A class a longer chain takes away

Symmetry survives a bigger problem. A symmetric matrix stays symmetric when a row and a column are added, and the theorem that its eigenvalues are real applies to the enlarged matrix without being re-checked, because no step of the proof mentions the size. That is what a structural class is worth: it is established once and then inherited, and a method that requires it can require it once.

Hyperbolicity reads like a class of that kind. The essay that gives it a certificate states it as a property of the coefficients — M, C and K symmetric with M positive definite, and (xᵀCx)² > 4(xᵀMx)(xᵀKx) for every non-zero x — and proves it with a single factorisation that either completes or does not. Nothing in that sentence mentions a size either.

The chain says otherwise, and it says so at one mass. Take the damped chain of the field’s first essay with C = βK, set the damping to β = 5.7, and ask for the certificate at seven masses. It is there: a μ at which Q(μ) is negative definite, at μ = −0.4339, with seven eigenvalues above it and seven below and a gap of 0.3796 between the two groups. Every one of the fourteen eigenvalues is real, and the overdamping margin — the smallest value over the modes of (βκ)² − 4κ — is 0.1441, comfortably positive.

Add one mass. The same damping, the same springs, the same construction, eight of them instead of seven, and there is no certificate at all. The margin is −0.0098. The spectrum has left the real axis and the class has been lost by lengthening the object, which is the one thing a class is supposed to survive.

The quantity that decides it has a closed form, and the whole of this essay is what that closed form does as the chain grows.

The boundary is a function of the length, and it has no ceiling #

For the chain with C = βK the quantifier over every x collapses. In the basis of K’s eigenvectors the hyperbolicity condition becomes β²κ > 4 for every stiffness eigenvalue κ, so the binding constraint is the smallest κ — the longest wavelength, the mode that is last to become overdamped — and the critical damping is β* = 2/√κ_min = 1/sin(π/2(n+1)), exactly.

That is a decreasing sine in a denominator, so β* increases with n, and the rate is the first thing to measure. Read off the closed form at the sizes a chain is actually built at:

n      3        4        6        8        10       12       14       16
β*   2.6131   3.2361   4.4940   5.7588   7.0267   8.2962   9.5668   10.8380

Between n = 3 and n = 16 the required damping rises by a factor of 4.15 while the chain grows by 5.33, and the two rates converge. Expanding the sine for small argument gives β* ≈ 2(n+1)/π, and the ratio of the exact value to that approximation runs 1.0262 at n = 3, 1.0166 at n = 4, 1.0084 at n = 6, 1.0051 at n = 8, 1.0034 at n = 10, 1.0024 at n = 12 and 1.0014 at n = 16. It is 1.000097 at n = 64 and 1.0000062 at n = 256. So the growth is linear in the length with slope 2/π = 0.6366, and it is linear from very nearly the first size anybody would build: at eight masses the linear formula is already right to half a per cent.

Linear growth with no ceiling is the finding. There is no β that makes every member of this family hyperbolic, because for any β the closed form eventually exceeds it — at n = 100 the class demands 64.30 and at n = 1000 it demands 637.26, and neither number is an asymptotic estimate but the value of a sine. The class is not a property of the construction. It is a property of one member of it, and the sentence “this chain is overdamped” is incomplete without the length.

What a fixed damping buys, in masses #

The dual reading is the one a designer would use, and it is a single inversion of the closed form. Given a damping β, the longest chain it certifies is the largest n with 1/sin(π/2(n+1)) ≤ β, which is the floor of π/(2 arcsin(1/β)) − 1. Measured:

β       5.7    6     8    10    20    50    100   1000
masses    7    8    11    14    30    77    156   1569

The first column is the refusal at the top of this page, arrived at from the other direction: β = 5.7 certifies exactly seven masses and not eight, which is why the chain of seven above has a certificate and the chain of eight has none. Nothing was tuned to produce that boundary; 5.7 sits between β*(7) = 5.1258 and β*(8) = 5.7588, and the arithmetic of the eighth mass does the rest.

