Reddit reviews Numerical Methods for Scientists and Engineers (Dover Books on Mathematics)

Thanks for the link. That's an excellent article!

Edit: re this hype around interval arithmetic see Kahan's "How Futile are Mindless Assessments of Roundoff in Floating-Point Computation?": http://www.cs.berkeley.edu/~wkahan/Mindless.pdf

There he notes the following:
> Interval Arithmetic approximates every variable by an interval whose ends straddle the variable’s true value. Used naively, this scheme is cursed by excessively wide intervals that undermine its credibility when wide intervals are deserved. Swollen intervals can often be curbed by combining Interval Arithmetic with ordinarily rounded arithmetic in a computation artfully recast as the determination of the fixed-point of a sufficiently contractive mapping. “Artful” is far from “Mindless”. Far less art may coax success from extendable-precision Interval Arithmetic, though its price may be high and its performance slow.

Kahan is of course a father of IEEE, and an author of some seminal work on interval arithmetic.

Mildly related, though somewhat out of date, I can't recommend Hamming's Numerical Methods book enough (http://www.amazon.com/Numerical-Methods-Scientists-Engineers-Richard/dp/0486652416). Its really great for the thousand mile view of what numerical methods are all about.

Be very careful with Numerical Recipes. The C/C++ code is written in the style of FORTRAN77 by someone with no clue about basic software engineering principles. The code is also prohibitively licensed to the point where it can't be used in any work. You can never share your code that uses any NR code with anyone or let them run it (regardless of your choice of license), unless you share it with someone on the same IP block as you and that is only if you spend thousands of dollars in licensing costs a year).

You are much better starting with a good algorithms book (e.g., CLRS or DPV) for basics, maybe a classic text like Hamming's 1987 book and using modern libraries (e.g., GSL, LAPACK), wikipedia, and if necessary delve deeper into books on the specific subtopic you want to learn about.

It depends on what the company does. Math does come up in embedded (e.g. control theory, modeling, DSP, etc.) but usually it's not super theoretical, and you can look up what you need when you need it. It's rare that you would have to derive something from scratch generally (and even then you can work up to it).

If you're doing a lot of computational stuff, some reading on numerical methods might be more helpful (something like this) - even if you have computers to do the heavy lifting for you, being aware of the basics can be helpful in avoiding common issues with precision, numerical stability, etc.

I'm not familiar with that book, but it does seem to cover a lot of useful material and be reasonably well-reviewed.

As to the simulations - the Wikipedia article mentions "Numerical computational approaches using computers are outside the scope of the book." So you might need to get some materials to learn that also. Maybe something like this (although I haven't read this book either): http://www.amazon.com/Numerical-Methods-Scientists-Engineers-Mathematics/dp/0486652416

It sounds like you are particularly interested in numerical approaches to differential equations. You might start by just reading the Wikipedia article on Runge-Kutta http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

If you have questions about them, feel free to post them here - I think people should be able to help you along.

So, you are gonna be an engineer/scientist, rather than a pure math major which, probably, means techniques will take precedence over ideas and rigor. To that end, you might like:

Engineering Mathematics

Advanced Engineering Mathematics

Numerical Methods for Scientists and Engineers

Mathematical Methods in the Physical Sciences

Basically, you need to put yourself through technical boot-camp that involves Calculus, Applied Linear Algebra, some Stats, Diff. Equations.

If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.

Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.

Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.

Statistical analysis of experimental data

Principals of Statistics

Information Theory

Statistics Manual

Some theory of sampling

Numerical Methods for Scientists and Engineers (Hamming)

Mathematical Handbook for Scientists Engineers

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables

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There is also the Data-Science Humble Bundle for more technical / practical skill building.