# Re-calculate solution after altering some elements in a linear system

• Problem

I have a linear system: $$Mx = b$$ $M$ is like a Band Matrix. And assume I have a solution $x_{init}$ at beginning.

There will be some operations which are going to alter some elements in the matrix $M$ and $b$. Usually this "change of elements" operation happens in consecutive 2 or 4 rows of the system.

Since the "change of elements" operation is local I have a strong feeling that I can use a "local modification" of initial solution $x_{init}$ to solve the new system.

More specially, I modify the component correspond to the changed row in the system. This will lead to some unsatisfied constraints, so I go to resolve that. As a result there are going to be more unsatisfied constraints spreading in the initial solution vector due to the local modification. Is there any hope that this "modify-resolve" loop will terminate in finite(and hopefully a small amount of) steps?

• Problem Background

I am a game developer and I am working on a curve(Cubic B-Spline) editor. I build the linear system correspond to the constraint of $C^0$, $C^1$, $C^2$ continuity. I notice that the linear system shows a strong locality. Therefore I try to find out if I can solve the system without solving a brand new system.

Any hint is appreciate. And I would like to know the mathematical glossary behind this problem so I can search it on Google. Thank you.

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