The rest of the row is the useful shape. A fixed damping buys a length that grows like π/2 = 1.571 masses per unit of β, so doubling the damping roughly doubles the chain it certifies, and the exchange rate never improves. That is a very poor return on a physical quantity: β is a damping coefficient, and a structure given ten times the damping it needs is a structure whose model has stopped describing the thing it was built for.

The consequence for a proof is exact rather than approximate. A hyperbolicity argument established at one size is not evidence at a larger one, in the same way that a parameter chosen on a smaller problem is not evidence about the larger problem it is then used on. Both are a decision made where it could be measured and carried to where it was not, and both fail in the direction that looks like success: the smaller problem behaved, so the transfer looks confirmed.

Both routes stay at thirteen digits, at every size #

If the boundary has to be located again at every length, the question is how well it can be located, and the site’s habit is to answer that with two computations that share no arithmetic.

One route is the sine. The other knows nothing about it: bisect on β, and at each trial value scan for a μ at which the Cholesky of −Q(μ) completes, keeping the half of the bracket where one exists. That is a search over a bracket of [1, 20] driven by nothing but whether a factorisation runs to the end, and forty-four bisections of it are what each of these figures reports. Neither route computes an eigenvalue. One is a trigonometric identity and the other is the same definiteness test that stands in for definiteness everywhere else on this site, asked forty-four times — and used here as a decision rather than as a factorisation, which is the reading a matrix that is definite on one machine shows can turn on the last digit when the matrix is near its own boundary. Every trial value of this bisection is near the boundary by construction, since that is what a bisection converges to, and the thirteen digits below are the answer to whether that matters here.

The agreement, in digits, at eight lengths:

n         3    4    6    8   10   12   14   16
digits   13   13   15   13   14   13   14   14

Thirteen digits is the floor and it does not fall as the chain lengthens. The variation between thirteen and fifteen is not a trend in n and should not be read as one: forty-four bisections of a bracket nineteen wide leave a half-width of 5.40·10⁻¹³, and every measured difference between the two routes is inside it. The bisection’s own resolution is a property of the bracket, not of the chain, so the relative accuracy of the boundary actually improves with n as β* grows inside a fixed bracket — which is the opposite of what a family that becomes harder to certify would look like.

That is worth stating plainly, because the essay is otherwise a catalogue of what lengthening takes away. Lengthening takes away the class and not the instrument. The certificate remains exact in the only sense a factorisation can be exact — it completes or it does not — and the closed form remains a sine. What changes is which side of the boundary a given damping is on.

Where the family is going #

The last size the bisection can be run at cheaply is sixteen, and it is worth drawing because the picture at sixteen is the picture at four with one number moved.

The margin curve has the same shape, the same third route agrees — the largest imaginary part in the computed spectrum falls to the rounding level exactly at the marked line, which is a fact about eigenvalues rather than about factorisations or sines — and the boundary sits at 10.8379514475 instead of 3.2360679775. The class has not changed its character over a factor of four in length. It has moved its wall, by a factor of 3.35, and it will keep moving it.

Standing on the boundary costs half the digits, and the cost also moves #

A proof that has to be re-earned at every size invites an obvious economy: certify once at a damping just above the boundary, and raise it only as far as each new length requires. That puts the problem permanently near β*, and near β* the smallest mode’s two eigenvalues are close to coinciding.

At β = β*(1 + ε) the colliding pair separates like a constant times √ε, and the constant belongs to the chain length rather than to the law: 1.750 at four masses, 1.260 at six, 0.9657 at eight, 0.8162 at ten, 0.6909 at twelve, 0.5600 at fourteen and 0.5162 at sixteen. Across twelve decades of ε each of those holds to within two per cent, which is the square-root law measured; across four factors of chain length it falls by 3.39, which is what makes it a constant of the chain. A longer chain approaches its own double root more slowly at the same relative distance from the boundary, so the two effects of lengthening pull opposite ways and the net cost has to be measured rather than argued.

