slice-staircase
At its defaults it draws ν(σ), the number of eigenvalues below σ, counted from the signs of an unpivoted ldlᵀ. The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 14 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
slice-staircase is one function in lib/figures/slice.js —
counting a spectrum — an answer that is an integer, and the band where it is wrong by a whole one. 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 14 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
k: 14
The arguments are the ones A condition number sent to infinity 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 14 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
k: 6
The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 6 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
k: 8
The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 8 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
kind: "graded", k: 12
The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 12 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
k: 10
The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.
The staircase is the number of negative pivots in an LDLᵀ factorisation of A − σI, taken with no pivoting at all. Congruence preserves the signs, so that count is the number of eigenvalues below σ — Sylvester's law of inertia, used as an algorithm. The vertical marks are the 10 eigenvalues a Jacobi decomposition returns, which is a completely different computation; every step of the staircase is at one of them and every one of them has a step. Over 2000 shifts placed at random across the interval the two answers disagree 0 times. That is not a statement about a tolerance: the output is a count, so it is exactly right or wrong by a whole eigenvalue, and there is nothing in between for a rounding error to land in.
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.
5 distinct claims across 6 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.
2000 shifts and no disagreement
a scan the figure can afford
a size the staircase can show every step of
a spectrum this figure builds
Jacobi needs a symmetric matrix
Against the rule
It draws a decomposition and prints its residual. It calls
countBelow, scanShifts,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 165
of 306 generators —
150 print a residual and
15 are exempt with a published reason;
141 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 condition number sent to infinity
An interior-point method manufactures an ill-conditioned matrix on every iteration, deliberately, because the separating of a diagonal is how it discovers which constraints are active. Written one way the answer keeps fifteen digits at a condition number of 3·10¹⁵. Written the other way — the way almost every code writes it — it has none left.
Eigenvalues, singular values, rankAn eigenvalue count that cannot be slightly wrong
Every spectral computation on this site returns floats with errors in them. Counting eigenvalues below a shift by the signs of an unpivoted elimination returns an integer, and an integer cannot be 6.9999999997 — so the answer is exactly right, or wrong by a whole eigenvalue, and where the second happens is a band of measurable width.
Sparsity, and what elimination costsAn ordering that does not wait for the numbers
A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.
The matrix a constraint makesThe regularisation that legalises every order
Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.
The matrix a constraint makesThe zero that is not a missing entry
A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.
The matrix a constraint makesTwo ways to remove a constraint
A constrained system can be reduced by eliminating the multipliers or by eliminating the constrained directions. Both give the same answer in exact arithmetic and inherit different condition numbers — one of them squares the constraint's, and the other does not contain it at all.