Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Centrality — where it appears

A score assigned to each vertex of a graph, usually as an eigenvector of some matrix built from it. What is read from it is an order rather than the values, so the question that matters is how many of its comparisons the data settles.

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

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 places settled, of 14the arithmetic14 of 14one missing link3 of 14α from 0.80 to 0.9011 of 14entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons14smallest gap6.3·10⁻⁵largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

A ranking whose order is not determined

Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.

graph · perron frobenius

Named alongside it

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

Backward errorCertificateCollatz wielandtConditioningEigenvectorPagerankParameter choicePerron frobeniusPower iterationRankingRun-to-run variationSpectral radius

All concepts