condition-amplify
At its defaults it draws how much a perturbation of the right-hand side is amplified, κ = 10⁶. The cumulative distribution of the amplification factor over two hundred random perturbation directions, with the condition number marked as the upper limit.
condition-amplify 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.
The cumulative distribution of the amplification factor over two hundred random perturbation directions, with the condition number marked as the upper limit.
kappa: 1000000
The arguments are the ones A bound that holds with probability 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.
The cumulative distribution of the amplification factor over two hundred random perturbation directions, with the condition number marked as the upper limit.
kappa: 100000000
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.
The cumulative distribution of the amplification factor over two hundred random perturbation directions, with the condition number marked as the upper limit.
kappa: 256
The arguments are the ones Where the hardware went 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.
The cumulative distribution of the amplification factor over two hundred random perturbation directions, with the condition number marked as the upper limit.
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.
7 distinct claims across 4 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 even the mildest of the two hundred is amplified substantially
and two hundred random directions find a third of it
LU is for square matrices
matmul shapes agree
no perturbation is amplified by more than κ
the median direction is within an order of magnitude of the worst case
while still being below it
Against the rule
It calls a factoriser without drawing a factorisation
(solve),
so the rule is written down as not applying, with the reason:
uses the SVD only to obtain κ for the axis
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 bound that holds with probability
Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.
Eigenvalues, singular values, rankA condition number for one eigenvalue
In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.
Structure, and the solver that cannot see itA 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.
Iterating, instead of factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.
Eigenvalues, singular values, rankSymmetry is worth more than precision
A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.
Elimination, and the swapThe bound that is never attained
Partial pivoting's stability guarantee permits the entries to double at every step — a factor of 5.5·10¹¹ at n = 40. The measured growth on random matrices of that size is about three. The gap is eleven orders of magnitude, and the guarantee is still worth having.
Two errors, and whose fault they areThe condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
Randomised, and the guarantee that changes kindThe dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
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.
Iterating, instead of factorisingThe rate the condition number predicts
Conjugate gradients converge at a rate governed by the square root of the condition number. That is a bound rather than an estimate, it is provable, and it is loose enough that provisioning iterations from it wastes nine out of ten.
Least squares, and the road not to takeThe valley with no bottom
A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.
The arithmetic underneathWhere the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.