Minimax polynomial approximation

### 10.2.8 Minimax polynomial approximation

The minimax command finds the minimax polynomial approximation of a continuous function on a bounded interval using the Remez algorithm.

- minimax takes three mandatory argument and one optional argument:
  - expr, a univariate expression representing a continuous function.
  - var=a..b, a variable and the interval.
  - n, a positive integer, the degree of the resulting polynomial.
  - Optionally limit=m, where m is a positive integer specifying the number of iterations of the Remez algorithm. (By default, the number of iterations is unlimited.)
- minimax(expr,var=a..b,n ⟨,limit=m ⟩) returns the minimax polynomial approximation of degree n or lower that approximates expr on [a,b]. The largest absolute error of the resulting polynomial will be printed in the message area.

Since the coefficients of p are computed numerically, you should avoid setting n unnecessary high as it may result in a poor approximation due to the roundoff errors.

#### Example

minimax(sin(x),x=0..2*pi,10)

max. absolute error: 5.85231264871e-06

| 5.85141928628×10−6+0.999777263417 x+0.001400152516 x2 |  |  |  |  |  |  |  |  |
| −0.170089663468 x3+0.0042684302281 x4+0.00525794778108 x5 |  |  |  |  |  |  |  |  |
| +0.00135760211866 x6−0.000570502070425 x7 |  |  |  |  |  |  |  |  |
+ | ⎛ ⎝ | 6.0729711779×10−5 | ⎞ ⎠ | x8 − | ⎛ ⎝ | 2.14787415748×10−6 | ⎞ ⎠ | x9
− | ⎛ ⎝ | 1.49765719172×10−15 | ⎞ ⎠ | x10