growth-factor
At its defaults it draws growth factor under partial pivoting to n = 40: the bound, the worst case, and reality. Growth factor against matrix size on a logarithmic vertical axis. The two-to-the-n bound rises as a straight line; Wilkinson's matrix sits exactly on it; random matrices stay near one.
growth-factor 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.
Growth factor against matrix size on a logarithmic vertical axis. The two-to-the-n bound rises as a straight line; Wilkinson's matrix sits exactly on it; random matrices stay near one.
nMax: 40
The arguments are the ones A reflection cannot stop being one 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.
Growth factor against matrix size on a logarithmic vertical axis. The two-to-the-n bound rises as a straight line; Wilkinson's matrix sits exactly on it; random matrices stay near one.
nMax: 24
The arguments are the ones The same arithmetic at a different price 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.
Growth factor against matrix size on a logarithmic vertical axis. The two-to-the-n bound rises as a straight line; Wilkinson's matrix sits exactly on it; random matrices stay near one.
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.
32 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.
and no random matrix exceeds it at n = 4 — asserted 13 times
Wilkinson's matrix attains the bound at n = 4 agree — asserted 13 times
the worst random growth at n = 40 is still single digits — asserted 2 times
and it is pivoting doing it — without the swaps the growth is far larger
and the gap between bound and reality grows with n — 4.9 at n = 4, 1.1·10¹¹ at n = 40
and the gap between bound and reality grows with n — 4.9 at n = 4, 2.3·10⁶ at n = 24
LU is for square matrices
Against the rule
It calls a factoriser without drawing a factorisation
(luFactor),
so the rule is written down as not applying, with the reason:
plots the growth factor, which is a property of U and is itself the 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 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.
Two errors, and whose fault they areA small residual is not a small error
Substituting the answer back and finding that it fits is the most natural check there is, and it verifies the wrong thing. A residual of 10⁻¹⁷ is entirely compatible with an answer whose second digit is wrong.
Sparsity, and what elimination costsA 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.
The arithmetic underneathCancellation takes the answer, not a digit
Subtracting two nearly equal numbers is exact. That is what makes it dangerous — the subtraction introduces no error at all, it exposes error the operands were already carrying, and the exposure can consume every significant figure at once.
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.
Orthogonality, measuredOrthogonal is a number
"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.
Sparsity, and what elimination costsStructure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
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 condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
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.
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.
Orthogonality, measuredTwo Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.