The width of a block of 4, at every step
At its defaults it draws the width of a block of 4, at every step. Three step functions against the step count. A block of 4 independent columns keeps its width at every step. A block with one column copied from another loses that column before the first product with A and keeps 3 for the rest of the run. A block whose columns all lie in the span of two eigenvectors starts at 2 and ends at 0 after one step, having found the whole subspace.
block-deflation 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.
Three step functions against the step count. A block of 4 independent columns keeps its width at every step. A block with one column copied from another loses that column before the first product with A and keeps 3 for the rest of the run. A block whose columns all lie in the span of two eigenvectors starts at 2 and ends at 0 after one step, having found the whole subspace.
block: 4
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three step functions against the step count. A block of 4 independent columns keeps its width at every step. A block with one column copied from another loses that column before the first product with A and keeps 3 for the rest of the run. A block whose columns all lie in the span of two eigenvectors starts at 2 and ends at 0 after one step, having found the whole subspace.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
7 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 block wide enough to have something to lose
a random block never narrows
a repeated column is dropped at once
and an invariant block runs out
enough steps for a width to change if it is going to
Jacobi needs a symmetric matrix
matmul shapes agree
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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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 equation whose unknown is a matrix
AX + XB = C is linear in X, so it has a coefficient matrix, and writing it down is the obvious thing to do. At n = 100 that matrix has a hundred million entries for a problem with ten thousand unknowns, and the algorithm everybody uses instead never forms it. Its conditioning is not the eigenvalue gap either, which is the number a reader is invited to consult.
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.