Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

How far the two residuals drift apart, against how large the iterates got on the way

One function in the resgap library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws how far the two residuals drift apart, against how large the iterates got on the way. Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.507 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 9·10⁴ across the sweep.

gap-vs-iterate is one function in lib/figures/resgap.js — residual gap — the number a method reports against the residual of its answer. 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.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.507 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 9·10⁴ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.51a bound of one, a walk of a halffitted slope0.51the bound's slope1smallest gap measured4.6·10⁻¹⁵largest gap measured5.1·10⁻¹⁰share of the bound, near end6.6·10⁻⁴share of the bound, far end7.3·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.507 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 9·10⁴ across the sweep.

n: 25

The arguments are the ones A walk needs a length passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.463 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 2.7·10⁵ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.46a bound of one, a walk of a halffitted slope0.46the bound's slope1smallest gap measured3.9·10⁻¹⁵largest gap measured1.5·10⁻¹⁰share of the bound, near end9.1·10⁻⁴share of the bound, far end3.4·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.463 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 2.7·10⁵ across the sweep.

n: 20

The arguments are the ones A walk needs a length passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.443 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 2.5·10⁵ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.44a bound of one, a walk of a halffitted slope0.44the bound's slope1smallest gap measured5·10⁻¹⁵largest gap measured2·10⁻¹⁰share of the bound, near end0.0012share of the bound, far end5·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.443 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 2.5·10⁵ across the sweep.

n: 30

The arguments are the ones A walk needs a length passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.483 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 1.2·10⁵ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.48a bound of one, a walk of a halffitted slope0.48the bound's slope1smallest gap measured3.8·10⁻¹⁵largest gap measured3.1·10⁻¹⁰share of the bound, near end7.8·10⁻⁴share of the bound, far end6.4·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.483 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 1.2·10⁵ across the sweep.

n: 40

The arguments are the ones A walk needs a length passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.494 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 1.3·10⁵ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.49a bound of one, a walk of a halffitted slope0.49the bound's slope1smallest gap measured4.9·10⁻¹⁵largest gap measured3.7·10⁻¹⁰share of the bound, near end8.1·10⁻⁴share of the bound, far end6.1·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.494 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 1.3·10⁵ across the sweep.

n: 60

The arguments are the ones A walk needs a length passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

How far the two residuals drift apart, against how large the iterates got on the waySix runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.507 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 7.5·10⁴ across the sweep.10³10⁶10⁹10¹²10¹⁵10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1largest iterate on the way, ‖x‖difference between the two residualsthe bound, linear in ‖x‖what it does, slope 0.51a bound of one, a walk of a halffitted slope0.51the bound's slope1smallest gap measured5.3·10⁻¹⁵largest gap measured7·10⁻¹⁰share of the bound, near end7·10⁻⁴share of the bound, far end9.3·10⁻⁹the bound is rightand loose by a square root

Six runs, differing only in the smallest eigenvalue of the matrix, which sets how far the iterates travel: from 9309 to 9.31·10¹³, ten decades of it. The upper line is the standard bound, u·‖A‖·max‖x‖, which every run stays inside. The lower series is what the drift actually does, and its fitted slope is 0.507 where the bound's is 1 — so the roundings are a random walk rather than a sum, the bound is correct, and it is loose by a square root that reaches 7.5·10⁴ across the sweep.

What it checked while drawing

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.

11 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.

a size the whole run is affordable at

a size the whole sweep is affordable at

an eigenvalue small enough to make the iterates large

and grows as the square root of the largest iterate, not in proportion to it

enough decades of iterate size to fit a slope

the gap at ε = 10⁻¹⁰ is inside its bound

the gap at ε = 10⁻¹² is inside its bound

the gap at ε = 10⁻¹⁴ is inside its bound

the gap at ε = 10⁻⁴ is inside its bound

the gap at ε = 10⁻⁶ is inside its bound

the gap at ε = 10⁻⁸ is inside its bound

Against the rule

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.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Iterating, instead of factorising

A walk needs a length

The gap between the two residuals grows as the square root of something, and a square root needs a length. Two quantities are candidates — how far the iterates travelled and how many steps were taken — and only a second sweep separates them. Across a fourfold change in size the iteration count goes from 39 to 96 and the gap goes from 5.04·10⁻¹⁵ to 5.33·10⁻¹⁵.

 Iterating, instead of factorising

The residual the method reports

Conjugate gradients prints a relative residual of 6.9·10⁻²¹. The unit roundoff is 1.1·10⁻¹⁶, so that is not a small residual and not a large one — it is not a residual. The vector the method is holding at that step has ‖b − Ax‖/‖b‖ = 5.1·10⁻¹⁰, and nothing in the run says so.

The whole library · All essays · What must fail