Quasi-definite — the series
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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.
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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.
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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.
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The residual turns before the error doubles
Regularise a saddle-point matrix past the Hessian's most negative eigenvalue and its pivot signs certify every shallow saddle as a minimum; refinement against the unregularised matrix keeps the test, as a rate, and a saddle at a hundredth of the regularisation grows by only 1.0014 a step — 485 steps to double. That was read as hundreds of steps before the history says anything. The residual, which is what a solver actually has, says it at step 62: a fit of its logarithm over the last ten steps turns positive there and stays positive, while the minimum's is negative from step 10. Across five regularisations the verdict comes at an eighth of the doubling time.
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The right-hand side that hides the saddle
Refinement against an unregularised saddle-point matrix gives the second-order verdict the pivot signs cannot, from step 62 on the shallowest saddle — for one right-hand side. The verdict depends on how much of the growing direction the right-hand side contains, and it can contain none. Each decade removed delays the verdict by a fixed number of steps, 147 on the shallowest saddle, set by the growth against the minimum's slowest contraction rather than by the growth alone, which predicted 1,611. Rounding stops the hiding at about 2,700 steps — but by then refinement has solved the system to a residual of 2.6 times ten to the minus thirteen, and any stopping test has already accepted it. A random kick to the starting point costs nothing and finds it.
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One hyperplane hides nothing
Iterative refinement against a saddle-point matrix gives its verdict when the growing direction overtakes the minimum's slowest decay, and a right-hand side with that direction removed delays it by a fixed number of steps a decade. With two negative curvatures, the prediction was that hiding both is paid for by the faster, and that hiding only the faster leaves the slower one's race to run. The first half holds to half a per cent on three pairs, and to one and a half on two whose curvatures differ by a factor of two and of 1.2. The second does not: hiding either direction alone moves the verdict by a fixed handful of steps, the same at two decades as at sixteen, because the direction left in view is already growing. Only the intersection of the two hyperplanes hides anything — and it hides less than either curvature can alone, 1,127 steps at worst against 2,907, with random right-hand sides' slowest at 79 against 417.