The three tolerances that do nothing

The tolerance that can only take away ends with a paragraph headed no cross terms:

secondSensitivity computes the mixed second derivatives as a by-product of the same two gradients … The population was measured directly instead — six hundred solves being cheaper than the argument — so the claim that the quadratic form is negative-definite over the whole tolerance box is evidence rather than demonstration.

This essay uses them. The demonstration costs twenty-two solves where the evidence cost six hundred, it turns the claim from none of six hundred went up into none can, and on the way it finds something the sampling could not have: three of the five tolerances do nothing at all.

What a curvature matrix is, and which one #

A response at a fixed frequency is a function of every component value. Expand it about the nominal design in the relative errors xᵢ = δpᵢ/pᵢ and the first two terms are

$$\frac{\Delta |H|}{|H|} \approx g \cdot x + \tfrac{1}{2} x^{\mathsf{T}} Q x$$

where g is the relative gradient and Q the relative curvature matrix. The rung below measured g — zero at a ripple peak, to ten digits, which is the whole of its finding — and the diagonal of Q, which is what a per-component sensitivity analysis reports. The off-diagonal entries are what happens when two parts are wrong at once, and they are not small: on this ladder the off-diagonal mass is larger than the diagonal mass.

The trap the rung below records had to be avoided again here. The adjoint identity differentiates the complex response, and at a ripple peak the complex derivative with respect to a reactance is not zero while the magnitude derivative is zero to ten digits. The magnitude’s second derivative is a combination of three complex quantities:

$$\frac{\partial^2 |H|}{\partial p\, \partial q} = \frac{\mathrm{Re}\!\left(\bar H \frac{\partial^2 H}{\partial p\, \partial q}\right) + \mathrm{Re}\!\left(\frac{\partial H}{\partial p}\overline{\frac{\partial H}{\partial q}}\right) - \frac{\partial |H|}{\partial p}\frac{\partial |H|}{\partial q}}{|H|}$$

and the middle term — the one that has no second derivative in it — is the one that carries most of the off-diagonal mass. Dropping it would have given a matrix that is not even symmetric in the right places.

Two of five #

Two eigenvalues are of order one — −0.9473 and −0.8051, both negative — and the other three are 3.9 × 10⁻⁹, which is zero at the precision the arithmetic has.

So the form is negative semi-definite of rank two, not negative definite of rank five. That is a stronger statement than the rung below’s in one direction and a weaker one in another, and both matter. Stronger: every direction the response can see is a maximum, so no combination of tolerances raises it, which is now demonstrated over the whole box rather than sampled. Weaker: there is a three-dimensional subspace of component errors that changes the response by nothing measurable, so three fifths of the tolerance budget is being spent on directions that do not exist.

The second route is the check that matters. Computing the same matrix by four full re-solves per pair — a five-point stencil on the magnitude itself, sharing no adjoint machinery, no transposed solve and no derivative identity — reproduces the two live eigenvalues to parts in ten thousand and places the three zeros about two decades higher, at 4 × 10⁻⁷. Both are zero against 0.95. The difference in where they sit is not a disagreement; it is the price of differencing a difference, and it is worth recording because it is exactly the situation in which a naive check would have concluded that the null space is not quite null.

The two directions are the ladder’s own symmetry #

The eigenvectors say what the two live directions are, and neither of them is a component.

The first is (0.385, 0.497, −0.459, 0.497, 0.385) — symmetric under reversing the ladder, with the two ends moving together and the middle moving against them. The second is (−0.372, 0.602, 0, −0.602, 0.372) — antisymmetric, with the middle element not participating at all.

That symmetry is not imposed. A doubly terminated ladder built from a symmetric prototype is symmetric under reversal, which is the property the reading that does not care which way round it is measures directly on the network. Here it appears in the statistics of its tolerances: the curvature matrix commutes with the reversal, so its eigenvectors are the reversal’s, so they are symmetric or antisymmetric and nothing in between.

The practical consequence is the one a component engineer would want. The part to tighten is not a part. Buying a one per cent middle inductor and leaving the rest at five buys nothing against the antisymmetric direction, which the middle element is not in. Tightening the two ends together buys against both.

