exact-ground-truth
At its defaults it draws the exact solution of a 13×13 hilbert system beside the computed one. Two columns of numbers: the exact answer, which is the integers one to thirteen, and the answer double-precision elimination returns, with the number of correct digits beside each.
exact-ground-truth is one function in lib/figures/error.js —
error — the backward one, the forward one, and the number between them. 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.
Two columns of numbers: the exact answer, which is the integers one to thirteen, and the answer double-precision elimination returns, with the number of correct digits beside each.
n: 13
The arguments are the ones A limit the matrix never reaches 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.
Two columns of numbers: the exact answer, which is the integers one to thirteen, and the answer double-precision elimination returns, with the number of correct digits beside each.
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.
4 distinct claims across 2 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.
and at least one component has no correct digits at all
LU is for square matrices
the right-hand side came from exact arithmetic agree
the solve is backward stable even here
Against the rule
It calls a factoriser without drawing a factorisation
(solve),
so the rule is written down as not applying, with the reason:
shows a solution vector against the exact one; the badge carries the backward error instead
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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.
A limit the matrix never reaches
Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.
Two errors, and whose fault they areA small residual is not a small error
Substituting the answer back and finding that it fits is the most natural check there is, and it verifies the wrong thing. A residual of 10⁻¹⁷ is entirely compatible with an answer whose second digit is wrong.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
Elimination, and the swapElimination is a sequence of choices
Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.
Two errors, and whose fault they areThe exact answer to a nearby problem
A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.
Least squares, and the road not to takeThe road that squares the problem
The normal equations are the first method every course teaches and the method no library uses. Forming AᵀA squares the condition number, and below ε = √u it does not degrade — it produces a matrix that is exactly singular, from data that was perfectly usable.
Elimination, and the swapThe swap that is not optional
Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.
The arithmetic underneathWhat a float can hold
The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.