Measured at ε = 0 — the damping set to exactly β*, where the two eigenvalues genuinely coincide — the worst relative error of the computed spectrum against the closed form runs 2.855·10⁻⁸ at four masses, 4.550·10⁻⁸ at six, 3.732·10⁻⁸ at eight, 5.690·10⁻⁸ at ten, 7.465·10⁻⁸ at twelve, 1.178·10⁻⁷ at fourteen and 1.085·10⁻⁷ at sixteen.

Roughly eight digits, at every length. The number is not monotone — fourteen masses is worse than sixteen — so the honest statement is a range and not a trend: the loss at the boundary rises by a factor of 3.80 across the whole family, from the smallest chain the closed form can be read at to the largest one drawn here.

Why the boundary moves, and what moves with it #

Both of the quantities this essay has been reporting come out of one eigenvalue, and writing them side by side turns the comparison above from a coincidence of two sweeps into an identity.

K is the chain’s stiffness matrix, and its smallest eigenvalue is κ_min = 4sin²(π/2(n+1)) — the longest wavelength, the mode with the least restoring force for a unit of displacement, and the one no norm of K can see, because a norm reads the largest eigenvalue and this is the smallest. It falls like 1/n². The largest eigenvalue is κ_max = 4 − κ_min, which approaches four from below at every length.

The critical damping is 2/√κ_min, so it rises like n, which is the linear growth measured at the top of this page. The conditioning is κ_max/κ_min = (4 − κ_min)/κ_min = 4/κ_min − 1, so it rises like n². And those are the same expression twice:

κ(K) = β*² − 1,   exactly

Measured against the numerically computed condition number at seven lengths: 5.828427125 against 5.828427125 at three masses, 9.472135955 at four, 32.16343748 at eight, 116.4611916 at sixteen, 440.6885604 at thirty-two. The two routes agree to the rounding level at every one of them, which is what an identity looks like rather than what a fit looks like.

Two things follow, and the second is why the sweep in the next section is a control rather than an anecdote. A long chain cannot be both marginally overdamped and well conditioned, because the damping that makes it hyperbolic and the conditioning of its stiffness are the same number squared — so the design freedom a reader might expect to have between the two does not exist on this family at all.

And the conditioning is therefore not an independent variable that happens to be large. It is the boundary, squared. Anything that scales with κ(K) scales as the square of the critical damping, and anything that scales with β* scales as its first power, so a measured effect can be told which of the two it is following by how fast it moves — which is precisely what the previous section’s seven chains were measured for.

The control that separates the mechanisms #

Eight digits lost is the kind of number that gets attributed to conditioning, and the family supplies the control that says it is not.

κ(K) for this chain is a function of the length alone: β does not appear in K, so the stiffness matrix is the same matrix at every stop of the sweep towards the boundary, and its condition number is 9.472 at four masses, 19.20 at six, 32.16 at eight, 48.37 at ten, 67.83 at twelve, 90.52 at fourteen and 116.46 at sixteen. Over the same range in which the loss at the boundary rises by 3.80, the conditioning of K rises by 12.30.

Three times the movement in the quantity that is supposed to be the cause, and the effect does not follow it. If the loss at a double root were a conditioning effect under another name it would track κ(K); it tracks nothing of the sort, and the two numbers are measured on the same seven chains with nothing else changed. The essay that establishes the class makes this claim at one size, where κ(K) is a single number and cannot move at all, and so cannot test it; a family in which the conditioning moves by an order of magnitude can.

The identity above sharpens the reading rather than merely supporting it. Across the seven chains the critical damping rises by 3.35 and its square less one, which is the conditioning, rises by 12.30. The loss rises by 3.80. That is within fourteen per cent of the first factor and a quarter of the second, on a quantity that is not monotone in n and is therefore not being fitted to either — the useful statement is the exclusion. Whatever the loss at a double root is following, it is not the square, and the square is the only thing in this problem a condition number is.

