n-vectors. I want to find j given b. And I have a book (Numerical Recipes 3rd edition) to help me do that. So far so good.

But now there's an interesting twist, and I don't know how to handle it. Suppose that each element in j has a bounds on it. Something like 0 <= j<sub>2</sub> <= 80 or something along those lines. This would represent something like a stack of blocks (each block can only push on other blocks, they can't pull on each other), or a bridge giving way (some strut can only handle 8000 Newtons of force (is force the right label here?) before it gives way).

So how do I address this case? The brute force solution I had was to invert the matrix and find j, then find the first element out of its bounds in j and clamping it to the bounds. Then rebuild a new matrix and solve again, etc. But this would have an absolutely terrible run time performance of O(n^4), and I'm not even sure it would give the right answer in the end.

2.

3. Okay, let me try to present the problem in a nicer format:

Let's say I have a system of equations: (where A is a matrix and x and b are vectors).

Let's also define a "window" (I'm sure there's a better math term for this, but I don't know it). A window is the n-dimensional box where a valid solution must live.

...

For some .

Suppose I define . And suppose that satisfies the window inequality, satisfies the system of equations, and is "minimized", in the sense that it's L2 norm is minimized.

I want to find . How do I go about doing that?

I think it's somewhat similar to iterative improvement used in numerical analysis. Something like first I solve for using standard methods. And then I define such that (that is, is the vector I had to add to to "push" the solution back into the window of valid solutions). And then some sort of math magic happens where I iteratively improve the solution somehow until is within some range, or until some maximum number of steps has been taken.

4. Actually, thinking about it some more, I'm trying to minimize that comes from computing .

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