Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

factor-tolerance

One function in the recompress library, called 11 times across 8 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 residual of a formatted cholesky, against the accuracy its blocks were compressed at. A leaf of 16 means 34 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.027 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.93, 1.83, 0.88, 0.91, 0.91 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

factor-tolerance is one function in lib/figures/recompress.js — arithmetic in the format — the one operation it is not closed under, and what a hundred roundings cost. 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 residual of a formatted Cholesky, against the accuracy its blocks were compressed atA leaf of 16 means 34 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.027 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.93, 1.83, 0.88, 0.91, 0.91 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations34residual at 10⁻⁴2.2·10⁻⁵residual at 10⁻¹⁰9.9·10⁻¹²slope1worst ratio to the representation1.8ten decadesand a slope of one

A leaf of 16 means 34 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.027 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.93, 1.83, 0.88, 0.91, 0.91 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

leaf: 16

The arguments are the ones A factorisation kept past its date 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 residual of a formatted Cholesky, against the accuracy its blocks were compressed atA leaf of 16 means 34 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.027 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.93, 1.83, 0.88, 0.91, 0.91 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations34residual at 10⁻⁴2.2·10⁻⁵residual at 10⁻¹⁰9.9·10⁻¹²slope1worst ratio to the representation1.8ten decadesand a slope of one

A leaf of 16 means 34 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.027 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.93, 1.83, 0.88, 0.91, 0.91 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

leaf: 32

The arguments are the ones The kernel with nothing to compress 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 residual of a formatted Cholesky, against the accuracy its blocks were compressed atA leaf of 32 means 10 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.028 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.94, 0.89, 0.92, 0.92, 0.93 — one excursion above one, at 0.01, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations10residual at 10⁻⁴1.1·10⁻⁵residual at 10⁻¹⁰9.6·10⁻¹²slope1worst ratio to the representation0.94ten decadesand a slope of one

A leaf of 32 means 10 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.028 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.94, 0.89, 0.92, 0.92, 0.93 — one excursion above one, at 0.01, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

leaf: 8

The arguments are the ones The rounding that was not the problem 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 residual of a formatted Cholesky, against the accuracy its blocks were compressed atA leaf of 8 means 98 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.043 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.83, 1.78, 0.85, 0.81, 0.86 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations98residual at 10⁻⁴2.2·10⁻⁵residual at 10⁻¹⁰10⁻¹¹slope1worst ratio to the representation1.8ten decadesand a slope of one

A leaf of 8 means 98 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.043 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.83, 1.78, 0.85, 0.81, 0.86 — one excursion above one, at 10⁻⁴, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

leaf: 64

The arguments are the ones The test that costs what it saves 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 residual of a formatted Cholesky, against the accuracy its blocks were compressed atA leaf of 64 means 2 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.022 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.95, 0.95, 0.96, 0.93, 0.93 — one excursion above one, at 10⁻⁶, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations2residual at 10⁻⁴5.2·10⁻⁶residual at 10⁻¹⁰9.5·10⁻¹²slope1worst ratio to the representation0.96ten decadesand a slope of one

A leaf of 64 means 2 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 9, 12, 14 across the sweep. The residual tracks the tolerance at a slope of 1.022 over ten decades: the knob does what a knob should, and nothing in the approximate arithmetic bends it. The ratio to the representation's own error stays at 0.95, 0.95, 0.96, 0.93, 0.93 — one excursion above one, at 10⁻⁶, where the tolerance sits between two singular values of one block and the rank it takes is a rounding of a decision rather than a measurement.

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 5 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 leaf size the tree is built from

the residual moves decade for decade with the accuracy the blocks were compressed at

the residual never runs away from the representation's error at ε = 0.01

the residual never runs away from the representation's error at ε = 10⁻¹⁰

the residual never runs away from the representation's error at ε = 10⁻⁴

the residual never runs away from the representation's error at ε = 10⁻⁶

the residual never runs away from the representation's error at ε = 10⁻⁸

Against the rule

It draws a decomposition and prints its residual. It calls factorSweep, 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 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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.

When the problem arrives again

A factorisation kept past its date

One Cholesky factor can serve five members of a drifting sequence and save 44 per cent of the work. Kept for twenty it does not lose accuracy — it stops converging altogether. The optimum and the cliff are four members apart, both move with the drift, and a rule written in a ratio the iteration has already computed finds them without being told what the drift is.

 Elimination, and the swap

A factorisation with nothing to pivot for

Cholesky's growth factor is not bounded by one. It is equal to one, at every size and every condition number, and the two-line reason is why the algorithm needs no pivoting at all — not "usually gets away without it". Its only failure is the square root of a non-positive number, which is exactly the test for definiteness, and in floating point that test moves with the precision.

 Neither sparse nor dense

A rank that is a number of digits

Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.

 Two errors, and whose fault they are

An accuracy that is a backward error

Every backward error on this site is something an algorithm produced and somebody then measured. This one is a line in the program. Solving with a compressed matrix gives a residual that is the compression's own error, at a slope of 1.000 over ten decades, so the knob that sets the storage sets the backward error directly.

 Eigenvalues, singular values, rank

Small compared to what

This site's own singular value routine has carried a sentence since the month it was written — that one-sided Jacobi computes the small singular values to high relative accuracy and the standard method does not. It has never been measured here, because measuring it needs a σ that is known rather than computed. A bidiagonal matrix and a Sturm count in exact rationals supply one.

 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.

 Neither sparse nor dense

The rounding that was not the problem

A rank-k block plus a rank-k block is a rank-2k block, exactly, so every arithmetic in this format truncates after every addition. A Cholesky performed inside it does ninety-eight of those and its residual is 1.14·10⁻⁹ against a representation error of 1.40·10⁻⁹ — the roundings cost nothing measurable.

 Neither sparse nor dense

The test that costs what it saves

The partition that refuses to compress a touching pair keeps every rank at five while the other lets them climb from nine to thirteen. It also stores more numbers at every size measured — 67,968 against 61,440 at n = 512 — and which of those two facts matters is a question about how large the problem is going to get.

The whole library · All essays · What must fail