I'll just make one brief comment in regard to wanting the math you've written by hand to closely match the syntax on the computer, and say that this is rarely t

I'll just make one brief comment in regard to wanting the math you've written by hand to closely match the syntax on the computer, and say that this is rarely the correct thing to do when writing high quality numerical software. Making your code performant, and doing a thorough numerical analysis, usually requires a significant rearrangement of your formulae.