Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Grounding — where it appears

Removing a Laplacian's kernel by deleting one vertex's row and column, so that what is left is positive definite. Which vertex is a free parameter no account of the method mentions, and it moves the condition number by up to three orders.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

1611162126110¹10²10³degree of the grounded vertexcondition number of what is left1, the best choicea parameter nobody setsvertices tried30best κ1at degree29worst κ898at degree1spread898one row and column deletedand it matters which

The vertex nobody solves for

A Laplacian is singular, so every solve with one has to remove its kernel first. There are three ways, they agree to fourteen digits, and the one everybody uses carries a free parameter that no account of the method mentions and that moves the condition number by nine hundred.

graph · Graph laplacian 110¹10²10³10⁴10⁵10⁶10¹10²10³10⁴10⁵spread of resistances, largest ÷ smallest possiblecondition number, unit diagonalleast-resistance treebreadth-first treegreatest-resistance treenode equationsspread resistances make the loops easyand the nodes hard

Spread resistances make the loops easy

Scaled to a unit diagonal, the loop equations on the least-resistance tree get easier as a network's resistances spread — from 120 to 5.44 over six decades — and stop depending on the grid's size, while the node equations of the same flow get harder, from 538 to 4.6·10⁴. The spread that ruins the range-space formulation rescues the null-space one, though the loops' density means the work saved is a factor of two, not the factor of nine the iteration counts suggest.

orthogonality · Null space basis 1357911131510⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1vertexrelative error against the exact answerthree routes, one numberworst, by elimination4.8·10⁻¹⁶worst, by spectrum10·10⁻¹⁶solve residual1.5·10⁻¹⁵Foster's sum15n − 115difference3.6·10⁻¹⁵the answer is a ratio of integersso the error is known

A distance computed by a solve

Effective resistance is the one quantity in this field with no combinatorial route to it — it is defined by a linear system. On a small unweighted graph the answer is a ratio of two integers, so for once the error is known rather than estimated, and every resistance in a graph has to add up to a number fixed in advance.

graph · Effective resistance groundarcs of the treearcs off the treethe loop worst servedleast-resistanceκ(Z)11κ(ZᵀHZ)143κ, unit diagonal9.3nonzeros in Z426longest loop, arcs26worst path ÷ own arc0.97every tree gives a basis of whole numbersthe resistances decide which one to want

The tree the resistances choose

On a network every basic set is a spanning tree and every null-space basis is a set of loops with entries 0 and ±1, so no tree can make Z badly conditioned. The tree with the best-conditioned Z still gives loop equations 4.8 times worse than the tree of least resistance: the basis has to be chosen against the Hessian, and pivoting finds it only when it pivots on the resistances too.

orthogonality · Null space basis

Named alongside it

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

Graph laplacianCondition numberNull spaceNull space methodPseudoinverseReduced hessianSpanning-treeColumn pivotingConditioningConjugate gradientsConnected componentsDeflation

All concepts