gap-vs-iterate
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.
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: 50
The arguments are the ones The number that is re-derived passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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: 30
The arguments are the ones The residual the method reports passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What it checked while drawing
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.
11 distinct claims across 3 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 113
of 238 generators —
98 print a residual and
15 are exempt with a published reason;
125 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.
The number that is re-derived
GMRES prints a residual it never computes from its answer either. On the matrix that sends a conjugate gradient recurrence 7.3·10¹⁰ wrong, and on two others chosen to be worse, its number is never more than a factor of 2.86 out — while the basis it is computed from has lost orthogonality entirely. The disease is not iterative methods, and it is not floating point.
Iterating, instead of factorisingThe 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 arithmetic underneathThree walks and one bound
A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.