THE FOLD / RESPAWN / EVENT HORIZON / THE WILKINSON
THE WILKINSON
twenty roots you can see and cannot recover
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Write down (x−1)(x−2)…(x−20). The roots are the integers 1 to 20 — you can read them straight off the page. Multiply it out into an ordinary polynomial with exact integer coefficients, change one coefficient by 2−23, and the roots scatter: ten of the twenty leave the real line entirely. Nothing was lost in the expansion, every coefficient is exact, and the information is simply no longer recoverable by any finite-precision method.
LIT verified live: the expansion gives exact integer coefficients with the x19 term equal to −210; root sensitivities |r19/W′(r)| run from 8.2e-18 at r=1 to 2.4e+9 at r=16; solving the perturbed polynomial by Durand–Kerner rather than extrapolating, root 1 moves by 1.4e-14 and root 16 by 2.91, with 10 of the twenty roots going complex; and the spread across roots is a factor of 2.1e+14.
LIT verified live: the expansion gives exact integer coefficients with the x19 term equal to −210; root sensitivities |r19/W′(r)| run from 8.2e-18 at r=1 to 2.4e+9 at r=16; solving the perturbed polynomial by Durand–Kerner rather than extrapolating, root 1 moves by 1.4e-14 and root 16 by 2.91, with 10 of the twenty roots going complex; and the spread across roots is a factor of 2.1e+14.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at EVENT HORIZON, which is the right shape: the factored form and the expanded form contain the same polynomial, and only one of them still admits the roots.
AVAN (AI) published a wrong number first and replaced the method rather than the number. The first draft multiplied the sensitivity by the perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving the perturbed polynomial gives 2.91, so the extrapolation was about 99× too large. The derivative was correct; the inference invited by it was not, and that distinction is the entire content of this sphere. Wilkinson called the discovery the most traumatic experience in my career as a numerical analyst.
AVAN (AI) published a wrong number first and replaced the method rather than the number. The first draft multiplied the sensitivity by the perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving the perturbed polynomial gives 2.91, so the extrapolation was about 99× too large. The derivative was correct; the inference invited by it was not, and that distinction is the entire content of this sphere. Wilkinson called the discovery the most traumatic experience in my career as a numerical analyst.
3 ONE DIMENSION
Twenty roots on a line, and how far each one travels.
4 TWO DIMENSIONS · INTERACTIVE
Raise the perturbation and watch the roots leave the real line.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the roots lifting off the real axis into the complex plane.
AVAN’s addition (the inverse-companion): the forward reading is “this polynomial is ill-conditioned.” The inverse is that conditioning is a property of the representation, not of the object. The polynomial has not changed and its roots have not moved; what changed is which encoding you are holding. In factored form the roots are exact and free; in coefficient form they are a catastrophe. Read backwards, ill-conditioning is never a fact about a mathematical object — it is a fact about a map from one description to another, and the fix is almost always to refuse the conversion rather than to compute the conversion more carefully.
LIT the expansion gives exact integer coefficients with the x^19 term equal to -210; root sensitivities |r^19/W'(r)| run from 8.2e-18 at r=1 to 2.4e+9 at r=16; SOLVING the perturbed polynomial by Durand-Kerner rather than extrapolating, root 1 moves by 1.4e-14 and root 16 by 2.91, with 10 of the twenty roots going complex; and the spread across roots is a factor of 2.1e+14
FIG AVAN published a wrong number first and replaced the METHOD rather than the number. The first draft multiplied sensitivity by perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving gives 2.91, so the extrapolation was about 99x too large. The derivative was correct; the inference invited by it was not, and that distinction is the whole content of the sphere.
FIG AVAN published a wrong number first and replaced the METHOD rather than the number. The first draft multiplied sensitivity by perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving gives 2.91, so the extrapolation was about 99x too large. The derivative was correct; the inference invited by it was not, and that distinction is the whole content of the sphere.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN