block-width-cost
At its defaults it draws products with a to return all 2 copies, against the block width. One bar per block width. Widths below 2 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 2, at 32 products with A; the widest drawn costs 84.
block-width-cost is one function in lib/figures/blocksize.js —
how wide the block should be — the multiplicity, and the deflation that never fires. 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.
One bar per block width. Widths below 2 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 2, at 32 products with A; the widest drawn costs 84.
mult: 4
The arguments are the ones A threshold between fill and growth 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.
One bar per block width. Widths below 4 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 4, at 64 products with A; the widest drawn costs 84.
mult: 3
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.
One bar per block width. Widths below 3 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 3, at 51 products with A; the widest drawn costs 84.
mult: 2
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.
One bar per block width. Widths below 2 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 2, at 32 products with A; the widest drawn costs 84.
mult: 1
The arguments are the ones Structure and stability stop being separable 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.
One bar per block width. Widths below 1 return fewer copies than the eigenvalue has, at every step count up to a basis of half the problem's dimension, and are marked rather than drawn. The cheapest width that works is 1, at 19 products with A; the widest drawn costs 78.
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 5 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 multiplicity the spectrum is built for
a size the dense reference is affordable at
and every width at or above it works
Jacobi needs a symmetric matrix
matmul shapes agree
only the widths below the multiplicity fail
the cheapest working width is the multiplicity itself
widths on both sides of the multiplicity
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 70
of 151 generators —
55 print a residual and
15 are exempt with a published reason;
81 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 threshold between fill and growth
One number decides how small a pivot an elimination will accept. At 0.001 the factor holds 172 entries and the matrix grows by 1,330; at 1 it holds 260 and grows by 1.2. The libraries ship 0.1, and the measurement says why.
Eigenvalues, singular values, rankAn 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, rankHow wide the block should be
A block narrower than the multiplicity does not converge slowly — it never returns the missing copy at all. Above the multiplicity every extra column buys iterations at about ten products with A each. And the mechanism that is supposed to make the choice unimportant never fires from a random start.
Sparsity, and what elimination costsStructure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
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 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.