Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Machine epsilon — where it appears

The spacing of the representable numbers near one, which is the unit every rounding error on this site is measured in.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

[½, 1)[1, 2)[2, 4)0.5124gap 0.125gap 0.25 — twice as wide8 values per octavespacing doubles at each power of two

What a float can hold

The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.

arithmetic · floating point significand bits10³10⁴10⁵10⁶10⁷10⁸10⁹10¹⁰κ(A)87892634456329625911109435761199127514162024every matrix positive definiteruns producing a false certificate14of runs in total72never above, in significand bits12first κ at eight bits10⁵the comparison was correctand what it proved was not true

Deciding that a zero has arrived

The previous tolerances were offers — accept this much error, save this much work. A detection threshold is not an offer, because both directions are failures. One matrix here has three genuinely near-invariant subspaces, and the constant somebody typed decides which of them the recurrence stops at; at eight significand bits the same kind of constant produces a proof of something false.

error · deliberate zero

Named alongside it

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

Backward errorCertificateCounterexampleError accumulationFloating pointHalf precisionInfinite eigenvalueInvariant subspaceKrylov subspaceNegative curvatureNumerical rankSignificand

All concepts