Why two, and not one, and not five #

A ripple peak is not an accident of the response curve. A doubly terminated lossless ladder delivers exactly half its source voltage at frequencies where its input reflection is zero, and the two resistors a ladder was designed between is the essay about why that half is a structural constant rather than a fitted one.

A reflection zero at the nominal design sits on the imaginary axis, and a point on the imaginary axis is two real numbers: where it is along the axis, and how far a perturbation has pushed it off. Five component errors map onto those two numbers. A map from five dimensions to two has a three-dimensional kernel, and the kernel is the null space above.

That is an explanation, so it is owed a measurement.

Perturbing along the symmetric direction by one per cent slides the peak 1.69 per cent down in frequency and leaves its height at exactly one half — 0.500000000, to nine figures. The zero moved along the axis and stayed on it.

Perturbing along the antisymmetric direction moves the peak by 336 parts per million — a fiftieth of the other — and drops its height by 1.01 × 10⁻⁴. The zero left the axis.

The three flat directions move the peak by tens of parts per million and drop it by parts in a billion. They do neither thing, which is why the response at a fixed frequency cannot see them.

One further reading falls out of the same measurement and is worth having. Along a flat direction the response falls as the fourth power of the perturbation: doubling the error multiplies the change by sixteen rather than by four. A direction that is flat to second order here is flat to third order as well, which is what one expects of a maximum whose leading correction is the square of a quantity that is itself second order.

Every corner, rather than six hundred draws #

A concave quadratic over a box takes its minimum at a vertex, so the true worst case of a ±1 per cent tolerance box is a corner — every part at one extreme or the other — and there are thirty-two of them. Enumerating them costs thirty-two solves, which is less than a twentieth of what sampling cost.

Every corner is below nominal, which is the rank-two negative semi-definite statement made concrete. The worst is −2.352 × 10⁻⁴, at the sign pattern ++−++ — which is the sign pattern of the symmetric eigenvector, as it has to be.

Two comparisons follow and both are the kind of number that decides a specification.

The worst of six hundred random draws is −1.397 × 10⁻⁴: the sample under-reports the worst case by a factor of 1.68. That is not a criticism of the sampling; it is a property of boxes. A uniformly drawn point in five dimensions is almost never near a vertex, and increasing the sample size fixes it only logarithmically.

The corner a worst-case analysis reaches for — every part at the same end of its tolerance — is −8.090 × 10⁻⁵, which is 2.91 times less bad than the true worst. That corner is nearly the symmetric direction and is not quite it, and the difference is a factor of three. It is the same mistake as tightening one component: the extremes that matter are a pattern, and the pattern has a minus sign in the middle of it.

What this changes about a tolerance budget #

The rung below’s design rule was about order: the spread at a ripple peak is second order in the tolerance, so halving the tolerance buys a factor of four there and a factor of two between the peaks. That rule survives — the worst corner fits an exponent of 2.009 across a factor of eight in tolerance, and the naive corner fits 2.005.

What is new is about allocation. Three of the five reactances’ worth of tolerance space does nothing at the peaks, so a budget that treats the five parts as five independent contributors is buying precision in directions the response is blind to.

That does not mean three parts can be loose. Each component appears in both live directions, so loosening any one of them loosens both; the flat directions are combinations, not parts, and there is no way to buy a component whose error is guaranteed to lie in a subspace. What it does mean is that the right figure of merit is the projection of the tolerance box onto two directions, and that the number to quote is not the sum of five per-component sensitivities.

The diagonal alone is not misleading about signs. It is misleading about count: five negative numbers look like five mechanisms, and there are two.

Between the peaks, where the form is not definite at all #

Everything above is at a ripple peak, which is a stationary point. Between two peaks the gradient is not zero — it is 0.40 in relative terms — and the curvature matrix is indefinite: its eigenvalues there are −2.27, −1.38, −0.063, +0.040 and +0.963.

So between the peaks the response can be moved up as well as down, at first order, by a single part. That is the rung below’s population result arriving as a spectral one: 325 of 600 ladders were above nominal between the peaks, and none was above it at them. The two facts are the same fact, and the matrix says which is which without building a population.

