The running estimate of a diagonal 40×40 matrix's trace, from the two probe distributions
At its defaults it draws the running estimate of a diagonal 40×40 matrix's trace, from the two probe distributions. Two curves of the running average against the number of probes. The ±1 probe returns 98.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 177.66 and is still 0.029 away after 60 of them.
diagonal-exact is one function in lib/figures/trace.js —
counting the diagonal without looking — two probes, and one of them is free. 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.
Two curves of the running average against the number of probes. The ±1 probe returns 98.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 177.66 and is still 0.029 away after 60 of them.
n: 40
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of the running average against the number of probes. The ±1 probe returns 98.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 177.66 and is still 0.029 away after 60 of them.
n: 8
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of the running average against the number of probes. The ±1 probe returns 18.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 14.259 and is still 0.099 away after 60 of them.
n: 20
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of the running average against the number of probes. The ±1 probe returns 48.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 96.927 and is still 0.051 away after 60 of them.
n: 80
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of the running average against the number of probes. The ±1 probe returns 198.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 236.16 and is still 0.0057 away after 60 of them.
n: 200
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of the running average against the number of probes. The ±1 probe returns 498.5 — the exact trace — from its first draw and never moves, because zᵀAz is Σ aᵢᵢ zᵢ² and every zᵢ² is 1. The normal probe starts at 620.39 and is still 0.0086 away after 60 of them.
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.
9 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 size the sweep can afford
and the first normal one is not
and the same spectrum in a general basis is not free either
enough probes for the running average to settle
every ±1 running average is the exact trace
matmul shapes agree
the ±1 probe's variance on a diagonal matrix is exactly zero
though its trace is the same
while the normal probe's is not
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.
A rule that reads only its own probes
A trace estimator is a mean of independent samples, so its own standard error is estimable from the samples and a stopping rule needs nothing the estimator does not already have. Over forty draws it is calibrated in the middle and not at the edge: at a target relative standard error of 1% the median error reached is 5.3·10⁻³ and the worst of forty is 2.9·10⁻² — three times the target. And the cost of the target is the estimator's own square root: tightening it from 3% to 1% takes the median probe count from 75 to 696.
Randomised, and the guarantee that changes kindCounting what cannot be looked at
The trace is n additions and one of the most expensive quantities in the subject to estimate, because the matrices whose trace is wanted are never stored. Hutchinson's estimator is unbiased with one line of algebra — and its variance depends on which random vector is used, by a factor that is a property of the matrix, and on a diagonal matrix one choice is exact from the first probe and the other is not.