Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The residual of a formatted Cholesky on 256 unknowns, against the accuracy its blocks were compressed at

One function in the recompress 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 16 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 on 256 unknowns, against the accuracy its blocks were compressed at. A leaf of 16 on 256 unknowns 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 on 256 unknowns, against the accuracy its blocks were compressed atA leaf of 16 on 256 unknowns 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 on 256 unknowns 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, n: 256

The arguments are the ones A knob calibrated in residuals passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The residual of a formatted Cholesky on 256 unknowns, against the accuracy its blocks were compressed atA leaf of 16 on 256 unknowns 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 on 256 unknowns 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: 64, n: 256

The arguments are the ones A knob calibrated in residuals passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The residual of a formatted Cholesky on 256 unknowns, against the accuracy its blocks were compressed atA leaf of 64 on 256 unknowns 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 on 256 unknowns 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.

leaf: 8, n: 256

The arguments are the ones A knob calibrated in residuals passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The residual of a formatted Cholesky on 256 unknowns, against the accuracy its blocks were compressed atA leaf of 8 on 256 unknowns 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 on 256 unknowns 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: 16, n: 128

The arguments are the ones A knob calibrated in residuals passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The residual of a formatted Cholesky on 128 unknowns, against the accuracy its blocks were compressed atA leaf of 16 on 128 unknowns means 10 truncations inside the factorisation, and the ranks of its blocks run to 3, 6, 8, 10, 12 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 0.97, 1.79, 0.90, 0.98, 0.97 — 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 factorisationtruncations10residual at 10⁻⁴2.1·10⁻⁵residual at 10⁻¹⁰9.9·10⁻¹²slope1worst ratio to the representation1.8ten decadesand a slope of one

A leaf of 16 on 128 unknowns means 10 truncations inside the factorisation, and the ranks of its blocks run to 3, 6, 8, 10, 12 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 0.97, 1.79, 0.90, 0.98, 0.97 — 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, n: 512

The arguments are the ones A knob calibrated in residuals passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The residual of a formatted Cholesky on 512 unknowns, against the accuracy its blocks were compressed atA leaf of 16 on 512 unknowns means 98 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 10, 13, 16 across the sweep. The residual tracks the tolerance at a slope of 1.031 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 0.91, 1.71, 0.86, 0.89, 0.87 — 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 10⁻⁴2.3·10⁻⁵residual at 10⁻¹⁰9.5·10⁻¹²slope1worst ratio to the representation1.7ten decadesand a slope of one

A leaf of 16 on 512 unknowns means 98 truncations inside the factorisation, and the ranks of its blocks run to 4, 7, 10, 13, 16 across the sweep. The residual tracks the tolerance at a slope of 1.031 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 0.91, 1.71, 0.86, 0.89, 0.87 — 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.

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.

16 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 kernel this file defines

a leaf size the tree is built from

a positive pivot, which is what positive definite means here

a power of two, so the bisection is exact at every level

a shift inside the range the matrix stays positive definite and the geometry stays the geometry

a size the dense reference below is affordable at

a size the whole ten-decade sweep is affordable at

an accuracy, not a rank

LU is for square matrices

matmul shapes agree

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 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 knob calibrated in residuals

A formatted Cholesky has two numbers in it and only one of them is an accuracy. Across twelve trees — three sizes by four leaf sizes — the leaf moves the truncation count from 0 to 258 and moves the ranks of the blocks not at all, while the residual follows the tolerance at slopes between 1.022 and 1.046 and sits at about a tenth of it throughout.

 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.

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