kronecker-size
At its defaults it draws the coefficient matrix of ax + xb = c against the answer, to n = 100. At n = 100 the unknown X has 10000 entries and the coefficient matrix of the linear map has 10⁸ — 0.80 gigabytes of doubles. Eliminating it costs 6.67·10¹¹ operations against the 6·10⁷ Bartels and Stewart's algorithm needs, a ratio of 11111. The Kronecker form is what the equation means and it is not a method.
kronecker-size is one function in lib/figures/sylvester.js —
matrix equations — the n²×n² coefficient matrix nobody forms, and sep. 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.
At n = 100 the unknown X has 10000 entries and the coefficient matrix of the linear map has 10⁸ — 0.80 gigabytes of doubles. Eliminating it costs 6.67·10¹¹ operations against the 6·10⁷ Bartels and Stewart's algorithm needs, a ratio of 11111. The Kronecker form is what the equation means and it is not a method.
nMax: 200
The arguments are the ones A reduction that changes the order 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.
At n = 200 the unknown X has 40000 entries and the coefficient matrix of the linear map has 1.6·10⁹ — 12.80 gigabytes of doubles. Eliminating it costs 4.27·10¹³ operations against the 4.8·10⁸ Bartels and Stewart's algorithm needs, a ratio of 88889. The Kronecker form is what the equation means and it is not a method.
nMax: 100
The arguments are the ones An equation whose unknown is a matrix 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.
At n = 100 the unknown X has 10000 entries and the coefficient matrix of the linear map has 10⁸ — 0.80 gigabytes of doubles. Eliminating it costs 6.67·10¹¹ operations against the 6·10⁷ Bartels and Stewart's algorithm needs, a ratio of 11111. The Kronecker form is what the equation means and it is not a method.
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.
3 distinct claims across 3 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 cost model is meant to cover
and the arithmetic ratio is well past ten at the largest size
the coefficient matrix has n⁴ entries
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 92
of 198 generators —
77 print a residual and
15 are exempt with a published reason;
106 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 reduction that changes the order
A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.
Structure, and the solver that cannot see itAn 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.
Iterating, instead of factorisingThe coarse problem is a different problem
In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.
Structure, and the solver that cannot see itThe matrix that is one row
A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.
Iterating, instead of factorisingThe same problem on a coarser grid
Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.
Structure, and the solver that cannot see itTwo dimensions, and the cluster that thins
The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.
Eigenvalues, singular values, rankTwo shifts that are never formed
The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.