Explain (and improve) accuracy

[jwuttke]

This code

#!/usr/bin/env python3

import numpy as np
import chebpy as ch

def g(x):
    return .3+.02*x+abs(x)**1.8

ch.core.settings.userPrefs.eps = 1e-18
R = ch.chebfun(g, [-1,1])

n = 17
for i in range(n):
    x = -1+2*i/(n-1)
    print(x, g(x), R(x), g(x)-R(x))

generates this output (on a x86_64 with Python 3.9.2, numpy 1.19.5-1 from Debian):

-1.0 1.28 1.280000000000984 -9.838796444228137e-13
-0.875 1.0688474729277848 1.0688474729268571 9.277023593767808e-13
-0.75 0.8808134105899568 0.8808134105891541 8.026912468039882e-13
-0.625 0.7166252122289523 0.716625212227441 1.511235581119763e-12
-0.5 0.5771745887492588 0.5771745887502417 -9.829914660031136e-13
-0.375 0.4636024711574641 0.4636024711595356 -2.0715096304968483e-12
-0.25 0.37746924442330587 0.37746924441954577 3.76010333980048e-12
-0.125 0.32118307135172497 0.32118307134976054 1.964428619771752e-12
0.0 0.3 0.3000000070720421 -7.072042118583255e-09
0.125 0.326183071351725 0.32618307134976054 1.964428619771752e-12
0.25 0.3874692444233059 0.3874692444195459 3.759992317498018e-12
0.375 0.4786024711574641 0.4786024711595357 -2.0715651416480796e-12
0.5 0.5971745887492588 0.5971745887502418 -9.829914660031136e-13
0.625 0.7416252122289522 0.7416252122274409 1.511235581119763e-12
0.75 0.9108134105899568 0.9108134105891542 8.025802245015257e-13
0.875 1.103847472927785 1.1038474729268573 9.277023593767808e-13
1.0 1.32 1.3200000000009837 -9.836575998178887e-13

Before setting eps to 1e-18, the output was

-1.0 1.28 1.279999999967151 3.2849056808004207e-11
-0.875 1.0688474729277848 1.0688474728987043 2.9080515773216575e-11
-0.75 0.8808134105899568 0.8808134106073755 -1.741873312255393e-11
-0.625 0.7166252122289523 0.71662521223807 -9.117706589734098e-12
-0.5 0.5771745887492588 0.5771745888149571 -6.569833566061334e-11
-0.375 0.4636024711574641 0.46360247122168646 -6.422234966052542e-11
-0.25 0.37746924442330587 0.37746924437881435 4.4491521578038373e-11
-0.125 0.32118307135172497 0.32118307153050574 -1.7878076796762343e-10
0.0 0.3 0.30000006714646454 -6.714646455519002e-08
0.125 0.326183071351725 0.32618307153050563 -1.7878065694532097e-10
0.25 0.3874692444233059 0.38746924437881436 4.4491521578038373e-11
0.375 0.4786024711574641 0.4786024712216866 -6.422246068282789e-11
0.5 0.5971745887492588 0.5971745888149573 -6.569855770521826e-11
0.625 0.7416252122289522 0.7416252122380699 -9.117706589734098e-12
0.75 0.9108134105899568 0.9108134106073754 -1.741862210025147e-11
0.875 1.103847472927785 1.1038474728987047 2.908029372861165e-11
1.0 1.32 1.3199999999671506 3.2849500897214057e-11

Suggestions for improvement:

• Document userPrefs (in README or whereever), advise on settings for eps.
• Why is accuracy lowest at x=0? I'd guess that this can be overcome by enforcing even (or odd?) order of the approximant.
• For eps << 1e-18, we get a "UserWarning: The Chebtech2 constructor did not converge: using 65537 points." Upon this warning, the user does not know whether the resulting polynomial can be trusted.
• The warning can be overcome by increasing core.settings.userPrefs.maxpow2. This needs documentation.
• The actual accuracy is much lower than eps. Is there a way to inform the user about the maximum error?