Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Saddle-point systems — where it appears

A block system whose lower-right block is zero because the constraints have no term of their own, which is where a zero diagonal arrives by construction. Its zero block is a modelling fact rather than a numerical accident, so a solver that refuses a zero pivot has refused the problem rather than found a defect.

Named by 15 essays across 10 fields — each of them below, with the objects they name alongside it.

-1-0.582271-0.1645420.2531860.6709151.088641.506370eigenvalue4 negative10 positivecounted before it was formedpositive10negative4at zero0innermost ratio39the zero block is a theoremand so is the count either side of it

The zero that is not a missing entry

A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.

constraint · saddle point system 01234510⁻¹⁷10⁻¹³10⁻⁹10⁻⁵10⁻¹10³10⁷10¹¹log₁₀ κ(A)condition number, and relative errorκ(S)κ(ZᵀHZ)range-space errornull-space erroragainst a BigInt answerκ(S) at κ(A) = 10⁵4·10¹⁰κ(ZᵀHZ), all stops21range-space forward error5.3·10⁻⁶null-space forward error5.8·10⁻¹²both are the same algebraand only one squares

Two ways to remove a constraint

A constrained system can be reduced by eliminating the multipliers or by eliminating the constrained directions. Both give the same answer in exact arithmetic and inherit different condition numbers — one of them squares the constraint's, and the other does not contain it at all.

constraint · saddle point system 00.3670080.7340171.101031.468031.835040eigenvalue of P⁻¹Kwritten down, then computeddistinct3at 16φ computed1.6off the closed form2.9·10⁻¹⁴1 − φ1φthe preconditioner's effect is a theoremand the golden ratio is in it

Three eigenvalues, and two are the golden ratio

Precondition a saddle-point system by the block diagonal of its own two definite pieces and the preconditioned matrix has exactly three distinct eigenvalues — 1, and the two roots of λ² − λ − 1. A minimal polynomial of degree three means three steps, at every conditioning, and the preconditioner nobody can afford turns out to be the statement the affordable ones are measured against.

constraint · block preconditioning 012345610⁻³10⁻²10⁻¹110¹log₁₀ κ(A)eigenvalue of P⁻¹Kthe constraint is not in itnontrivial6at one8movement, six decades7.1·10⁻⁶drift at one6·10⁻⁵2m at onethe marks are a pencil that never saw Aand the lines are the preconditioned matrix

A preconditioner that need not know the constraint

Keep the constraint block exactly and replace the objective block by anything positive definite on the null space. The preconditioned matrix then has 2m eigenvalues at exactly one, and its remaining n − m are the generalised eigenvalues of a pencil in which the constraint does not appear. Sweep its condition number over six decades and they do not move in six digits.

constraint · block preconditioning -14-12-10-8-6-4-2010⁻¹⁷10⁻¹³10⁻⁹10⁻⁵10⁻¹10³10⁷10¹¹10¹⁵log₁₀ μ — the barrier parametercondition number, and relative errorκ₂, condensedκ₂, augmentederror, condensederror, augmentedagainst a BigInt answerκ₂ augmented, μ = 10⁻¹⁴3·10¹⁵its relative error10⁻¹⁵κ₂ condensed2.4·10¹⁶its relative error0.31the same step, written two waysand only one of them is solvable

A condition number sent to infinity

An interior-point method manufactures an ill-conditioned matrix on every iteration, deliberately, because the separating of a diagonal is how it discovers which constraints are active. Written one way the answer keeps fifteen digits at a condition number of 3·10¹⁵. Written the other way — the way almost every code writes it — it has none left.

constraint · interior point conditioning D from PAPᵀ = LDLᵀ — the shaded pairs are 2×2 pivots10⁻⁶0.749······0.749·········1.2·10⁻⁶1.4······1.4·········2.1·10⁻⁶0.549······0.549·········3.6·10⁻⁶0.614······0.614·three rules, one matrix‖PAPᵀ − LDLᵀ‖, blocks5.8·10⁻¹⁷‖PAPᵀ − LDLᵀ‖, diagonal3.1·10⁻¹¹growth, blocks1.3growth, diagonal5·10⁵the zero block is what the problem saysand one rule does not need it to be nonzero

When symmetry is not enough

The matrix [[0, 1], [1, 0]] is symmetric, nonsingular and perfectly conditioned, and there is no diagonal entry to pivot on. Every factorisation restricted to symmetric interchanges and one-by-one pivots fails on it, at any depth of searching, because every entry it could search is zero. The repair is to take two variables at once.

elimination · cholesky -18-15-12-9-6-300log₁₀ ‖PKPᵀ − LDLᵀ‖ / ‖K‖share of orderingsbestworstexistence and stabilityfactorise1unregularised0.69worst growth6.4·10⁵growth × δ0.64the ordering is free to chooseand not free of consequence

The regularisation that legalises every order

Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.

constraint · quasi definite κ(A) = 10⁶ throughout · κ(H) = 100 · the answer is the same answer for every basisorthonormal — κ(Z)1κ(ZᵀHZ)25.6relative error1.07·10⁻¹⁵first m basic — κ(Z)1.99·10⁸κ(ZᵀHZ)3.8·10¹⁶relative error0.0518pivoted basic — κ(Z)2.06κ(ZᵀHZ)31.9relative error6.71·10⁻¹⁶what the choice costsdensity, orthonormal1density, fundamental0.5κ(ZᵀHZ) ÷ κ(Z)², naive0.96error, pivoted choice6.7·10⁻¹⁶every one of them is a basisand one of them loses fourteen digits

