Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Random walk — where it appears

3 essays name this object, across 3 fields. What follows is each of them, and the objects they name alongside it.
10²10³10⁴10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²steps taken‖RᵀR − AᵀA‖ ⁄ ‖AᵀA‖the bound, linear in the steps√k · uevery step safe, the chain notdrift after the run3.9·10⁻¹⁴the bound there3.3·10⁻¹³√k · u there6.1·10⁻¹⁵worst single amplification2.7worst leverage met0.69refactorisations1backward stable onceand three thousand times is a different claim

Stable once, and three thousand times

A sliding window adds a row and removes one at every step and never looks at the data again. No single step of it amplifies by more than 2.72, no downdate fails, and after three thousand steps the triangular factor in memory is 3.9·10⁻¹⁴ from the matrix it is supposed to be a factor of — six hundred times growth from a per-step bound that says nothing about chains.

sequence · sequence stability 110¹10²10³110¹10²10³the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

Three walks and one bound

A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.

arithmetic · summation 01428425670849811210⁻²²10⁻¹⁸10⁻¹⁴10⁻¹⁰10⁻⁶10⁻²conjugate gradient iterationrelative residualthe unit roundoff, 1.11·10⁻¹⁶the answer's residualthe residual reportedtwo residuals, one runreported, at its best6.9·10⁻²¹the answer's, at its best5.1·10⁻¹⁰unit roundoff1.1·10⁻¹⁶largest iterate on the way9.3·10¹³iterations drawn110the recurrence remembers every roundingand the stopping test is written in it

The residual the method reports

Conjugate gradients prints a relative residual of 6.9·10⁻²¹. The unit roundoff is 1.1·10⁻¹⁶, so that is not a small residual and not a large one — it is not a residual. The vector the method is holding at that step has ‖b − Ax‖/‖b‖ = 5.1·10⁻¹⁰, and nothing in the run says so.

iterative · residual gap

Named alongside it

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

Backward errorConjugate gradientsExact ground truthLow-rank updateUnit roundoffCancellationCholesky factorisationCompensated summationCondition numberGivens rotationHyperbolic rotationKrylov subspace

All concepts