hessenberg-reduction
At its defaults it draws householder reduction to hessenberg form, on a symmetric 6×6. Two matrices side by side. The first is full; the second has zeros everywhere below the first subdiagonal, and being symmetric is tridiagonal.
hessenberg-reduction is one function in lib/figures/qralg.js —
the qr algorithm — hessenberg, the shift, and what the libraries actually run. 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 matrices side by side. The first is full; the second has zeros everywhere below the first subdiagonal, and being symmetric is tridiagonal.
n: 6
The arguments are the ones The algorithm the libraries actually run 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.
Two matrices side by side. The first is full; the second has zeros everywhere below the first subdiagonal, and being symmetric is tridiagonal.
n: 7
The arguments are the ones The form a real matrix can reach 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.
Two matrices side by side. The first is full; the second has zeros everywhere below the first subdiagonal, and being symmetric is tridiagonal.
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.
9 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.
two 6×6 grids leave enough width for a two-figure entry — asserted 2 times
and a symmetric matrix came out tridiagonal
and every eigenvalue survived the reduction
and QHQᵀ is the matrix it reduced
Jacobi needs a symmetric matrix
matmul shapes agree
the matrix is large enough for a subdiagonal to be a structure
the reduction produced Hessenberg form
Against the rule
It draws a decomposition and prints its residual. It calls
jacobiEigSym, hessenberg,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
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.
The 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 form a real matrix can reach
A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.
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.
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.
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.