This program compares iterated modifications
of a polynomial approximation of a function, with the function,
selecting the approximation which best minimizes maximum error.
It's only practically useful when the number of coefficients to
find optimal values for is very low, as in at most 4. Something
else (e.g. the Remez algorithm) is recommended for more general
uses. This program does not search for the form of a polynomial
and requires it to be pre-entered by editing polapt.c.
When the starting point has the correct form and the search for better coefficients doesn't take too long, it's possible to get the optimal minimax result (except for floating point precision limits). Simple code can also be used to transform that result, e.g. to instead minimize end-point error primarily, afterwards.
The program presents more than one solution when asked for more than one coefficient value: one per coefficient. That's because the program does a search first for one, then two, etc. values; the search time increases in polynomial time with the number to find values for, so the earlier searches take a negligible time compared to the last.
Each time a search for modifying coefficients for the approximation completes, they are printed (for easy use in a program), along with the maximum and end-point error when using them.
When searches are complete, a full picture of the error (difference curve) for the final resulting approximation is written as a table, to a text file for plotting with e.g. gnuplot.
See the article "Modifying Taylor polynomials for better accuracy" for background and results for a Taylor polynomial of degree 7 (4 terms).
make
./polapt
gnuplot
plot 'plot.txt' w l
exit
Various commits change which function is tested, and the input range (i.e. function domain) and other parameters along with that.
Taylor polynomials and other approximations can be made much more accurate for a given computational cost by limiting input to the smallest range practical to reduce to.
For approximating sin() or sinf() (single-precision version), as starting points mainly Taylor polynomials are provided. Changing the coefficients of the degree 5 (3 coeffs) or degree 7 (4 coeffs) version can produce a minimax polynomial. By default, the domain (input range) to try to minimize maximum error for is -PI/2 <= x <= PI/2 for sin().
Some other functions are also provided.