A prediction from three numbers, exact at one end of the spectrum and wrong by n² at the other
At its defaults it draws a prediction from three numbers, exact at one end of the spectrum and wrong by n² at the other. The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
tropical-prediction is one function in lib/figures/qepscale.js —
the units a polynomial is written in — two lines of scaling, a prediction that is half right, and a format's edge. 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.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
alpha: 6, beta: 0.5
The arguments are the ones An estimate that does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.5K. The large root is within 5.7 per cent of the largest modulus at every size. The small root is out by 5.36, 17.1, 60.8 and 229.6 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
alpha: 6, beta: 0.2
The arguments are the ones An estimate that does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 0.2K. The large root is within 4.4 per cent of the largest modulus at every size. The small root is out by 5.76, 18.6, 66.7 and 252 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
alpha: 100, beta: 1
The arguments are the ones An estimate that does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 100M + 1K. The large root is within 1.9 per cent of the largest modulus at every size. The small root is out by 6.04, 19.5, 69.8 and 263.8 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
alpha: 6, beta: 4
The arguments are the ones An estimate that does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 6M + 4K. The large root is within 31 per cent of the largest modulus at every size. The small root is out by 3.09, 8.59, 29 and 107.8 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
alpha: 1, beta: 0.5
The arguments are the ones An estimate that does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The tropical roots of the max-plus quadratic with coefficients ‖M‖, ‖C‖ and ‖K‖, divided by the extreme moduli they are supposed to estimate, for chains of 4 to 32 masses with C = 1M + 0.5K. The large root is within 10 per cent of the largest modulus at every size. The small root is out by 1.81, 8.78, 33.1 and 126.5 — a factor growing like the square of the size. The reason is structural: a norm is a maximum, the small end of this spectrum is set by the SMALLEST eigenvalue of K, which is 4sin²(π/2(n+1)), and no norm of K contains that number. A quantity built out of maxima is exact where the answer is a maximum and silent where it is a minimum.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
19 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 chain long enough to have a spectrum and short enough to draw
a damping the eigenvalue picture is drawn at
a damping the sweep runs
a family of cubic this file builds
a family the dial draws
a group of eigenvalues
a linearisation has as many eigenvalues as it has rows
a positive mass
a quadratic with both outer coefficients present
a reduction this file knows
a separation the sweep runs
and it stays within a factor of two and a half of the largest modulus
and the small one drifts
damping that removes energy rather than adding it
damping the tropical roots can be taken of
LU is for square matrices
matmul shapes agree
the large tropical root's ratio does not move with the size
two tropical corners, so two groups to separate
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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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.
An estimate that does not move
The tropical roots are said to miss the bottom of a spectrum by a factor growing like n². The estimate does not get worse. It changes by four per cent between four masses and sixty-four while the modulus it names falls by a factor of a hundred and sixty, and an estimate that is constant in the variable the answer depends on is a different defect from an inaccurate one.
The eigenvalue problem that is not linearThe middle group needs no reduction of its own
A quadratic whose two groups of eigenvalues sit far apart needs two reductions, one per end. A cubic can have three groups, and the middle one is inverted by neither reduction, so the worry was that it would need a third route — a shift-and-invert at the middle tropical root. Build order-four cubics with three groups up to eight decades apart and measure every eigenvalue: the leading reduction keeps the middle group at 10⁻¹¹ or better at every separation, the trailing reduction loses it to 5·10⁻⁹ by eight decades, and the reversed polynomial's leading reduction, which inverts the same constant coefficient as the trailing one, keeps it at 2·10⁻¹⁴. The middle group's loss belongs to the pencil, not to the end. Two reductions still give all twelve eigenvalues to 2·10⁻¹¹. And the tropical scaling at the small root, which rescues the small group when every coefficient shares one diagonal form, does nothing for it once the coefficients are perturbed out of that form.
The eigenvalue problem that is not linearThe 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 eigenvalue problem that is not linearTwo groups need two reductions
When a quadratic eigenproblem's tropical roots are far apart its eigenvalues fall into two groups, and the remedy offered is a second scaling — one per tropical root, each run keeping the group its root predicts. Raise the damping on an eight-mass chain until the roots are 10²³ apart and measure every eigenvalue: no scaling rescues the small group. Its backward error grows as the unit roundoff times the separation under Fan–Lin–Van Dooren's scaling, under both tropical scalings and under none, to 10⁻⁵ at a separation of 4·10¹¹. What rescues it is the other reduction of the same pencil — inverting the constant term instead of the leading one — which keeps the small group at rounding at every separation and loses the large one instead. Both reductions, each keeping its own group, give all sixteen eigenvalues to 10⁻¹¹ at worst. On the way, the closed form that served as the exact answer turned out to lose the small roots to cancellation, from the same separation.