reorth-cost
At its defaults it draws what each reorthogonalisation costs and what it leaves, n = 40. Two quantities for three variants on one logarithmic axis: the number of projections performed, and the departure from orthogonality left behind. Doing nothing costs nothing and leaves a basis that is not one. Full reorthogonalisation costs 820 projections and leaves 1.9·10⁻¹⁵. Selective costs 137 — 6.0 times fewer — and leaves 10·10⁻⁸, which is √u and is enough.
reorth-cost is one function in lib/figures/lanczos.js —
lanczos — three terms, lost orthogonality, and an eigenvalue that arrives twice. 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 for three variants on one logarithmic axis: the number of projections performed, and the departure from orthogonality left behind. Doing nothing costs nothing and leaves a basis that is not one. Full reorthogonalisation costs 820 projections and leaves 1.9·10⁻¹⁵. Selective costs 137 — 6.0 times fewer — and leaves 10·10⁻⁸, which is √u and is enough.
n: 40
The arguments are the ones An eigenvalue that arrives twice 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 for three variants on one logarithmic axis: the number of projections performed, and the departure from orthogonality left behind. Doing nothing costs nothing and leaves a basis that is not one. Full reorthogonalisation costs 820 projections and leaves 1.9·10⁻¹⁵. Selective costs 137 — 6.0 times fewer — and leaves 10·10⁻⁸, which is √u and is enough.
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.
8 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 ghosts appear at within n steps
and both repairs remove them
at the threshold it was aimed at
for a fraction of the projections
Jacobi needs a symmetric matrix
leaving a basis that is only semi-orthogonal
matmul shapes agree
the plain run has ghosts
Against the rule
It draws a decomposition and prints its residual. It calls
lanczos,
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 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.
An 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.
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.