sylvester-spy
At its defaults it draws the two 6×6 matrices of a sylvester equation, and the 36×36 matrix it means. A and B are 6×6 with 11 nonzeros each. The coefficient matrix of the map X ↦ AX + XB is 36×36 with 96 nonzeros, drawn at the same scale so the ratio is visible rather than quoted. Bartels and Stewart's algorithm never forms it: two Schur reductions and a back-substitution give the same X to 1.7·10⁻¹⁶, and that X satisfies AX + XB = C to 2·10⁻¹⁶.
sylvester-spy 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.
A and B are 6×6 with 11 nonzeros each. The coefficient matrix of the map X ↦ AX + XB is 36×36 with 96 nonzeros, drawn at the same scale so the ratio is visible rather than quoted. Bartels and Stewart's algorithm never forms it: two Schur reductions and a back-substitution give the same X to 1.7·10⁻¹⁶, and that X satisfies AX + XB = C to 2·10⁻¹⁶.
n: 6
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.
A and B are 6×6 with 11 nonzeros each. The coefficient matrix of the map X ↦ AX + XB is 36×36 with 96 nonzeros, drawn at the same scale so the ratio is visible rather than quoted. Bartels and Stewart's algorithm never forms it: two Schur reductions and a back-substitution give the same X to 1.7·10⁻¹⁶, and that X satisfies AX + XB = C to 2·10⁻¹⁶.
n: 3
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.
A and B are 3×3 with 5 nonzeros each. The coefficient matrix of the map X ↦ AX + XB is 9×9 with 21 nonzeros, drawn at the same scale so the ratio is visible rather than quoted. Bartels and Stewart's algorithm never forms it: two Schur reductions and a back-substitution give the same X to 5.5·10⁻¹⁷, and that X satisfies AX + XB = C to 8.8·10⁻¹⁷.
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.
7 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 n²×n² grid can be drawn at
a superdiagonal the pair is defined for
and it satisfies the equation
and the coefficient matrix is n² by n²
LU is for square matrices
matmul shapes agree
the Kronecker solve and Bartels–Stewart give one X 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 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.
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.
Structure, and the solver that cannot see itThe circulant that cannot be indefinite
The previous essay found a preconditioner taking 117 steps against an unpreconditioned 59, because its smallest eigenvalue was −0.173. Average the two diagonals instead of choosing between them and the count is 7, 8, 9, 10, 10 across a factor of sixteen in size.
Eigenvalues, singular values, rankThe form a real matrix can reach
A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.
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.
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.