What is unbounded near a double root is the derivative of the map from a coefficient to a root — O(√ε) rather than O(ε) — and that is a property of the question rather than of any matrix in it. It is the same object the condition number is an amplifier has to exclude, since the amplifier it defines is a statement about simple eigenvalues and is infinite at a repeated one, and the same wall a condition number for one eigenvalue reaches from the matrix side at a defective matrix. Here it arrives with the length as a second axis, and the second axis is what turns an assertion into a control.

What follows for anything that inherits a proof #

The practical content is a short list, and each item is a habit rather than an algorithm.

Re-certify at every size, because it costs one Cholesky. The certificate is a single symmetric matrix factorised at a single μ, and on this family μ is even predictable, so the price of the honest answer is one factorisation against a solve that costs many. A structural claim that is cheap to check and is checked once is a claim that has been converted into an assumption for no saving at all.

Do not carry a damping forward. A value that certified the model at seven masses is a decision that was right last time and is being used where it was never measured. The failure is silent: the eigenvalues go complex, a solver returns them, and nothing in the output says that a class was assumed and has been lost.

Refinement is a lengthening. Anything computed on a coarser grid and then recomputed on a finer one has increased n, so a hyperbolicity argument made at the coarse resolution is void at the fine one by exactly the arithmetic above. The physical model has not changed and the class has.

Symmetry is the contrast, not the analogue. Symmetry is worth more than precision measures what a symmetric matrix gives back — its eigenvalues to full accuracy however ill conditioned it is — and the property that buys that is checkable by inspection and survives enlargement. Hyperbolicity is checkable only by computation, at a cost that is small but not zero, and it does not survive enlargement at all. Both are called structural properties of the coefficients and only one of them is inherited, which is the distinction this family exists to make.

The integer the certificate hands over is safe. Once a certificate exists at a given size, the inertia of Q(μ) counts eigenvalues relative to μ exactly, and that count is an eigenvalue count that cannot be slightly wrong at every length. Nothing in this essay weakens it. What the length changes is whether there is a certificate to read it against.

And a chain near its own boundary is a chain whose modes are crowded. The primary group of a long chain sits close to zero with neighbouring pairs a fraction apart, so anything that wants an individual eigenvector out of it is subject to the gap that decides the eigenvector, and the class does not help with that either. Realness is not separation, and a longer chain buys less of the second while demanding more damping for the first.

The economy the boundary suggests — damp as little as the class permits — is therefore the expensive one twice over: it puts the problem where half the digits go, and it puts it there at the size where the modes are least separated. Damping well clear of β* costs a physically implausible model; damping just above it costs eight digits and cannot be repaired by buying the accuracy back without solving for the pair rather than the roots. Both of those are design decisions, and both need the length before they can be made.

Why this is the class’s second essay and not a footnote to the first #

The first essay about this class establishes it at a single length: a certificate, a boundary located twice, an integer read two ways, and a loss of eight digits with nothing ill conditioned anywhere. Every one of those statements is true at eight masses and every one of them is about eight masses.

Making the length a variable changes two of them and leaves two alone, which is the reason this is worth a second reading rather than a longer first one. The certificate and the closed form are unchanged: two routes, no eigenvalues, thirteen digits at every size, and a floor set by the bisection rather than by the chain. The class itself and the price of standing on its edge are not: no damping serves the whole family, and the price rises by a factor that is measurably not the conditioning’s factor.

The second of those needed a family to state at all. A control is a quantity that moves while the effect does not, or moves differently, and at one size κ(K) is a constant. The claim that the loss at a double root is not a conditioning effect was made there on the strength of κ(K) being small; here it is made on the strength of κ(K) being three times more mobile than the loss it is supposed to explain. That is a better argument, and it is available only because a chain can be made longer — the same operation that takes the class away.