wilkinson-growth
At its defaults it draws wilkinson's 7×7 matrix and its upper triangular factor. The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to sixty-four.
wilkinson-growth is one function in lib/figures/elim.js —
elimination — the swap, the growth factor, and the matrix with a known answer. 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 matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to sixty-four.
n: 7
The arguments are the ones Elimination is a sequence of choices 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.
The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to sixty-four.
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.
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.
and all of the growth is in the final column agree
and the growth is exactly the bound agree
LU is for square matrices
matmul shapes agree
partial pivoting swaps no row of Wilkinson's matrix
the factorisation is still exact to rounding
Against the rule
It draws a decomposition and prints its residual. It calls
luFactor,
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.
Elimination 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.
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.
Where the flop count stopped predicting the timeThe same arithmetic at a different price
A blocked and an unblocked elimination perform 72,568 operations each — the same operations, associated differently — choose the same pivots, and return a factorisation identical to the last bit: ‖PA − LU‖/‖A‖ = 4.487946226420872·10⁻¹⁶ in both. One of them moves 41,332 words between fast and slow memory and the other moves 19,476.