Different trig algorithms in CAS and Home?

[Claudio L.]

(01-05-2018 12:35 AM)Tim Wessman Wrote:  1. Because it is basically useless for numerical calculations.
2. Causes much more memory to be used for no benefit. (think matrices and other items with potentially thousands of numbers)

I'm surprised to see this argument. I wouldn't say useless, some problems have thousands of intermediate operations and are potentially unstable or ill conditioned (matrices, actually, are a good example of where extra precision *IS* useful). While not necessarily too sensitive to the initial argument, the errors accumulated over thousands of operations can and do have a big impact.

Examples where additional digits are crucial:
* Finding all roots of a high-degree polynomial using a numerical root method and polynomial deflation.
* Finding eigenvectors/eigenvalues with iterative methods. This is inherently unstable, and getting those first values in high precision is absolutely required if you want to get even decently close to the higher eigenvalues.

For instance, a couple of months ago I wrote the polynomial root solver for newRPL using Laguerre method and polynomial deflation. After degree 10 it gets hard to get a good answer on the higher roots unless you increase the precision. Old and slow machines didn't stand a chance against these problems, but the Prime and newer calculators are fast enough where these problems are solvable and actually a good fit for the calculator, so we better make sure the calculator gives a decent answer.