Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Two condition numbers of one 8×8 system, as its rows are put into different units

One function in the scaling library, called 9 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 36 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 condition numbers of one 8×8 system, as its rows are put into different units. Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10¹⁰ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

unit-ladder 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 condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10¹⁰ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.0246810110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 10 decades1.9·10¹⁰cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10¹⁰ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

maxSpread: 8

The arguments are the ones The inverse that is never formed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10⁸ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.02468110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 8 decades1.9·10⁸cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10⁸ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

maxSpread: 10

The arguments are the ones The units the matrix is measured in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10¹⁰ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.0246810110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 10 decades1.9·10¹⁰cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10¹⁰ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

maxSpread: 2

The arguments are the ones The units the matrix is measured in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 223 while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.012110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 2 decades223cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 223 while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

maxSpread: 4

The arguments are the ones The units the matrix is measured in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 2·10⁴ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.01234110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 4 decades2·10⁴cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 2·10⁴ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

maxSpread: 6

The arguments are the ones The units the matrix is measured in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two condition numbers of one 8×8 system, as its rows are put into different unitsFour curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10⁶ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.0123456110²10⁴10⁶10⁸10¹⁰10¹²spread of the row units (decades)condition numberκ∞(DA)cond(DA)Hilbert κ∞Hilbert condone system, two numbersκ∞ at no spread9.8κ∞ at 6 decades1.9·10⁶cond, either end7Hilbert, equilibrated1.3·10¹⁰the solution is the same at every spreadand one of these curves knows it

Four curves against the spread of the row units, in decades. κ∞ of the scaled matrix rises from 9.83 to 1.9·10⁶ while the componentwise condition number stays at 6.98 throughout — the same system, the same solution, and one of the two numbers is a fact about the units. Hilbert's two numbers are drawn flat beside them at 3.4·10¹⁰ and 1.2·10¹⁰: a matrix whose sensitivity no scaling repairs.

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.

36 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.

the componentwise number is unchanged at a spread of 0.0 decades — checked 29 times

a size the two inversions stay accurate at

a spread of units the axis has room for

and equilibrating Hilbert barely moves its condition number

and is never above the normwise one

LU is for square matrices

matmul shapes agree

the units alone take κ up by nearly the whole spread

Against the rule

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

Elimination, and the swap

The inverse that is never formed

x = A⁻¹b is how the solution of a linear system is written and it is not how it is computed. The usual reason given is cost — three times the arithmetic. The real reason is that one of the two routes is backward stable and the other is not, and at κ = 10¹⁴ they differ by twelve orders of magnitude in the number that says whose fault a wrong answer is.

 Elimination, and the swap

The pivot that reads the units

Partial pivoting compares the entries of a column and takes the largest. Those entries carry units, so the comparison depends on them — and there is a row scaling, on the standard two-by-two that pivoting exists to fix, which makes partial pivoting perform the identical catastrophic elimination it was introduced to prevent, with no interchange at all.

 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.

The whole library · All essays · What must fail