qep-backward-error

At its defaults it draws one quadratic eigenvalue problem in 9 systems of units: what the solver reports and what the answer is worth. An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 7.59·10⁻¹³ at γ = 108 — it moves by a factor of 1928 while the other two move by 1.68·10¹⁰. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 8.76·10¹⁰ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 0.00126. Nothing went wrong with the solver at any stop.

qep-backward-error is one function in lib/figures/qepbe.js — the price of linearising — a residual that stays flat while the answer loses eleven orders. 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.

An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 7.59·10⁻¹³ at γ = 108 — it moves by a factor of 1928 while the other two move by 1.68·10¹⁰. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 8.76·10¹⁰ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 0.00126. Nothing went wrong with the solver at any stop.

An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 7.59·10⁻¹³ at γ = 108 — it moves by a factor of 1928 while the other two move by 1.68·10¹⁰. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 8.76·10¹⁰ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 0.00126. Nothing went wrong with the solver at any stop.

An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 7.81·10⁻¹⁵ at γ = 104 — it moves by a factor of 19.8 while the other two move by 1.14·10⁵. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 1.02·10⁵ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 8.57·10⁻⁹. Nothing went wrong with the solver at any stop.

An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 2.2·10⁻¹⁰ at γ = 1010 — it moves by a factor of 5.59·10⁵ while the other two move by 3.99·10¹⁵. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 2.44·10¹⁴ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 299. Nothing went wrong with the solver at any stop.

An overdamped chain of 8 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 6.76·10⁻¹⁶ at γ = 1 and 1.26·10⁻¹³ at γ = 106 — it moves by a factor of 320 while the other two move by 7.31·10⁷. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 1.86·10⁸ across the same sweep, and the forward error follows it: 7.49·10⁻¹⁴ to 5.47·10⁻⁶. Nothing went wrong with the solver at any stop.

An overdamped chain of 6 masses, with λ replaced by γμ so that the coefficients become (γ²M, γC, K). That substitution is exact in both directions and divides the spectrum by γ exactly, so the closed form is still available and every error here is measured against it. The backward error of the eigenpair for the LINEARISED MATRIX — the residual a solver's own error analysis is about — is 7.1·10⁻¹⁶ at γ = 1 and 2.57·10⁻¹² at γ = 108 — it moves by a factor of 7022 while the other two move by 7.37·10¹¹. The backward error for the QUADRATIC, which is what the person who posed the problem is entitled to, grows by 7.03·10¹¹ across the same sweep, and the forward error follows it: 9.14·10⁻¹⁵ to 0.00674. Nothing went wrong with the solver at any stop.

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.

18 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.