Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 8 decades of units

One function in the scaling library, called 8 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 14 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 two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 8 decades of units. Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁷ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

componentwise-bound is one function in lib/figures/scaling.js — units — the condition number that is a choice of them, and the one that is not. 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.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 8 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁷ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)1.9·10⁸cond(A, x)4.8ratio of the bounds4·10⁷both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁷ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

spread: 8

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 8 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁷ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)1.9·10⁸cond(A, x)4.8ratio of the bounds4·10⁷both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁷ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

spread: 4

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 4 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4120 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)2·10⁴cond(A, x)4.8ratio of the bounds4120both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4120 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

spread: 0

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 0 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 2.1 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)9.8cond(A, x)4.8ratio of the bounds2.1both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 2.1 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

spread: 2

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 2 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 47 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)223cond(A, x)4.8ratio of the bounds47both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 47 times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

spread: 6

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two perturbation bounds and the error that was measured, on a 8×8 matrix spread over 6 decades of unitsThree curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁵ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)1.9·10⁶cond(A, x)4.8ratio of the bounds4·10⁵both curves above the data are boundsand only one of them is a measurement

Three curves against the size of an entrywise relative perturbation. The normwise bound κ∞·ε is a valid bound and sits 4·10⁵ times above the componentwise one cond(A, x)·ε, which is also a bound and is nearly attained by the worst of forty random perturbations at each size.

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.

14 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 size the sweep can afford

a spread of units the inversions survive

and so does the componentwise one

and the componentwise one nearly attained

LU is for square matrices

matmul shapes agree

the normwise bound holds at ε = 10⁻¹⁰

the normwise bound holds at ε = 10⁻¹¹

the normwise bound holds at ε = 10⁻¹²

the normwise bound holds at ε = 10⁻¹³

the normwise bound holds at ε = 10⁻¹⁴

the normwise bound holds at ε = 10⁻⁹

with no spread of units the two bounds are close together

with the two bounds orders of magnitude apart

Against the rule

It draws a decomposition and prints its residual. It calls solve, condInf, condSkeelAt, 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.

Two errors, and whose fault they are

A condition number scaling cannot move

Skeel's componentwise condition number is invariant under any row scaling — exactly, before any norm is taken, because two diagonal factors cancel entry by entry. It is never larger than the normwise one and can be arbitrarily smaller, and the ratio between them is a diagnostic for which kind of ill-conditioning a matrix has.

 Structure, and the solver that cannot see it

The condition number of the model

Describe a 40×40 Toeplitz matrix by its 1,600 entries and its condition number is 78,800. Describe it by the one number it actually contains and the condition number is 61.9. The three decades in between are not an approximation or a bound — they are what κ has been over-stating, and the drop is not where the linear algebra is.

 Two errors, and whose fault they are

The units the matrix is measured in

One linear system, written twice. The rows of the second are the rows of the first in different units, the solution is identical to the last bit, and the condition number has moved by eight orders of magnitude. One of those two numbers is a fact about the problem and the other is a fact about the notation.

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