elimination-steps
At its defaults it draws gaussian elimination on a 4×4, one step at a time. Four copies of the same matrix: as given, and after each of the three elimination steps. The pivot in use is outlined and the entries reduced to zero are greyed.
elimination-steps 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.
Four copies of the same matrix: as given, and after each of the three elimination steps. The pivot in use is outlined and the entries reduced to zero are greyed.
A: [2,1,-1,3, -3,-1,2,1, -2,1,2,-4, 4,3,-1,2]
The arguments are the ones A threshold between fill and growth 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.
Four copies of the same matrix: as given, and after each of the three elimination steps. The pivot in use is outlined and the entries reduced to zero are greyed.
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.
21 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.
the last frame is the U that luFactor returns (0,0) — asserted 16 times
and no multiplier exceeds one, which is what pivoting buys
LU is for square matrices
matmul shapes agree
PA = LU to rounding
the determinant by pivots and by cofactors agree
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.
A threshold between fill and growth
One number decides how small a pivot an elimination will accept. At 0.001 the factor holds 172 entries and the matrix grows by 1,330; at 1 it holds 260 and grows by 1.2. The libraries ship 0.1, and the measurement says why.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
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.
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.
Two errors, and whose fault they areThe exact answer to a nearby problem
A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.
Sparsity, and what elimination costsThe factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.
Sparsity, and what elimination costsThe order decides the memory
Four elimination orderings on one matrix give factors of 1,739, 1,354, 1,413 and 1,026 entries. All four factorisations are exact, all four return the same answer, and the one with the better asymptotics is not the one that wins.
The arithmetic underneathThe order they are added in
Addition is associative in the algebra and is not associative in the arithmetic. The same million numbers, added in a different order, give answers that differ in the third significant figure — and the fix is not a wider float, it is a different order.
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.
Elimination, and the swapThe swap that is not optional
Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.
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.