What it cost, and whether the zeros are real #

Twenty-two solves for the whole matrix, and thirty-two for every corner of the box. Six hundred for the population the rung below built. The adjoint route is what makes the first number small: every derivative, and the one that is zero established that one transposed solve gives the derivative with respect to every component at once, and the mixed second derivatives come out of the same two gradients rather than needing a solve of their own. Five parameters cost 2 + 4×5 solves and would cost 2 + 4×20 for twenty, which is linear where a direct difference is quadratic.

The eigenvalues at 4 × 10⁻⁹ deserve one paragraph of suspicion, because a small number produced by subtraction is exactly what the matrix that is ill, and the answer that is not is about. Three things say they are real rather than numerical. The Jacobi rotation used to diagonalise never subtracts nearly-equal quantities, so an eigenvalue nine decades below the largest still comes back with its own sign. The second route, which shares none of that arithmetic, also puts them at zero — higher, at 4 × 10⁻⁷, but zero against 0.95 either way. And the direct test is independent of both: perturbing the ladder along a null eigenvector and re-solving gives a change of 5 × 10⁻¹⁰ in the response, which is a measurement of the actual network and not of a matrix.

The same suspicion applied to the derivative of a root is what made that rung use two routes as well. A derivative is a difference; a second derivative is a difference of differences; and this is the third place in this anchor where the check that the answer is not arithmetic has been worth more than the answer.

Where the same shape has already appeared #

A quantity that depends on fewer combinations of its parameters than it has parameters is not a curiosity of this ladder. It is what a structural constraint looks like when it is differentiated, and this collection has met it twice before without naming it.

Exact outside and wrong within is the same statement about a Thévenin equivalent: two numbers describe everything a two-terminal network does to the outside, so however many components are inside, the outside sees a map into two dimensions and everything in its kernel is invisible. Here the two numbers are the position of a reflection zero rather than a source and a resistance, and the invisibility is second order rather than exact — but the arithmetic is the same arithmetic.

The tolerance that is not on any part is the other half of it. There the quantity that matters is a ratio, so the direction in which all parts move together is the flat one and a data sheet’s absolute tolerance is the wrong number to quote. That is a one-dimensional null space arrived at by inspection; this is a three-dimensional one arrived at by diagonalising.

The general rule is worth stating because it decides where to look. Count the outputs before counting the inputs. A response at one frequency is one number; a stationary point makes its first derivative vanish; what is left is a quadratic form whose rank is set by how many independent things a perturbation can actually do to the object the response is pinned to. On this ladder the object is a reflection zero and it can do two things.

What is not in this model #

No terminations. The two resistors are excluded from the matrix, and the rung below measured why they must be treated separately: their first derivatives are ±0.5 rather than zero, so they are not at a stationary point at all and a second-order argument about them is the wrong argument. The stationarity is bought by the matching, and the parts that do the matching are exempt from the property they buy.

No third order. The flat directions are flat to second order and measured flat to third, but the quartic term is real and is what the measurement above sees at 10⁻⁹. For a tolerance of ten per cent rather than one, the fourth-order term in a flat direction is 10⁻⁴ of the response, which is no longer nothing.

And only one frequency at a time. A filter specification is about a band, and the matrix here is about a point in it. The right object is a family of matrices over the passband, and the right question is the worst corner over the whole band rather than at one peak — which is thirty-two corners times however many frequencies, and is still cheaper than sampling.

The habit this belongs to #

The rung below did the honest thing available to it: it could not demonstrate the claim, so it measured a population and said that a population is what it had. The demonstration turned out to cost a twentieth as much, and to contain a fact — three directions of nothing — that no population of any size would have shown, because the thing it is a fact about has measure zero.

Two routes to the matrix, an explanation for its rank, and a measurement that the explanation predicts: the peak slides for one direction, drops for another, and does neither for the rest.

What that leaves for a manufacturer is narrower than it sounds and is worth stating plainly. Three directions of component variation cost nothing, but a real batch of components does not vary along a chosen direction — it varies along all five at once, and the two expensive directions are the ones a random spread lands in. So the null space is a fact about the design rather than an economy available to production: it says which two elements are worth trimming and which three are not, and that is a screening decision rather than a tolerance decision.