jacobi-convergence
At its defaults it draws jacobi sweeps on the 6×6 hilbert matrix. A plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
jacobi-convergence is one function in lib/figures/spectra.js —
spectra — sensitivity, the symmetric easy case, and rank as a decision. 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 plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
n: 6
The arguments are the ones A condition number for one eigenvalue 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 plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
n: 7
The arguments are the ones Two shifts that are never formed 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 plot of the off-diagonal norm against sweep number, falling from about one to ten to the minus seventeen in five sweeps, with the matrix shown at three stages beneath it.
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 3 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.
swept diagonal 0 is the eigenvalue — asserted 7 times
a 6×6 grid leaves a cell wide enough for an entry — asserted 2 times
in at most eight sweeps
Jacobi needs a symmetric matrix
matmul shapes agree
the off-diagonal norm reaches rounding
the trace is unchanged by the rotations agree
Against the rule
It calls a factoriser without drawing a factorisation
(jacobiEigSym),
so the rule is written down as not applying, with the reason:
plots the off-diagonal norm, which is the convergence measurement
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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 condition number for one eigenvalue
In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.
Iterating, instead of factorisingA rate that is known in advance
On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.
Orthogonality, measuredA reflection cannot stop being one
Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.
Eigenvalues, singular values, rankAn eigenvalue that arrives twice
A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.
Elimination, and the swapElimination is a sequence of choices
Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.
Eigenvalues, singular values, rankSymmetry is worth more than precision
A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.
Eigenvalues, singular values, rankThe algorithm the libraries actually run
Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.
Eigenvalues, singular values, rankThe best approximation there is
The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.
Elimination, and the swapThe bound that is never attained
Partial pivoting's stability guarantee permits the entries to double at every step — a factor of 5.5·10¹¹ at n = 40. The measured growth on random matrices of that size is about three. The gap is eleven orders of magnitude, and the guarantee is still worth having.
Eigenvalues, singular values, rankThe form that makes it affordable
One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.
Least squares, and the road not to takeThe road that squares the problem
The normal equations are the first method every course teaches and the method no library uses. Forming AᵀA squares the condition number, and below ε = √u it does not degrade — it produces a matrix that is exactly singular, from data that was perfectly usable.
Eigenvalues, singular values, rankTwo shifts that are never formed
The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.