Version 6 (modified by sam, 8 years ago) (diff)

fix tutorial for upcoming LolRemez 0.2

# Remez tutorial 5/5: additional tips

This part of the tutorial is constantly updated. Check its history to know what is new.

## What to do when the solver doesn’t converge?

It may happen that the solver doesn’t converge. There are many possible reasons for that:

• the function being approximated is not smooth enough; try to increase the polynomial degree, or reduce the approximation range.
• the solver cycles between a finite number (usually 3 or 4) of solutions; in this case, experiments indicate that you may safely choose the solution with the smaller error.

## Floating point precision

The Remez algorithm performs high precision computations to find its solutions. As a result, the polynomial coefficients found are printed with far more decimals than the usual `double` or `float` numbers can store. So when you write:

```float a = 8.333017134192478312472752663154642556843e-3;
```

The exact value of `a` is actually:

```8.333017118275165557861328125e-3
^-- this decimal changes!
```

See “fixing lower-order parameters” for both an explanation of why this can be a problem, and a method to reduce the introduced error.

## When not to use Remez?

There are cases when you should expect the Remez algorithm to potentially perform badly:

There are cases where you should not try to use the Remez algorithm at all:

• when the range is not finite, eg. [0,+∞]
• when the function to approximate has an infinite derivative at a point contained in or near the approximation range, eg. sqrt(x) or cbrt(x) in 0

## What if I want to use Remez anyway?

If you need to approximate a function f(x) over [a,+∞] and for some reason you want to use the Remez exchange algorithm, you can still through a change of variable: y = 1/x. The function to approximate becomes f(1/y) and the new range is [0,1/a] (see “changing variables” for how to deal with 1/x in 0). The minimax polynomial will use 1/x as its variable; please be aware that computing 1/x at runtime may be expensive.