Two Krylov methods against products with A, at a kernel shift of 1
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: 1
The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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 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.
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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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.
An operator with no entries
At the sizes where linear algebra is expensive the matrix does not exist. What exists is a subroutine that returns Av. Every Krylov method survives that unchanged; every algorithm that reads an entry disappears. And the derivative such a code computes is accurate to ten digits instead of sixteen, which turns out to cost nothing at all.
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.