filter-identity-gap
At its defaults it draws the filter description, and the orthogonality it rests on. Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.
filter-identity-gap is one function in lib/figures/krylovreg.js —
iterative regularisation — a step count as the parameter, and the filter it applies. 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 quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.
steps: 30
The arguments are the ones A parameter that counts steps 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 quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.
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 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 size the SVD is affordable at
and have parted company by the end
enough steps for the identity to break inside the figure
Jacobi needs a symmetric matrix
matmul shapes agree
the two routes agree at the early steps
Against the rule
It draws a decomposition and prints its residual. It calls
cgls,
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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 parameter that counts steps
The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.
Eigenvalues, singular values, rankAn eigenvalue that arrives twice
A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.
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.
Orthogonality, measuredTwo Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.