Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Row scaling — where it appears

Multiplying each equation of a system through by a constant, which changes the matrix, the condition number and the pivot order, and not the solution. It is invisible in the solution and decisive in the pivot order and the condition number, which is why a pivot rule that compares scaled entries is comparing units.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

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

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.

elimination · Pivoting 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

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.

error · Scaling 10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1relative size of the entrywise perturbationrelative forward errorκ∞ · εcond(A,x) · εmeasuredboth bounds holdκ∞(A)1.9·10⁸cond(A, x)4.8ratio of the bounds4·10⁷both curves above the data are boundsand only one of them is a measurement

A condition number scaling cannot move

Skeel's componentwise condition number is invariant under any row scaling — exactly, before any norm is taken, because two diagonal factors cancel entry by entry. It is never larger than the normwise one and can be arbitrarily smaller, and the ratio between them is a diagnostic for which kind of ill-conditioning a matrix has.

error · Scaling 0510152025303540012345matrix size ngrowth factorpartial pivotingrook pivotingcomplete pivotingsolid: median of 30 · dashed: worst of themat n = 40partial pivoting, median3.3rook pivoting, median2complete pivoting, median1.6rook, worst of the draw2.8the same matrices under every rulerook 3.3× partial's search

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.

elimination · Pivoting error at the largest strainas given4.6·10⁻⁴multipliers to the size of x3.3·10⁻⁸first-solve scale, never below one1.8·10⁻⁸null-space route1.8·10⁻⁸10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³strain δ (left: none)relative error in x10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²noneas givenmultipliers to the size of xfirst-solve scale, never below onenull-space routethe loss was the multipliers' sizeand it scales away

The scale that only moved a pivot

Multiply the constraint rows of a saddle-point system until its multipliers are the size of its solution, and the extra error the route was blamed for — 4.6·10⁻⁴ against the null-space route's 1.8·10⁻⁸ — falls to 3.3·10⁻⁸. The prediction holds and its reason does not. A scale of ten does what a scale of 6·10⁵ does; hold the elimination's row order fixed and nine decades of scale move the error by less than a factor of five. What the scale changed was which row partial pivoting took at the second step, and taking the constraint rows first does the same job with no scale at all.

leastsquares · Constrained least-squares

Named alongside it

The objects these essays reach for when they reach for this one.

Partial pivotingBackward errorComplete pivotingComponentwise condition numberCondition numberEquilibrationExact ground truthForward errorGaussian eliminationGrowth factorHilbert matrixPerturbation

All concepts