Generator

pairing-use

One function in the palindromic library, called 15 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 14 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws what a reciprocal pairing is worth: three ways to the small half of a palindromic spectrum. A palindromic quadratic of size 6 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 5.52·10⁻¹⁵ at every stop and the small ones to 1.041·10⁻⁷ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.89·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 6.538·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

pairing-use is one function in lib/figures/palindromic.js — reciprocal pairs — a symmetry the solver discards, and the division that is worth seven orders. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

A palindromic quadratic of size 6 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 5.52·10⁻¹⁵ at every stop and the small ones to 1.041·10⁻⁷ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.89·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 6.538·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

`n: 6`

The arguments are the ones A perturbation that keeps the symmetry passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

`n: 3`

A palindromic quadratic of size 3 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 2.54·10⁻¹⁵ at every stop and the small ones to 3.242·10⁻¹³ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 2.62·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 5.871·10⁻⁷, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

`n: 10`

A palindromic quadratic of size 10 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 7.31·10⁻¹⁵ at every stop and the small ones to 7.925 at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.56·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 2.468·10⁻⁵, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

`n: 8`

A palindromic quadratic of size 8 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 4.94·10⁻¹⁵ at every stop and the small ones to 4.533·10⁻⁴ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 2.82·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 9.185·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

`n: 4`

The arguments are the ones A spectrum that comes in reciprocal pairs passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

A palindromic quadratic of size 4 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 3.46·10⁻¹⁵ at every stop and the small ones to 1.626·10⁻¹¹ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 1.56·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 3.473·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

14 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

a Cholesky factor of the second matrix, which requires it to be positive definite

a coupling for which both Cayley blocks are definite

a linearisation has as many eigenvalues as it has rows

a real reciprocal pair

a reduction this file knows

a size the closed form can be read at

a z that the Cayley map does not send to infinity

a z² that a real square root can be taken of

and the small half is behind it by a factor the division recovers

Jacobi needs a symmetric matrix

LU is for square matrices

matmul shapes agree

no eigenvalue at zero, which a palindromic problem cannot have

the large half is exact at every spread

Against the rule

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Structure, and the solver that cannot see it

A perturbation that keeps the symmetry

The smallest perturbation that makes a computed answer exact is the backward error. Ask for the smallest one that also keeps the problem's structure and the number can only go up — and measured on a palindromic quadratic it goes up by 1.17, while the structure the computed spectrum has lost is not in either number.

The eigenvalue problem that is not linear

A spectrum that comes in reciprocal pairs

A palindromic quadratic reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. A general solver discards that, computes the large half of the spectrum perfectly and the small half to seven digits — and the small half is a division away from being perfect too.

The eigenvalue problem that is not linear

Every eigenvalue real, and a test that says so

A quadratic eigenvalue problem has no reason to have real eigenvalues. One class does, as a property rather than an outcome, and the proof is a Cholesky that completes. The boundary of the class has a closed form, and at the boundary the arithmetic loses half its digits with nothing ill conditioned anywhere.

The eigenvalue problem that is not linear

Six routes to one spectrum

Three linearisations of one quadratic, each reduced to a standard eigenvalue problem two ways. All six have exactly the same eigenvalues in exact arithmetic. On a well-scaled problem they differ by noise; on a badly scaled one by a factor of forty; and two of the six are the same matrix.

The eigenvalue problem that is not linear

The scaling that buys ten orders

Two lines computed from three norms, a change of variable that is exact in both directions, and the whole of the loss the previous essay measured comes back — flat, at every stop, because after scaling every stop is the same problem.

The whole library · All essays · What must fail