root-conditioning

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.

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.

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 1.127·10⁻¹⁰ at degree 10 — 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. Every coefficient here is still exactly representable, which is what makes the failure a conditioning one rather than a storage one.

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 7.19·10⁻⁷ at degree 14 — 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. Every coefficient here is still exactly representable, which is what makes the failure a conditioning one rather than a storage one.

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 0.3333 at degree 22 — 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.

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 7.174·10⁻⁴ at degree 18 — 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. Every coefficient here is still exactly representable, which is what makes the failure a conditioning one rather than a storage one.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

5 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 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 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.