The projected matrix after 3 cycles, keeping 4 vectors
At its defaults it draws the projected matrix after 3 cycles, keeping 4 vectors. A 8×8 matrix drawn as a grid. The first 4 diagonal entries are the retained Ritz values; the row and column at index 4 are their couplings to the residual vector; the remaining 4×4 block is an ordinary Lanczos tridiagonal. 22 entries are above rounding out of 64.
arrowhead-projection is one function in lib/figures/thick.js —
keeping the vectors — a third of the products, and the bound that stops bounding. 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 8×8 matrix drawn as a grid. The first 4 diagonal entries are the retained Ritz values; the row and column at index 4 are their couplings to the residual vector; the remaining 4×4 block is an ordinary Lanczos tridiagonal. 22 entries are above rounding out of 64.
k: 4
The arguments are the ones Keeping the vectors, and losing the bound passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A 8×8 matrix drawn as a grid. The first 4 diagonal entries are the retained Ritz values; the row and column at index 4 are their couplings to the residual vector; the remaining 4×4 block is an ordinary Lanczos tridiagonal. 22 entries are above rounding out of 64.
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.
11 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.
nothing outside the diagonal and the border in row 0 — checked 4 times
a cycle at which something has been retained
a number of retained vectors the picture has room for
a projected matrix small enough to read
enough new steps a cycle for a tail to exist
Jacobi needs a symmetric matrix
matmul shapes agree
the projected matrix is an arrowhead with a tridiagonal tail
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.
Keeping the vectors, and losing the bound
Thick restarting keeps the Ritz vectors instead of filtering the starting vector — the same eigenvalues for a third of the products with A. Its residual bound reaches 9.4·10⁻⁴¹ while the residual it bounds sits at 5.7·10⁻⁵, and the eigenvalues are correct to 4.3·10⁻¹⁴ the whole time, so nothing reports it.
Sparsity, and what elimination costsTwo ends of the same arrow
One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.