Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Subnormal numbers — where it appears

The floating-point numbers below the smallest normal one, represented with a leading zero and progressively fewer significant bits. They keep subtraction of nearby numbers exact and are where relative accuracy is lost gradually rather than at once.

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

10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁸10⁻⁶magnitudespacing to the next numberthe smallest normalgradualflush to zerosmallest normal6.1·10⁻⁵smallest subnormal6·10⁻⁸octaves of subnormals10pairs that lie under FTZ10the spacing stops halving and stays putwhich is what makes x − y = 0 mean x = y

The numbers below the smallest one

Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.

arithmetic · subnormals 1611162126313610⁻²²10⁻¹⁹10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹vertex, clique first then tail|entry| relative to the largestone rounding of the largest entrya bound that is provedPerron root11bracket, low11bracket, high11bracket width1.2·10⁻⁴smallest entry-1.5·10⁻²⁰entries below zero4every entry is positiveand the picture disagrees

An eigenvector that must not change sign

Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the one two-sided bound on this site that is proved rather than estimated.

graph · perron frobenius

Named alongside it

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

CentralityCertificateCollatz wielandtEigenvectorFlush-to-zeroGradual underflowHalf precisionIEEE 754OverflowPerron frobeniusPower iterationRelative error

All concepts