Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Relative error of three summation algorithms in binary32

One function in the arith library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 3 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 relative error of three summation algorithms in binary32. A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

summation-error is one function in lib/figures/arith.js — arithmetic — what a float holds, and what it loses holding it. 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.

Relative error of three summation algorithms in binary32A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensatedbinary32 · terms are 1/icompensated: 3·10⁻⁸

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

bits: 24

The arguments are the ones A problem with no answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Relative error of three summation algorithms in binary32A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensatedbinary32 · terms are 1/icompensated: 3·10⁻⁸

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

bits: 16

The arguments are the ones The order they are added in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Relative error of three summation algorithms in 16-bitA log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensated16-bit · terms are 1/icompensated: 8.1·10⁻⁶

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

bits: 22

The arguments are the ones The order they are added in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Relative error of three summation algorithms in 22-bitA log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensated22-bit · terms are 1/icompensated: 2.1·10⁻⁷

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

bits: 31

The arguments are the ones The order they are added in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Relative error of three summation algorithms in 31-bitA log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensated31-bit · terms are 1/icompensated: 2.8·10⁻¹⁰

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

bits: 40

The arguments are the ones The order they are added in passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Relative error of three summation algorithms in 40-bitA log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensated40-bit · terms are 1/icompensated: 6·10⁻¹³

A log–log plot of relative error against the number of terms for naive, pairwise and compensated summation, each measured against the exactly rounded sum.

What it checked while drawing

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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.

3 distinct claims across 6 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 compensation beats them both

and the naive error grows with the count, which is the thing being shown

the tree beats the line at a million terms

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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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.

Eigenvalues, singular values, rank

A problem with no answer

If two matrices share a null vector then det(A − λB) is identically zero and every λ is an eigenvalue, which means none of them is. Perturb such a pencil by a ten-billionth and a solver returns six numbers with residuals below 10⁻⁹. Change the seed and it returns six different numbers, spread over forty-four, with residuals just as small.

 The arithmetic underneath

The 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.

The whole library · All essays · What must fail