Every essay — page 17
The answer that depends on the machine
Every other field here asks how wrong an answer is. This one asks how many answers there are. A parallel reduction adds a vector up in however many pieces there are workers, in whatever order they finish — so the same program, on the same data, at the same precision, returns a different number on a different machine, and every one of those numbers satisfies the published bound. The disagreement is a quarter of κu and the bound is ten thousand times larger, which is why nothing reports it. It matters where a number is compared to something: a stopping test, a rank test and a definiteness test each turn a real number into a verdict, and a verdict has no last digits for a disagreement to hide in. One matrix here has three different numerical ranks and one solve has thirteen different bills. And the smallest instance needs no parallelism at all — a multiply the compiler was allowed to fuse, which is one rounding, and which decides the sign of a determinant whose value is one.
The variation that comes with a seed
A randomised low-rank approximation's error moves by 31% between draws and by 10⁻¹⁵ between partitionings of one draw. In the one field on this site whose answer already comes as a band, the machine is inside the width of the line — and it is still there.
Two machines, one certificate
Nothing a solver returns says which of its answers you got. Four things could be reported instead — the summation condition number, the partition count, an exactly accumulated residual and a directed-rounding interval — and each costs about one pass over data the routine already has in hand.
The licence is not the boundary
Cholesky QR is licensed by κ²u ≪ 1, which reaches equality at κ = 9.5·10⁷ in double precision. At 10⁸ the factor it returns is already 0.37 away from orthogonal, and it goes on returning factors as far as 10¹³ — refusing at scattered condition numbers in between, at different ones for eight columns and for six.
The reading that never moves
Across thirty runs — five grids from 36 to 196 unknowns, six working precisions from 53 significand bits down to 8 — the residual conjugate gradients stops on stays between 1.10·10⁻¹³ and 9.95·10⁻¹³. Over the same thirty runs the error of the answer spans a factor of 2.39·10¹¹, and the step count more than doubles. The one number the run publishes is the only one that responds to neither axis.
The matrix a constraint makes
Every difficult matrix in the other fields was difficult for a reason the arithmetic supplied: a Hilbert matrix arrives ill conditioned, a Wilkinson matrix grows under elimination, a kernel matrix is dense. Ask a problem to minimise something subject to a constraint and the matrix that results is difficult for a reason the algebra supplies. Its zero block is the second derivative of a Lagrangian with respect to its own multipliers, so no pivot order removes it and no precision changes that; its inertia is known before anything runs; and Cholesky does not fail somewhere on it, it fails at the first constraint row, on a number the problem already contained. Then the field's second surprise, which is the opposite one: an interior-point method drives the condition number of this matrix to 10¹⁵ deliberately, and the answer keeps fifteen digits — because the number that describes the error is not the one every library prints.
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.
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.
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.
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.
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.
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.
A minimum the Hessian cannot see
A Hessian with four negative eigenvalues can sit at a constrained minimum, and a Cholesky of it stops at the third row. One symmetric indefinite factorisation of the saddle-point matrix settles the question anyway — ten positive pivots and four negative — without a basis for the null space ever being formed. The count is exact in the algebra and blind in floating point, in a band that grows like κ(A)²; the route through the null space is blind in one that grows like κ(A).
One eigenvalue and two steps
Put the off-diagonal block back into a block-diagonal saddle-point preconditioner and every eigenvalue of the preconditioned matrix becomes exactly one. GMRES still needs two steps, because the matrix is the identity plus a nilpotent part of norm 54, and a computed eigenvalue at one comes back as a ring of radius 8·10⁻⁸ — the square root of the rounding, not the rounding. With an approximate Schur complement the triangular form leaves one copy of each value where the diagonal form leaves two, and the step count halves.
