Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁷, before and after its first row is scaled by 1/ε

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 four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁷, before and after its first row is scaled by 1/ε. Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

pivot-under-scaling 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.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁷, before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given1 0 interchangesno pivoting · rows scaled1 0 interchangespartial · as given0 1 interchangepartial · rows scaled1 0 interchangesscaled partial · as given0 1 interchangescaled partial · rows scaled0 1 interchangecomplete · as given0 1 interchangecomplete · rows scaled0 1 interchangethe same problem twicepartial, as given10⁻¹⁸partial, rows scaled1its relative residual10⁻¹⁷complete, rows scaled10⁻¹⁸the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

logEps: -16

The arguments are the ones A pivot that searches one row and one column passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁶, before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given1.2 0 interchangesno pivoting · rows scaled1 0 interchangespartial · as given1.1·10⁻¹⁶ 1 interchangepartial · rows scaled1 0 interchangesscaled partial · as given1.1·10⁻¹⁶ 1 interchangescaled partial · rows scaled2.2·10⁻¹⁶ 1 interchangecomplete · as given2.2·10⁻¹⁶ 1 interchangecomplete · rows scaled2.2·10⁻¹⁶ 1 interchangethe same problem twicepartial, as given1.1·10⁻¹⁶partial, rows scaled1its relative residual5·10⁻¹⁷complete, rows scaled2.2·10⁻¹⁶the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

logEps: -17

The arguments are the ones The pivot that reads the units passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁷, before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given1 0 interchangesno pivoting · rows scaled1 0 interchangespartial · as given0 1 interchangepartial · rows scaled1 0 interchangesscaled partial · as given0 1 interchangescaled partial · rows scaled0 1 interchangecomplete · as given0 1 interchangecomplete · rows scaled0 1 interchangethe same problem twicepartial, as given10⁻¹⁸partial, rows scaled1its relative residual10⁻¹⁷complete, rows scaled10⁻¹⁸the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

logEps: -12

The arguments are the ones The pivot that reads the units passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹², before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given1.3·10⁻⁴ 0 interchangesno pivoting · rows scaled1.2·10⁻⁴ 0 interchangespartial · as given2.2·10⁻¹⁶ 1 interchangepartial · rows scaled1.2·10⁻⁴ 0 interchangesscaled partial · as given2.2·10⁻¹⁶ 1 interchangescaled partial · rows scaled2.2·10⁻¹⁶ 1 interchangecomplete · as given1.1·10⁻¹⁶ 1 interchangecomplete · rows scaled2.2·10⁻¹⁶ 1 interchangethe same problem twicepartial, as given2.2·10⁻¹⁶partial, rows scaled1.2·10⁻⁴its relative residual1.2·10⁻¹⁶complete, rows scaled2.2·10⁻¹⁶the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

logEps: -14

The arguments are the ones The pivot that reads the units passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁴, before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given8·10⁻⁴ 0 interchangesno pivoting · rows scaled0 0 interchangespartial · as given0 1 interchangepartial · rows scaled0 0 interchangesscaled partial · as given0 1 interchangescaled partial · rows scaled0 1 interchangecomplete · as given0 1 interchangecomplete · rows scaled0 1 interchangethe same problem twicepartial, as given10⁻¹⁸partial, rows scaled0its relative residual0complete, rows scaled10⁻¹⁸the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

logEps: -18

The arguments are the ones The pivot that reads the units passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four pivot rules on [[ε, 1], [1, 1]] with ε = 10⁻¹⁸, before and after its first row is scaled by 1/εEight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.forward error, relative to a solution of exactly (1, 1)no pivoting · as given1 0 interchangesno pivoting · rows scaled1 0 interchangespartial · as given0 1 interchangepartial · rows scaled1 0 interchangesscaled partial · as given0 1 interchangescaled partial · rows scaled0 1 interchangecomplete · as given0 1 interchangecomplete · rows scaled0 1 interchangethe same problem twicepartial, as given10⁻¹⁸partial, rows scaled1its relative residual10⁻¹⁸complete, rows scaled10⁻¹⁸the two systems have the same solutionand one pivot rule cannot see it

Eight bars of forward error. On the matrix as given, no pivoting loses the answer entirely and all three pivot rules return it to rounding. Multiply the first row by 1/ε — which does not change the solution — and partial pivoting makes no interchange at all and returns the same wrong answer that no pivoting does. Scaled partial pivoting and complete pivoting are unmoved.

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 corner entry small enough for the elimination to lose

a pivot rule this routine implements

a power of ten rather than an exponent literal

and makes none on the row-scaled one

and performs the same elimination that no pivoting does

at this ε the scaled elimination loses nothing at all

by making one interchange

complete pivoting is unmoved by the scaling

landing orders of magnitude from where the same rule landed on the same problem unscaled

LU is for square matrices

partial pivoting solves the unscaled system

scaled pivoting is unmoved by the scaling

so the scaling has made this one more accurate rather than less

while the unscaled elimination without pivoting still does

Against the rule

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

A pivot that searches one row and one column

Rook pivoting looks down a column for its largest entry, along that entry's row for a larger one, and back down that entry's column, until it finds an entry largest in both. On Gaussian matrices of size 64 it keeps the median growth factor at 2.53 against partial pivoting's 4.06 and complete pivoting's 1.88, and it compares 6,987 entries against 2,080 and 89,440. On Wilkinson's matrix it holds the growth at exactly 2 where partial pivoting reaches 9.2·10¹⁸. And on a matrix built to make it walk it compares 113,376 entries — more than complete pivoting.

 Sparsity, and what elimination costs

The order that was right last time

A pivot order computed once and reused across a sequence saves the symbolic phase, and the price is that a pivot which was large may now be small. Replacing it with √u·‖A‖ costs eight orders of backward error and iterative refinement recovers a factor of 8.8 of them. Divide each row by its largest entry first and the same reuse costs nothing at all.

 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.

The whole library · All essays · What must fail