Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

ordering-fill

One function in the quasidef library, called 13 times across 6 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 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 fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then counted. A stiffness block of 24 unknowns with 4 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 113, minimum-degree 63, reverse-cuthill-mckee 63: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

ordering-fill is one function in lib/figures/quasidef.js — quasi-definite — every ordering legal, and the fill a symbolic phase can promise exactly. 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 24 unknowns with 4 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 113, minimum-degree 63, reverse-cuthill-mckee 63: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural113 predicted · 113 countedminimum-degree63 predicted · 63 countedreverse Cuthill–McKee63 predicted · 63 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural113minimum-degree63reverse-cuthill-mckee63predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 24 unknowns with 4 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 113, minimum-degree 63, reverse-cuthill-mckee 63: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

touch: 3

The arguments are the ones An ordering that does not wait for the numbers 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 24 unknowns with 4 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 113, minimum-degree 63, reverse-cuthill-mckee 63: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural113 predicted · 113 countedminimum-degree63 predicted · 63 countedreverse Cuthill–McKee63 predicted · 63 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural113minimum-degree63reverse-cuthill-mckee63predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 24 unknowns with 4 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 113, minimum-degree 63, reverse-cuthill-mckee 63: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

touch: 5

The arguments are the ones An ordering that does not wait for the numbers 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 24 unknowns with 4 constraints each touching 5 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 117, minimum-degree 71, reverse-cuthill-mckee 71: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural117 predicted · 117 countedminimum-degree71 predicted · 71 countedreverse Cuthill–McKee71 predicted · 71 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural117minimum-degree71reverse-cuthill-mckee71predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 24 unknowns with 4 constraints each touching 5 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 117, minimum-degree 71, reverse-cuthill-mckee 71: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

touch: 2

The arguments are the ones An ordering that does not wait for the numbers 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 24 unknowns with 4 constraints each touching 2 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 111, minimum-degree 59, reverse-cuthill-mckee 59: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural111 predicted · 111 countedminimum-degree59 predicted · 59 countedreverse Cuthill–McKee59 predicted · 59 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural111minimum-degree59reverse-cuthill-mckee59predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 24 unknowns with 4 constraints each touching 2 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 111, minimum-degree 59, reverse-cuthill-mckee 59: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

touch: 4

The arguments are the ones An ordering that does not wait for the numbers 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 24 unknowns with 4 constraints each touching 4 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 115, minimum-degree 67, reverse-cuthill-mckee 67: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural115 predicted · 115 countedminimum-degree67 predicted · 67 countedreverse Cuthill–McKee67 predicted · 67 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural115minimum-degree67reverse-cuthill-mckee67predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 24 unknowns with 4 constraints each touching 4 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 115, minimum-degree 67, reverse-cuthill-mckee 67: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

n: 30, m: 6

The arguments are the ones An ordering that does not wait for the numbers 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.

Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then countedA stiffness block of 30 unknowns with 6 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 182, minimum-degree 83, reverse-cuthill-mckee 83: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.natural182 predicted · 182 countedminimum-degree83 predicted · 83 countedreverse Cuthill–McKee83 predicted · 83 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural182minimum-degree83reverse-cuthill-mckee83predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

A stiffness block of 30 unknowns with 6 constraints each touching 3 of them. Each panel is the pattern of PKPᵀ under one ordering, with the entries the factorisation adds shown against the entries the matrix arrived with. The counts are natural 182, minimum-degree 83, reverse-cuthill-mckee 83: computed from the graph before any number was touched, and matched exactly by the factorisation of the regularised matrix, because quasi-definiteness means nothing during the numeric phase is allowed to move a pivot. That is the guarantee sparse LU cannot make — there the pivot order depends on the values, so the symbolic phase can only bound the fill and the allocation has to be able to grow.

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.

11 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 constraint block the pattern can show

a pivot rule this routine implements

a size the spy plots can hold

and the orderings are worth choosing between

matmul shapes agree

minimum-degree: the count matches the prediction exactly

minimum-degree: the regularised matrix factorises

natural: the count matches the prediction exactly

natural: the regularised matrix factorises

reverse-cuthill-mckee: the count matches the prediction exactly

reverse-cuthill-mckee: the regularised matrix factorises

Against the rule

It draws a decomposition and prints its residual. It calls fillUnderOrderings, 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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

An ordering that does not wait for the numbers

A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.

 Orthogonality, measured

The basis nobody chose on purpose

A method that eliminates a constraint has to pick a basis for its null space, and every basis is correct. Their condition numbers are eight orders apart, the reduced problem inherits the square, and the choice is usually made by a one-line rule nobody thought of as a numerical decision.

 The matrix a constraint makes

The regularisation that legalises every order

Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.

 The matrix a constraint makes

The zero that is not a missing entry

A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.

 The matrix a constraint makes

Two ways to remove a constraint

A constrained system can be reduced by eliminating the multipliers or by eliminating the constrained directions. Both give the same answer in exact arithmetic and inherit different condition numbers — one of them squares the constraint's, and the other does not contain it at all.

 When the problem arrives again

What survives one step of the barrier

An interior-point method solves the same system dozens of times with the same pattern and different numbers, and exactly p entries change between one step and the next. The pattern is reusable for ever. The factorisation is reusable for none of them, and the threshold that says so is a reduction factor of about a per cent against schedules that use ten.

The whole library · All essays · What must fail