restart-convergence
At its defaults it draws 12 restarts keeping 4 of 8, on a 40×40 matrix. The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.
restart-convergence is one function in lib/figures/restart.js —
restarting and blocks — a filter on the starting vector, and what one vector cannot see. 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.
The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.
cycles: 12
The arguments are the ones An eigenvalue one vector cannot see 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.
The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.
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 number of eigenvalues worth asking for
and a basis the restart can afford
and the residual bound has fallen by orders of magnitude
enough cycles to converge and few enough to draw
Jacobi needs a symmetric matrix
matmul shapes agree
on a basis of k + p vectors at every cycle
the 4 wanted eigenvalues are found
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 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.
An eigenvalue one vector cannot see
A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.
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.
Eigenvalues, singular values, rankRestarting is a filter
A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.
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.
Eigenvalues, singular values, rankThe form that makes it affordable
One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.