Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

verified-boundary

One function in the interval 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 23 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 a bound can be proved, against κu. Every combination of 5 working precisions and 6 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.45 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.

verified-boundary is one function in lib/figures/interval.js — intervals — a bound that is proved rather than measured, and what proving it costs. 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 a bound can be proved, against κuEvery combination of 5 working precisions and 6 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.45 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.10⁻⁹10⁻⁶10⁻³110³10⁶13212937κ · usignificand bitsκu = 1a bound was provedthe method refusedit never returns a wrong boundlargest κu with a proof0.45smallest κu without one0.89cases refused, of the grid13a refusal is not a wide bound — it is no bound at alland it is the only failure mode here

Every combination of 5 working precisions and 6 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.45 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.

precisions: [16, 20, 24, 32, 40], sizes: [3, 4, 5, 6, 8, 10]

The arguments are the ones A bound that is proved 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 a bound can be proved, against κuEvery combination of 5 working precisions and 6 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.45 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.10⁻⁹10⁻⁶10⁻³110³10⁶13212937κ · usignificand bitsκu = 1a bound was provedthe method refusedit never returns a wrong boundlargest κu with a proof0.45smallest κu without one0.89cases refused, of the grid13a refusal is not a wide bound — it is no bound at alland it is the only failure mode here

Every combination of 5 working precisions and 6 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.45 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.

precisions: [16, 24, 32], sizes: [4, 6, 8, 10]

The arguments are the ones Where the hardware went 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 a bound can be proved, against κuEvery combination of 3 working precisions and 4 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.24 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.10⁻⁵10⁻²10¹10⁴10⁷132129κ · usignificand bitsκu = 1a bound was provedthe method refusedit never returns a wrong boundlargest κu with a proof0.24smallest κu without one0.89cases refused, of the grid8a refusal is not a wide bound — it is no bound at alland it is the only failure mode here

Every combination of 3 working precisions and 4 Hilbert sizes, placed by the product of the condition number and the unit roundoff. Filled marks are the cases where a bound was proved; open ones are the cases the method refused. The two sets separate at κu ≈ 1 — the largest verified is 0.24 and the smallest declined is 0.89 — which is the threshold iterative refinement's convergence sits at.

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.

23 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.

the bound holds at 16 bits, n = 3 — asserted 17 times

and everything declined is above κu = 0.1

everything verified is below κu = 1

Hilbert sizes the exact answer is available at

LU is for square matrices

precisions the arithmetic can round outward at

the boundary is inside the grid

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 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.

The arithmetic underneath

A bound that is proved

Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.

 The arithmetic underneath

Where the hardware went

bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.

The whole library · All essays · What must fail