🌱 Padé approximation

• you can approximate a function using a "rational function" R(x)=P(x)/Q(x). P(x) is a polynomial of order N, and Q(x) is a polynomial of order M. N&M are chosen by us, but the paper i'm learning this from says error will be lowest when N is M, or one higher.
• we need a "Maclaurin series" of the function, which is just a Taylor series with a=0. but we can still "simulate the a-slider" by putting it inside the function we're deriving.
• a[i] is the ith coefficient of the Maclaurin series.
• p[] and q[] are the coefficients of P(x) and Q(x). these are unknown and must be determined from a[i]. to do that, we need a way to solve a set of linear equations…

this uses "gaussian elimination" and "back substitution". the example finds a solution for 3 equations with 3 variables. both the code and example equations were taking from the wiki page on gaussian elimination.

earlier we stole a Padé approximated sine, so maybe we can make one ourself? N=2
M=2
a=1.500
auto-calibrate
reshape x as triangle wave

it works pretty well. the P(x)/Q(x) coefficients are printed below the code; notice how half of them are zero when a=0 (again - we saw this back in the article on taylor series too). also, linsolve() fails sometimes, e.g. for N=M=3. not sure what's going on. judging by the polynomials in the previous best stolen sine, it has N=M=6, and if we use those parameters, we see the same max error.

how about log2 N=2
M=2
a=1.500
auto-calibrate

here i see max error about 0.00002 for N=3,M=2, while a 6th order Taylor series last time had an error of about 0.00007. N=3 suggests a 3rd order polynomial, but the numerator coefficients are still only second order for some reason (the coefficient for x**3 is zero). not sure what's going on here.