krylov-turn
At its defaults it draws two krylov methods against products with a, at a kernel shift of 1. Two error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.1532 after 4 products and is 9.18 by the end of the run. The bidiagonal method reaches 0.1367 after 42 and degrades far more slowly.
krylov-turn 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 error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.1532 after 4 products and is 9.18 by the end of the run. The bidiagonal method reaches 0.1367 after 42 and degrades far more slowly.
shift: 3
The arguments are the ones A preconditioner that changes sign 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 error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.2126 after 13 products and is 24.7 by the end of the run. The bidiagonal method reaches 0.1356 after 32 and degrades far more slowly.
shift: 1
The arguments are the ones A reflection cannot stop being one 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 error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.1532 after 4 products and is 9.18 by the end of the run. The bidiagonal method reaches 0.1367 after 42 and degrades far more slowly.
shift: 0
The arguments are the ones An orthogonalisation nobody calls one 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 error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.1477 after 3 products and is 98.9 by the end of the run. The bidiagonal method reaches 0.1426 after 40 and degrades far more slowly.
shift: 2
The arguments are the ones The form a real matrix can reach 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 error curves against the number of products with A, on a logarithmic vertical axis. The Arnoldi method reaches 0.1724 after 5 products and is 4.46 by the end of the run. The bidiagonal method reaches 0.1374 after 36 and degrades far more slowly.
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 kernel shift the operator survives
a size the reference solve is affordable at
and in fewer steps
and leaves it far behind
enough steps for both methods to turn
the Arnoldi method reaches its best for fewer products
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 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.
A preconditioner that changes sign
Strang's circulant preconditioner takes Toeplitz conjugate gradients from 179 steps to 10 at n = 256. At n = 64 on the same family it takes 66 steps to 109 — worse than doing nothing. Between those rows the preconditioner's smallest eigenvalue crosses zero, and nothing in the published account of the method mentions that it can be negative.
Orthogonality, measuredA reflection cannot stop being one
Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.
Iterating, instead of factorisingAn orthogonalisation nobody calls one
Conjugate gradients are derived as a minimisation and behave as an orthogonalisation, which is why the finite-termination property in every textbook is not a property the method has in floating point.
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.
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.
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.
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.