Null space basis — where it appears
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
The basis nobody chose on purpose
A method that eliminates a constraint has to pick a basis for its null space, and every basis is correct. Their condition numbers are eight orders apart, the reduced problem inherits the square, and the choice is usually made by a one-line rule nobody thought of as a numerical decision.
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.
The condition number that does not know
Two constrained fits with the same size, the same number of constraints and the same κ(A) to twelve figures. One returns 4.7·10⁻¹⁶ and the other 3.0·10⁻⁴. What separates them is the conditioning of A restricted to the constraint's null space — 1.00 against 10¹² — which every solver computes on the way and none reports.
The reference was a method
The optimality conditions of a constrained fit contain AᵀA, so solving them is the road that squares the problem wearing a block structure. At κ(A) = 10¹¹ the route that never forms a cross-product returns 1.89·10⁻⁹ and the route that does returns 4.64·10⁻⁴ — and forming AᵀA and then solving it in exact rationals returns 3.45·10⁻⁴, so nearly all of the loss happens before any elimination begins.
The budget cannot finish the tree
The least-resistance spanning tree makes a network's loop equations well conditioned and its loops long, and the long loops cost a direct solver up to twice the breadth-first tree's factor work. A tree that refused any arc whose loop would run past a budget — 24 arcs, twice the breadth-first tree's longest — was predicted to keep the conditioning within twice the least tree's and save a third of the extra work. It keeps the conditioning and saves nothing. Every budget that binds leaves parts of the network it cannot join, the arcs it refused are needed to finish the tree, and they close loops as long as the least tree's. Half the least tree's loops are already as short as the property behind its conditioning allows; the slack is in the other half, and a greedy budget cannot reach it.
Named alongside it
The objects these essays reach for when they reach for this one.
Condition numberQR factorisationSaddle-point systemsColumn pivotingEquality-constrained least squaresExact ground truthNull spaceNull space methodReduced hessianSpanning-treeSparsityBasis