Fill under three orderings of a sparse saddle-point matrix, predicted symbolically and then counted

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.

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.

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.

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.

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.

A stiffness block of 32 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 145, minimum-degree 79, reverse-cuthill-mckee 79: 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.

A stiffness block of 24 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 152, 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.

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.

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.

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 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.