filter-misfit
At its defaults it draws the weights 8 steps apply, on an operator 8.6% asymmetric. Two sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 1.6·10⁻¹². The Arnoldi method's lie on a general polynomial to 0.063 — which on a symmetric operator is the level of rounding and on this one is 6%.
filter-misfit is one function in lib/figures/lsqrgmres.js —
the other two krylov regularisers — one sequence, two recurrences, and a weight that stops being a function of σ. 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 sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 1.6·10⁻¹². The Arnoldi method's lie on a general polynomial to 0.063 — which on a symmetric operator is the level of rounding and on this one is 6%.
asym: 0
The arguments are the ones Rank is a decision 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 sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 8.7·10⁻¹³. The Arnoldi method's lie on a general polynomial to 7.2·10⁻¹³ — which on a symmetric operator is the level of rounding and on this one is 0%.
asym: 0.518
The arguments are the ones Symmetry is worth more than precision 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 sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 1.5·10⁻¹². The Arnoldi method's lie on a general polynomial to 0.36 — which on a symmetric operator is the level of rounding and on this one is 36%.
m: 8
The arguments are the ones The basis decides what a filter is 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 sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 1.6·10⁻¹². The Arnoldi method's lie on a general polynomial to 0.063 — which on a symmetric operator is the level of rounding and on this one is 6%.
asym: 0.086
The arguments are the ones The matrix that is one row 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 sets of points against the singular-value index, each with the best polynomial of its own parity drawn through it. The bidiagonal method's weights lie on an even polynomial in σ to 1.6·10⁻¹². The Arnoldi method's lie on a general polynomial to 0.063 — which on a symmetric operator is the level of rounding and on this one is 6%.
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.
6 distinct claims across 5 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 reference SVD is affordable at
a step count both methods are still informative at
and on a normal operator the Arnoldi method's are a polynomial too
and the Arnoldi method's are not a function of σ once A is not normal
one of the three operators this family has
the bidiagonal method's weights are an even polynomial in σ
Against the rule
It draws a decomposition and prints its residual. It calls
lstsqQR,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 70
of 151 generators —
55 print a residual and
15 are exempt with a published reason;
81 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.
Rank is a decision
A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.
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.
Regularisation, and the answer that is chosenThe basis decides what a filter is
The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².
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.
Eigenvalues, singular values, rankThe form a real matrix can reach
A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.
Structure, and the solver that cannot see itThe matrix that is one row
A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.
Iterating, instead of factorisingThe spectrum that predicts nothing
For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.