Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

pivot-ties

One function in the rrqr library, called 3 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 16 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws where the same pivot rule lands on kahan's matrix, with and without a tie tolerance, at c = 0.5. Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 0, 9, 14, 6 interchanges at n = 20, 30, 40, 50 and lands 4.5·10⁶ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.

pivot-ties is one function in lib/figures/rrqr.js — the cheap rank — a greedy pivot rule, and the matrix it has nothing to choose on. 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.

Where the same pivot rule lands on Kahan's matrix, with and without a tie tolerance, at c = 0.5Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 0, 9, 14, 6 interchanges at n = 20, 30, 40, 50 and lands 4.5·10⁶ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.18283848110²10⁴10⁶10⁸size of the matrix|r_nn| ÷ σ_minequal treated as equalstrict comparisona choice between equal numbersties found, at n = 5049interchanges, tie-tolerant0interchanges, strict6ratio between the two4.5·10⁶the improvement is real and is not a repairit is the ulps, and they are not reproducible

Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 0, 9, 14, 6 interchanges at n = 20, 30, 40, 50 and lands 4.5·10⁶ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.

c: 0.5

The arguments are the ones Aggregating what the matrix calls strong 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.

Where the same pivot rule lands on Kahan's matrix, with and without a tie tolerance, at c = 0.5Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 0, 9, 14, 6 interchanges at n = 20, 30, 40, 50 and lands 4.5·10⁶ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.18283848110²10⁴10⁶10⁸size of the matrix|r_nn| ÷ σ_minequal treated as equalstrict comparisona choice between equal numbersties found, at n = 5049interchanges, tie-tolerant0interchanges, strict6ratio between the two4.5·10⁶the improvement is real and is not a repairit is the ulps, and they are not reproducible

Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 0, 9, 14, 6 interchanges at n = 20, 30, 40, 50 and lands 4.5·10⁶ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.

c: 0.35

The arguments are the ones The cheap rank and what it cannot see 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.

Where the same pivot rule lands on Kahan's matrix, with and without a tie tolerance, at c = 0.35Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 8, 11, 7, 16 interchanges at n = 20, 30, 40, 50 and lands 8.8·10⁵ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.18283848110¹10²10³10⁴10⁵10⁶10⁷size of the matrix|r_nn| ÷ σ_minequal treated as equalstrict comparisona choice between equal numbersties found, at n = 5049interchanges, tie-tolerant0interchanges, strict16ratio between the two8.8·10⁵the improvement is real and is not a repairit is the ulps, and they are not reproducible

Two curves of |r_nn| ÷ σ_min against n. Every trailing column norm of this matrix is exactly equal at every step, so a rule that interchanges on any strict improvement is comparing two rounding errors: it makes 8, 11, 7, 16 interchanges at n = 20, 30, 40, 50 and lands 8.8·10⁵ times nearer the truth at the largest size. It lands in the SAME place at the three largest sizes while the other rule's answer grows by orders of magnitude across them, which is what a floor set by rounding looks like rather than a rule that is finding something.

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.

16 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 never lands further away at n = 20 — asserted 4 times

the tie-tolerant rule makes no interchange at n = 20 — asserted 4 times

a Kahan parameter strictly inside (0, 1)

a size the decay has room in

and lands orders nearer the truth at the largest size drawn

having found a tie at nearly every step

landing at the same place at the three largest sizes

matmul shapes agree

while a strict comparison interchanges at more than one size

while the rule that treats equal numbers as equal grows across the same range

Against the rule

It draws a decomposition and prints its residual. It calls kahanVerdicts, 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 90 of 174 generators — 75 print a residual and 15 are exempt with a published reason; 84 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.

Iterating, instead of factorising

Aggregating what the matrix calls strong

The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.

 Eigenvalues, singular values, rank

The cheap rank and what it cannot see

Almost nobody computes singular values to decide a rank. The standard substitute is QR with column pivoting, read off the diagonal of R — and there is a triangular matrix on which the greedy rule makes no interchange at all, has no better column available at any step, and reports a matrix eight orders of magnitude further from singular than it is.

The whole library · All essays · What must fail