Mathematicians call Complex Analysis "The Disneyland of Mathematics". Everything you might want to be true is, and stuff you want seems to fall out of the air by magic.
Numerical Linear Algebra is math's Exurban Sprawl. Everybody's problem falls into one of a few cookie-cutter categories, and nobody minds the sameness because the solutions we have are good enough and cheap enough.
NLA is an amazing field if you think about it -- but you have to think about it.
> Many practical problems - optimization, differential equations, signal processing, etc. - boil down to solving linear systems, even when the original problems are non-linear. Finite element software, for example, spends nearly all its time solving linear equations.
If I remember correctly, it's not that all these problems are actually sets of linear sub-problems. We just are a lot more apt to think in terms of linear equations, making it so that we tend to break down any problem we have to a linear problem. If done incorrectly of course this has dire consequences to your results.
This is similar to how in physics we try to reduce or approximate every problem to a harmonic oscillator, because everything else is just too damn complicated to actually solve.
I find the below extremely fascinating in terms of both mathematical beauty and applicable practicality:
"There are many theorems bounding the error in solutions produced on real computers. That is, the theorems don’t just bound the error from hypothetical calculations carried out in exact arithmetic but bound the error from arithmetic as carried out in floating point arithmetic on computer hardware."
I my girlfriend's numerical linear algebra textbook once, fully expecting it to be a very annoying subject dealing with implementation of floating-point numbers. It turns out, the discipline is mathematically rigorous and looks a lot like standard analysis. It was eye-opening both to me, as a mathematician, and to her, as a computer scientist, how directly the fully-abstract theorems applied to real-world floating point operations.
My favorite piece of numerical linear algebra is the pseudospectrum. In some sense, this is a set that gives you an indication of whether your eigenvalue computations will be accurate or not. Understanding non-normal operators really requires an understanding of this object.
It also has a surprising connection to the local solvability of pdes and pseudodifferential operators by helping to construct quasimodes ("almost solutions").
Numerical Linear Algebra is math's Exurban Sprawl. Everybody's problem falls into one of a few cookie-cutter categories, and nobody minds the sameness because the solutions we have are good enough and cheap enough.
NLA is an amazing field if you think about it -- but you have to think about it.