block-multiplicity
At its defaults it draws the six largest ritz values, against a doubled eigenvalue at 10. Distance from 10 for each of the six largest Ritz values, on a logarithmic vertical axis. The single-vector run of 16 steps has one value at 10 to 2.6·10⁻¹² and its second is 1.696 away — it has found the eigenvalue once. The block of two has two values at 10, to 1.1·10⁻⁹ and 4.7·10⁻⁹.
block-multiplicity 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.
Distance from 10 for each of the six largest Ritz values, on a logarithmic vertical axis. The single-vector run of 16 steps has one value at 10 to 2.6·10⁻¹² and its second is 1.696 away — it has found the eigenvalue once. The block of two has two values at 10, to 1.1·10⁻⁹ and 4.7·10⁻⁹.
steps: 16
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.
Distance from 10 for each of the six largest Ritz values, on a logarithmic vertical axis. The single-vector run of 16 steps has one value at 10 to 2.6·10⁻¹² and its second is 1.696 away — it has found the eigenvalue once. The block of two has two values at 10, to 1.1·10⁻⁹ and 4.7·10⁻⁹.
steps: 12
The arguments are the ones The gap decides the eigenvector 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.
Distance from 10 for each of the six largest Ritz values, on a logarithmic vertical axis. The single-vector run of 12 steps has one value at 10 to 4.6·10⁻⁸ and its second is 1.730 away — it has found the eigenvalue once. The block of two has two values at 10, to 1.1·10⁻⁹ and 4.7·10⁻⁹.
steps: 20
The arguments are the ones The plane survives what its vectors do not 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.
Distance from 10 for each of the six largest Ritz values, on a logarithmic vertical axis. The single-vector run of 20 steps has one value at 10 to 5.5·10⁻¹⁴ and its second is 1.696 away — it has found the eigenvalue once. The block of two has two values at 10, to 1.1·10⁻⁹ and 4.7·10⁻⁹.
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 4 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 dense block recurrence is affordable at
a step count the recurrence has room for
and two starting vectors return two
Jacobi needs a symmetric matrix
matmul shapes agree
one starting vector returns at most one copy
Against the rule
It draws a decomposition and prints its residual. It calls
blockLanczos,
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.
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, rankThe gap decides the eigenvector
A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.
Eigenvalues, singular values, rankThe plane survives what its vectors do not
At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.