The active set before the digits
An interior-point method takes fifteen iterations on a quadratic programme with forty constraints, and its iterate has eight correct digits at the eleventh. Take the constraints its diagonal calls active at the first iterate, solve the equality problem they define once, and check the answer against the conditions for optimality. It passes, to thirteen digits. The step's matrix had a condition number of 43 at that iterate, and 7·10¹⁵ at the last.
Where the augmentation puts the cost
Add γAᵀA to the objective block of a saddle-point system and its Schur complement tends to I/γ, so the cheapest possible approximation becomes the right one and the golden-ratio spectrum arrives — within 7.6·10⁻⁶ at γ = 10⁶. MINRES falls from 21 steps to 6. The inner solve with the augmented block rises from 14 conjugate gradient steps to 43, their product does not fall at all, and the answer loses seven and a half digits on the way.
A shift that certifies a saddle
On a constrained problem whose Hessian has four negative eigenvalues, a saddle-point matrix is quasi-definite only once H + δI is positive definite — past δ = 5.08 here. Its pivot signs then count the curvature of ZᵀHZ + δI rather than of ZᵀHZ, so a saddle with a negative curvature of −1 is certified a minimum from δ = 1.05 on, and every saddle shallower than δ goes the same way. Iterative refinement against the unregularised matrix keeps the second-order test the count gave up: it contracts on the minimum at δ/(μ + δ), 0.980 a step at δ = 5, and on the saddle it grows at exactly 1.25.
The perturbation that does the work
A saddle-point matrix made quasi-definite is perturbed in both blocks, and the laws measured for it moved both together. Moved apart, the laws all belong to one block. The zero block's perturbation γ decides whether every ordering factorises, sets the worst ordering's growth at 0.51/γ, and costs the answer 1,451 per unit — the reciprocal of the smallest eigenvalue of AH⁻¹Aᵀ to three figures. The perturbation of H moves none of the first two and costs 19 per unit. Refinement removes each block's perturbation at the rate its own Schur complement sets, so γ's limit sits fifty times nearer than δ's.
Two repairs for one symptom
Rescale a quadratic programme's constraint rows over six decades and the crossover that certified its answer at iterate 1.5 first certifies at 69.8, with two of six programmes never certifying at all. Normalising the rows removes the spread completely — the same numbers at 10¹, 10² and 10³ either way. Starting the method at the magnitudes the rows imply repairs the iteration count completely and the identification only halfway. They are two repairs and they fix different halves.
A test with no tolerance in it
An interior-point method's own stopping test is a tolerance on μ, and at the tightest it can be set to it stops after 15 iterations with 1.6·10⁻¹². A crossover from the iterate at 1.5 returns a point whose error is 1.8·10⁻¹⁴ — ten times sooner and a hundred times better, from an iterate carrying one correct digit. One attempt costs an eighth of a step, and the guess's own margin says which iterate to spend it on.
The shift that stops at the first right count
A nonconvex solver that finds the wrong inertia adds δI to H and tries again, and the δ it settles on is used as though it measured the curvature it corrects. It does not. On a well-conditioned constraint it is the schedule's number — 1.8·10⁴ times the need at a curvature of 5.6·10⁻⁹, between one and 7.3 times above 10⁻² — and on an ill-conditioned one the loop stops wherever the count first reads right: 63 of 152 saddles at κ(A) = 10⁸, the deepest with curvature 56. Refining the shift by bisection removes the first error and adds to the second.
A constraint the count stops seeing
Let one constraint drift towards being a combination of the others and the inertia of the saddle-point matrix keeps its promise only while σ²/|h| can be resolved — σ the constraint's smallest singular value, h the curvature along the direction it barely constrains. At h = −1 the count stops seeing the constraint at σ = 1.4·10⁻⁹, six decades before any rank test would drop it, and below that it reports a genuine minimum as a saddle on three to six draws in eight. No shift of H brings the constraint back: the correction loop shifts a problem that needed nothing by as much as 2,620. A perturbation of the constraint block does not bring it back either — it decides, at σ = √(|h|δ).