Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The singular values of the fill between a separator's two halves, and of the same block renumbered

One function in the hfill library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 9 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws the singular values of the fill between a separator's two halves, and of the same block renumbered. The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

fill-spectrum is one function in lib/figures/hfill.js — the fill — dense, and not independent, on the separator nested dissection leaves. 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 singular values of the fill between a separator's two halves, and of the same block renumberedThe block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.02468101210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.062σ₄ ⁄ σ₁1.3·10⁻⁴σ₆ ⁄ σ₁1.5·10⁻⁸renumbered σ₄ ⁄ σ₁0.63rank at 10⁻⁸6the same entriesin two orders

The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

k: 23

The arguments are the ones The cliff behind the count passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of the fill between a separator's two halves, and of the same block renumberedThe block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.02468101210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.062σ₄ ⁄ σ₁1.3·10⁻⁴σ₆ ⁄ σ₁1.5·10⁻⁸renumbered σ₄ ⁄ σ₁0.63rank at 10⁻⁸6the same entriesin two orders

The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

k: 11

The arguments are the ones The cliff behind the count passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of the fill between a separator's two halves, and of the same block renumberedThe block is 5 × 6, every entry of it nonzero, and its singular values fall by a factor of 23.0 at the first step and keep falling: 1, 0.043, 8.3·10⁻⁴, 3.1·10⁻⁶, 1.4·10⁻⁹. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.87, 0.58, 0.36, 0.0034, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.024610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.043σ₄ ⁄ σ₁3.1·10⁻⁶σ₆ ⁄ σ₁1.2·10⁻¹⁶¹renumbered σ₄ ⁄ σ₁0.36rank at 10⁻⁸4the same entriesin two orders

The block is 5 × 6, every entry of it nonzero, and its singular values fall by a factor of 23.0 at the first step and keep falling: 1, 0.043, 8.3·10⁻⁴, 3.1·10⁻⁶, 1.4·10⁻⁹. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.87, 0.58, 0.36, 0.0034, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

k: 15

The arguments are the ones The cliff behind the count passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of the fill between a separator's two halves, and of the same block renumberedThe block is 7 × 8, every entry of it nonzero, and its singular values fall by a factor of 19.1 at the first step and keep falling: 1, 0.052, 0.0019, 2.2·10⁻⁵, 7.9·10⁻⁸. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.65, 0.62, 0.53, 0.49, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.0246810⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.052σ₄ ⁄ σ₁2.2·10⁻⁵σ₆ ⁄ σ₁6.3·10⁻¹¹renumbered σ₄ ⁄ σ₁0.53rank at 10⁻⁸5the same entriesin two orders

The block is 7 × 8, every entry of it nonzero, and its singular values fall by a factor of 19.1 at the first step and keep falling: 1, 0.052, 0.0019, 2.2·10⁻⁵, 7.9·10⁻⁸. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.65, 0.62, 0.53, 0.49, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

k: 23, kind: "random"

The arguments are the ones The cliff behind the count passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The fill left by a random-valued operator, and the same block renumberedThe block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 14672.3 at the first step and keep falling: 1, 6.8·10⁻⁵, 6·10⁻⁹, 4.6·10⁻¹¹, 1.2·10⁻¹⁴. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of this random-valued operator is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The values here are not a Laplacian's — the five-point pattern is kept and the entries replaced, symmetrically, with a diagonal large enough to keep the matrix definite — so the cliff is a property of the separator's geometry rather than of the operator, and it is the steeper for it. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.51, 0.42, 0.41, 0.35, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.02468101210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁6.8·10⁻⁵σ₄ ⁄ σ₁4.6·10⁻¹¹σ₆ ⁄ σ₁7.6·10⁻¹⁷renumbered σ₄ ⁄ σ₁0.41rank at 10⁻⁸2the same entriesin two orders

The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 14672.3 at the first step and keep falling: 1, 6.8·10⁻⁵, 6·10⁻⁹, 4.6·10⁻¹¹, 1.2·10⁻¹⁴. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of this random-valued operator is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The values here are not a Laplacian's — the five-point pattern is kept and the entries replaced, symmetrically, with a diagonal large enough to keep the matrix definite — so the cliff is a property of the separator's geometry rather than of the operator, and it is the steeper for it. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.51, 0.42, 0.41, 0.35, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

k: 23, kind: "anisotropic", eps: 0.01

The arguments are the ones The cliff behind the count passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The fill left by an anisotropic operator, and the same block renumberedThe block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 657.5 at the first step and keep falling: 1, 0.0015, 2.8·10⁻⁶, 5.5·10⁻¹⁰, 2.3·10⁻¹⁴. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of this anisotropic operator is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.82, 0.58, 0.58, 0.58, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.02468101210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.0015σ₄ ⁄ σ₁5.5·10⁻¹⁰σ₆ ⁄ σ₁2.7·10⁻¹⁹renumbered σ₄ ⁄ σ₁0.58rank at 10⁻⁸3the same entriesin two orders

The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 657.5 at the first step and keep falling: 1, 0.0015, 2.8·10⁻⁶, 5.5·10⁻¹⁰, 2.3·10⁻¹⁴. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of this anisotropic operator is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.82, 0.58, 0.58, 0.58, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

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.

9 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.

a grid the two dense half-factorisations are affordable at — the left half is a 253-square and its elimination is cubic in that

a grid whose cross block has room for a spectrum

an anisotropic operator with an anisotropy in it

an odd grid, so the separator is a column and the two halves are equal

an operator this file defines

and the same block renumbered does not

LU is for square matrices

matmul shapes agree

the block's spectrum falls off a cliff

Against the rule

It draws a decomposition and prints its residual. It calls svd, crossSpectrum, separatorSchur, 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 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.

Sparsity, and what elimination costs

The cliff behind the count

The fill's rank is an integer between three and six across every separator two dense half-eliminations can afford, and this field has already recorded that a handful of such integers cannot carry a law. The singular values underneath are real numbers. They say the cliff's first step is 23.0 at a separator of eleven, 19.1 at fifteen and 16.2 at twenty-three — and that a control with no differential operator behind it gives 14,672.

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