Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The singular values of one off-diagonal block, for four kernels on the same 96 points

One function in the kernel library, called 8 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 6 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 one off-diagonal block, for four kernels on the same 96 points. Two intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

kernel-spectra is one function in lib/figures/kernel.js — one block — where a matrix with no zero entries turns out to have five useful columns. 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 one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸5log r rank at 10⁻⁸5noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.024σ₆ ⁄ σ₁2.8·10⁻⁹the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

n: 96

The arguments are the ones A block nobody can call sparse passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸5log r rank at 10⁻⁸5noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.024σ₆ ⁄ σ₁2.8·10⁻⁹the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

gap: 0.05

The arguments are the ones A block nobody can call sparse passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [1.05, 2.05] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 7 a column and is below 10⁻⁸ after 10; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸10log r rank at 10⁻⁸9noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.19σ₆ ⁄ σ₁8·10⁻⁵the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [1.05, 2.05] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 7 a column and is below 10⁻⁸ after 10; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

gap: 0.25

The arguments are the ones A block nobody can call sparse passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [1.25, 2.25] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 15 a column and is below 10⁻⁸ after 7; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸7log r rank at 10⁻⁸7noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.085σ₆ ⁄ σ₁1.5·10⁻⁶the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [1.25, 2.25] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 15 a column and is below 10⁻⁸ after 7; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

gap: 2

The arguments are the ones A block nobody can call sparse passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [3, 4] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 127 a column and is below 10⁻⁸ after 4; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸4log r rank at 10⁻⁸4noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.0098σ₆ ⁄ σ₁3.2·10⁻¹¹the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [3, 4] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 127 a column and is below 10⁻⁸ after 4; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

gap: 4

The arguments are the ones A block nobody can call sparse passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of one off-diagonal block, for four kernels on the same 96 pointsTwo intervals that do not touch — [0, 1] and [5, 6] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 368 a column and is below 10⁻⁸ after 4; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸4log r rank at 10⁻⁸3noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.0034σ₆ ⁄ σ₁1.6·10⁻¹³the block has full rankand five useful columns

Two intervals that do not touch — [0, 1] and [5, 6] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 368 a column and is below 10⁻⁸ after 4; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

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.

6 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 block size the decomposition is affordable at

a block size the dense SVD below is affordable at

a kernel this file defines

and the block of independent draws is not

the smooth kernel's block is numerically low rank at eight digits

two intervals that do not touch

Against the rule

It draws a decomposition and prints its residual. It calls spectraByKernel, 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.

Neither sparse nor dense

A block nobody can call sparse

A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.

 Neither sparse nor dense

The kernel with nothing to compress

Hold the geometry fixed at q = ½, fix the wavelength, and scale the picture up by sixteen. A smooth kernel needs six columns at every scale. An oscillatory one needs twelve, sixteen, twenty-two, thirty-three, fifty-three, and there is no scale at which it stops.

The whole library · All essays · What must fail