‖e^tA‖₂ for a 10×10 matrix with every eigenvalue at −1 and 4 above the diagonal
At its defaults it draws ‖e^ta‖₂ for a 10×10 matrix with every eigenvalue at −1 and 4 above the diagonal. The norm rises to 3.68·10⁴ at t = 9.00 — which is (n − 1)/|λ| = 9.0 — before turning over and decaying to 1.8·10⁻¹⁸. The spectral prediction e^λt, drawn beneath it, is 1.2·10⁻⁴ at the peak: wrong by a factor of 3·10⁸. Every value is from the closed form, so the hump is a property of the matrix and not of a method.
exp-hump is one function in lib/figures/funm.js —
matrix functions — the definition that is not a method, and the vector that was wanted. 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 norm rises to 3.68·10⁴ at t = 9.00 — which is (n − 1)/|λ| = 9.0 — before turning over and decaying to 1.8·10⁻¹⁸. The spectral prediction e^λt, drawn beneath it, is 1.2·10⁻⁴ at the peak: wrong by a factor of 3·10⁸. Every value is from the closed form, so the hump is a property of the matrix and not of a method.
mu: 4
The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The norm rises to 3.68·10⁴ at t = 9.00 — which is (n − 1)/|λ| = 9.0 — before turning over and decaying to 1.8·10⁻¹⁸. The spectral prediction e^λt, drawn beneath it, is 1.2·10⁻⁴ at the peak: wrong by a factor of 3·10⁸. Every value is from the closed form, so the hump is a property of the matrix and not of a method.
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.
6 distinct claims across 2 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 closed form's factorials stay representable at
a superdiagonal between none and the largest drawn
and it comes back below one, as the spectrum says it must
and the spectral prediction is wrong there by orders of magnitude
at the time the closed form puts the peak, (n − 1)/|λ|
the norm rises before it decays
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 spectral radius that grows first
ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.
Iterating, instead of factorisingThe spectrum that predicts nothing
For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.