fill-pattern
At its defaults it draws the 12×12 grid laplacian and its cholesky factor, ordered by natural. Two square sparsity plots side by side. The left shows the nonzeros of the matrix; the right shows the nonzeros of its Cholesky factor, with the entries created by elimination marked in a second colour.
fill-pattern is one function in lib/figures/sparse.js —
sparsity — fill counted two ways, and the ordering that decides the memory. 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.
Two square sparsity plots side by side. The left shows the nonzeros of the matrix; the right shows the nonzeros of its Cholesky factor, with the entries created by elimination marked in a second colour.
ordering: 3
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.
Two square sparsity plots side by side. The left shows the nonzeros of the matrix; the right shows the nonzeros of its Cholesky factor, with the entries created by elimination marked in a second colour.
ordering: 0
The arguments are the ones Changing the condition number on purpose 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.
Two square sparsity plots side by side. The left shows the nonzeros of the matrix; the right shows the nonzeros of its Cholesky factor, with the entries created by elimination marked in a second colour.
ordering: 2
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.
Two square sparsity plots side by side. The left shows the nonzeros of the matrix; the right shows the nonzeros of its Cholesky factor, with the entries created by elimination marked in a second colour.
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.
4 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.
and the factorisation is exact to rounding
matmul shapes agree
the factor is denser than the matrix
the fill counted symbolically and numerically 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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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 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.
Iterating, instead of factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.
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.
Sparsity, and what elimination costsThe factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.
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.
Sparsity, and what elimination costsThe order decides the memory
Four elimination orderings on one matrix give factors of 1,739, 1,354, 1,413 and 1,026 entries. All four factorisations are exact, all four return the same answer, and the one with the better asymptotics is not the one that wins.
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.
Sparsity, and what elimination costsTwo ends of the same arrow
One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.
Sparsity, and what elimination costsWhat the symbolic phase can only bound
Without pivoting, the fill can be computed from the graph and the count is exact — 233 predicted, 233 measured. With pivoting it is 233 predicted and 242 measured, and what survives is a bound that is right at every threshold and loose by 1.7 times at the largest grid drawn.