Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The same first pivot decision, with the corner entry at 10⁻¹²

One function in the sparselu library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 same first pivot decision, with the corner entry at 10⁻¹². Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

pivot-conflict is one function in lib/figures/sparselu.js — sparse lu — where the fill argument and the stability argument disagree. 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 same first pivot decision, with the corner entry at 10⁻¹²Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest227 entries, growth 1.9·10¹¹largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest3.8·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted5.4·10⁻¹⁷forward error, sparsest3·10⁻⁵forward error, pivoted4.8·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

logTiny: -12

The arguments are the ones Structure and stability stop being separable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same first pivot decision, with the corner entry at 10⁻¹²Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest227 entries, growth 1.9·10¹¹largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest3.8·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted5.4·10⁻¹⁷forward error, sparsest3·10⁻⁵forward error, pivoted4.8·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

logTiny: -9

The arguments are the ones Structure and stability stop being separable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same first pivot decision, with the corner entry at 10⁻⁹Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest233 entries, growth 1.9·10⁸largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest4.8·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted8.2·10⁻¹⁷forward error, sparsest5.3·10⁻⁹forward error, pivoted1.8·10⁻¹⁵red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

logTiny: -3

The arguments are the ones Structure and stability stop being separable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same first pivot decision, with the corner entry at 0.001Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest233 entries, growth 188largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest8.4·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted6.1·10⁻¹⁷forward error, sparsest1.1·10⁻¹⁵forward error, pivoted5.1·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

logTiny: -6

The arguments are the ones Structure and stability stop being separable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same first pivot decision, with the corner entry at 10⁻⁶Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest233 entries, growth 1.9·10⁵largest first — safe242 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest6.6·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted6·10⁻¹⁷forward error, sparsest3·10⁻¹¹forward error, pivoted8·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

logTiny: 0

The arguments are the ones Structure and stability stop being separable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The same first pivot decision, with the corner entry at 1Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.the matrix105 entriescorner first — sparsest233 entries, growth 1largest first — safe241 entries, growth 1.19the middle factor is the smaller one, and its answer has no correct digitsboth factorisations reproduce the matrix‖PA − LU‖/‖A‖, sparsest7.5·10⁻¹⁷‖PA − LU‖/‖A‖, pivoted8·10⁻¹⁷forward error, sparsest1.6·10⁻¹⁶forward error, pivoted8.9·10⁻¹⁶red marks are entries elimination createdthe fill argument and the stability argument disagree

Three sparsity patterns side by side: the matrix, the factor obtained by eliminating the corner first, and the factor obtained by pivoting on the largest entry instead.

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.

8 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 tiny corner pivot multiplies the matrix by a great deal

and takes the answer with it

and the pivoted answer is right at every position of the slider

and the sparsest elimination is simply the right one

growth is a growth factor

matmul shapes agree

the sparsest-pivot elimination is never the larger factor

while an ordinary corner pivot costs nothing

Against the rule

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

Sparsity, and what elimination costs

Structure and stability stop being separable

The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.

The whole library · All essays · What must fail