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.

orthogonality · null space basis 0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹10⁴10⁷10¹⁰log₁₀ τ — the weight on the constraintrelative error against the exact answerτ = 1/√uGram–Schmidtnormal equationsHouseholder QRthe ceiling is the method'sHouseholder, τ = 10¹⁴4.8·10⁻¹⁵normal equations14Gram–Schmidt4.9·10¹⁰1/√u6.7·10⁷a constraint is a weight at infinityand the solver decides how far infinity is

A constraint is a weight at infinity

Stack an equality constraint on top of a least-squares problem with a large weight and the answer approaches the constrained one like 1/τ². The limit is takeable to any accuracy — and how far it can be taken is a property of the solver, not of the problem. One of them stops at the square root of the precision, and one of them does not stop.

leastsquares · constrained least squares -1-0.75-0.5-0.25010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹10⁴log₁₀ σ — the barrier's reduction factorresidual after one reused stepconvergedthe pattern free, the factors notentries moved6off-diagonal0survived at σ = 0.996survived at σ = 0.106 steps0 stepsthe few entries that movedare the ones that dominate

What survives one step of the barrier

An interior-point method solves the same system dozens of times with the same pattern and different numbers, and exactly p entries change between one step and the next. The pattern is reusable for ever. The factorisation is reusable for none of them, and the threshold that says so is a reduction factor of about a per cent against schedules that use ten.

sequence · reuse natural113 predicted · 113 countedminimum-degree63 predicted · 63 countedreverse Cuthill–McKee63 predicted · 63 countedthe shaded entries are fill: zeros of K that the factorisation makes nonzeroallocated before the numbersnatural113minimum-degree63reverse-cuthill-mckee63predicted minus counted0the symbolic phase decides the memoryand nothing later is allowed to argue

An ordering that does not wait for the numbers

A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.

sparsity · sparse pivoting 2468101210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²log₁₀ τ‖Bx − d‖ / ‖d‖sketched with the objectiveweighted, not sketchedkept out of the sketchone power instead of twokept out — feasibility1.7·10⁻¹⁶sketched at τ = 10⁸1.7·10⁻⁸unsketched at τ = 10⁸1.7·10⁻¹⁶objective ÷ optimum1.2a sketch preserves a normand a constraint is not one

The half of a problem a sketch may touch

A sketch guarantees that a norm is preserved to within a factor. An equality constraint is a statement that a quantity is zero, and no multiplicative guarantee says anything about zero. Sketch a constrained problem written as a weighted one and the constraint is not destroyed — it is demoted, from a violation of 1/τ² to one of ε/τ, exactly half the exponent.

randomised · sketching -12-10-8-6-4-2010⁻¹⁷10⁻¹³10⁻⁹10⁻⁵10⁻¹10³10⁷10¹¹10¹⁵log₁₀ μcondition number, and relative errorκ₂, augmentedcomponentwise, condensedcomponentwise, augmentederror, augmentedone matrix, two numbersκ₂ at μ = 10⁻¹²3·10¹³componentwise, same matrix13their ratio2.3·10¹²measured error9.4·10⁻¹⁶every library prints the top lineand the error obeys the third

Two condition numbers of one matrix

κ₂ is a worst case over perturbations of a given norm, and a normwise perturbation may put its whole budget on the smallest entry. The componentwise number is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. On one matrix they are 3·10¹³ and 13.3, and the error obeys the second.

error · scaling 081624324000.250.50.7511.25angle of the strong direction (degrees)ratio, and convergence factorenergy ratio of oneenergy ratioclassicalminimiserwith the constraintwhere the formula is optimalenergy ratio at 0°1energy ratio at 45°1.2classical rate at 45°0.35the minimiser's rate there0.28a ratio of exactly one while the assumption holdsand a diagnostic when it stops

The formula that was already optimal

Ask for the interpolation that minimises the energy of its own columns and the answer is the classical AMG formula — to zero at every row of the one-dimensional Laplacian, and to four digits in two dimensions. On the operator rotated to 45° the two part company, and the gap between them is a diagnostic that needs no reference solution.

iterative · algebraic multigrid -4-2.35371-0.7074290.9388572.585144.231435.8777102468101214shift σν(σ)an answer that is an integershifts2000disagreements0eigenvalues14steps14the marks are a Jacobi decompositionand the staircase never saw one

An eigenvalue count that cannot be slightly wrong

Every spectral computation on this site returns floats with errors in them. Counting eigenvalues below a shift by the signs of an unpivoted elimination returns an integer, and an integer cannot be 6.9999999997 — so the answer is exactly right, or wrong by a whole eigenvalue, and where the second happens is a band of measurable width.

spectra · inertia

Named alongside it

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

Condition numberExact ground truthConstrained minimisationInertiaLDLᵀ factorisationForward errorGrowth factorIndefinite matrixInterior point methodNormal equationsNull spacePreconditioning

All concepts