The roots of ∏(x − i) up to degree 20, computed from the coefficients that define them

At its defaults it draws the roots of ∏(x − i) up to degree 20, computed from the coefficients that define them. A polynomial whose roots are the integers 1 to n, expanded exactly in BigInt and then handed to the route every library takes: the eigenvalues of the companion matrix. The measured relative error runs 1.711·10⁻¹³ at degree 6 to 0.007614 at degree 20 — the third digit of an answer whose truth is an integer. The other curve is the root condition number times the unit roundoff, which is a prediction made from the coefficients alone, and it tracks the measurement to within two orders at every degree. Above degree 19 the coefficients are integers larger than 2⁵³, so the polynomial handed to the routine is not the polynomial that was written down — the problem is damaged before any algorithm runs.

root-conditioning is one function in lib/figures/qepscale.js — the units a polynomial is written in — two lines of scaling, a prediction that is half right, and a format's edge. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

A polynomial whose roots are the integers 1 to n, expanded exactly in BigInt and then handed to the route every library takes: the eigenvalues of the companion matrix. The measured relative error runs 1.711·10⁻¹³ at degree 6 to 0.007614 at degree 20 — the third digit of an answer whose truth is an integer. The other curve is the root condition number times the unit roundoff, which is a prediction made from the coefficients alone, and it tracks the measurement to within two orders at every degree. Above degree 19 the coefficients are integers larger than 2⁵³, so the polynomial handed to the routine is not the polynomial that was written down — the problem is damaged before any algorithm runs.

monomial, integers: 1.7·10⁻¹³ at 6, 1.1·10⁻¹⁰ at 10, 7.2·10⁻⁷ at 14, 7.2·10⁻⁴ at 18, 0.0076 at 20, 0.16 at 24, 0.21 at 28, 0.19 at 32, 1 at 40. monomial, Chebyshev points: 2.5·10⁻¹³ at 6, 1.5·10⁻¹⁰ at 10, 2.4·10⁻⁸ at 14, 1.8·10⁻⁵ at 18, 0.0042 at 20, 0.048 at 24, 0.064 at 28, 0.13 at 32, 1 at 40. Chebyshev, integers: 1.3·10⁻¹⁵ at 6, 1.8·10⁻¹⁴ at 10, 1.2·10⁻¹⁴ at 14, 2.4·10⁻¹³ at 18, 4.4·10⁻¹³ at 20, 5.8·10⁻¹¹ at 24, 1.6·10⁻⁹ at 28, 6.8·10⁻⁸ at 32, 2.8·10⁻⁴ at 40. Chebyshev, Chebyshev points: 4.1·10⁻¹⁵ at 6, 8.4·10⁻¹⁵ at 10, 10⁻¹⁴ at 14, 1.9·10⁻¹⁴ at 18, 2.6·10⁻¹⁴ at 20, 1.9·10⁻¹⁴ at 24, 2.7·10⁻¹⁴ at 28, 8.7·10⁻¹⁴ at 32, 9.1·10⁻¹⁴ at 40. At degree twenty on the integers the Chebyshev route is 1.7·10¹⁰ times more accurate.

With integer roots the polynomial's humps between neighbouring roots fall from 1 at the ends of the interval to about 3.8·10⁻⁵ in the middle. With roots at Chebyshev points every hump reaches the same height. A root's sensitivity to the coefficients goes like the polynomial's largest value over its slope at the root, so the middle integer roots are the sensitive ones.

Chebyshev route, degree 30: 7.8·10⁻¹⁴, 10⁻¹³, 7.8·10⁻¹³, 1.9·10⁻¹², 1.9·10⁻¹¹, 2.6·10⁻¹¹, 9.8·10⁻¹², 4.2·10⁻¹⁰, 1.8·10⁻⁹, 3.7·10⁻⁹, 6.3·10⁻⁹, 1.1·10⁻⁸, 1.9·10⁻⁸, 2.6·10⁻⁸, 3·10⁻⁸, 2.8·10⁻⁸, 2.2·10⁻⁸, 1.4·10⁻⁸, 7.2·10⁻⁹, 3.5·10⁻⁹, 1.7·10⁻⁹, 7.2·10⁻¹⁰, 1.5·10⁻¹⁰, 2.9·10⁻¹², 6.3·10⁻¹², 3.6·10⁻¹², 2.8·10⁻¹³, 8.3·10⁻¹⁴, 7.2·10⁻¹⁵, 2.6·10⁻¹⁵. Monomial route, degree 16: 10⁻¹³, 1.2·10⁻¹¹, 3.4·10⁻¹⁰, 4.6·10⁻⁹, 3.5·10⁻⁸, 1.7·10⁻⁷, 5.3·10⁻⁷, 1.1·10⁻⁶, 1.5·10⁻⁶, 1.1·10⁻⁶, 2.5·10⁻⁷, 4·10⁻⁷, 4.3·10⁻⁷, 1.8·10⁻⁷, 2.9·10⁻⁸, 2.2·10⁻¹⁰. Both are worst near the middle of the interval and best at its ends.

Integer roots: the largest ratio is 1329, at k = 2. Chebyshev points: 0.002 at most below the leading term — the polynomial is the leading term to within the 1/1024 grid the roots were rounded to.

n = 6: error 1.3·10⁻¹⁵, κu 6.6·10⁻¹⁷, the coefficient norm over the leading coefficient 1, product 6.6·10⁻¹⁷; n = 10: error 1.8·10⁻¹⁴, κu 4.2·10⁻¹⁶, the coefficient norm over the leading coefficient 2.2, product 9·10⁻¹⁶; n = 14: error 1.2·10⁻¹⁴, κu 3.7·10⁻¹⁵, the coefficient norm over the leading coefficient 6.9, product 2.5·10⁻¹⁴; n = 18: error 2.4·10⁻¹³, κu 3.8·10⁻¹⁴, the coefficient norm over the leading coefficient 24, product 9.1·10⁻¹³; n = 20: error 4.4·10⁻¹³, κu 1.3·10⁻¹³, the coefficient norm over the leading coefficient 46, product 5.8·10⁻¹²; n = 24: error 5.8·10⁻¹¹, κu 1.5·10⁻¹², the coefficient norm over the leading coefficient 172, product 2.5·10⁻¹⁰; n = 28: error 1.6·10⁻⁹, κu 1.8·10⁻¹¹, the coefficient norm over the leading coefficient 668, product 1.2·10⁻⁸; n = 32: error 6.8·10⁻⁸, κu 2.3·10⁻¹⁰, the coefficient norm over the leading coefficient 2659, product 6.2·10⁻⁷; n = 40: error 2.8·10⁻⁴, κu 4·10⁻⁸, the coefficient norm over the leading coefficient 4.4·10⁴, product 0.0018. From degree 14 the error is 0.08 to 0.46 times the product.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

7 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.