Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Backward error, forward error and the condition number, with measured values

One function in the error library, called 16 times across 7 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 5 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 backward error, forward error and the condition number, with measured values. Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

error-triangle is one function in lib/figures/error.js — error — the backward one, the forward one, and the number between them. 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.

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H10b = A·(1, 2, …, 10)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 2.3·10⁻¹⁷the answer you wantedx = (1, 2, …, 10), exactlythe answer you gotx̂, wrong by 2.7·10⁻⁴ relativebackward error 2.3·10⁻¹⁷forward error 2.7·10⁻⁴κ = 1.6·10¹³κ · η = 3.6·10⁻⁴, and the measured forward error is 2.7·10⁻⁴.The algorithm is not at fault. The problem is.H10, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

n: 12

The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H12b = A·(1, 2, …, 12)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 1.8·10⁻¹⁷the answer you wantedx = (1, 2, …, 12), exactlythe answer you gotx̂, wrong by 0.02 relativebackward error 1.8·10⁻¹⁷forward error 0.02κ = 1.8·10¹⁶κ · η = 0.33, and the measured forward error is 0.02.The algorithm is not at fault. The problem is.H12, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

n: 13

The arguments are the ones A small residual is not a small error passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H13b = A·(1, 2, …, 13)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 2.2·10⁻¹⁷the answer you wantedx = (1, 2, …, 13), exactlythe answer you gotx̂, wrong by 0.75 relativebackward error 2.2·10⁻¹⁷forward error 0.75κ = 1.7·10¹⁸κ · η = 38, and the measured forward error is 0.75.The algorithm is not at fault. The problem is.H13, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

n: 10

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

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H10b = A·(1, 2, …, 10)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 2.3·10⁻¹⁷the answer you wantedx = (1, 2, …, 10), exactlythe answer you gotx̂, wrong by 2.7·10⁻⁴ relativebackward error 2.3·10⁻¹⁷forward error 2.7·10⁻⁴κ = 1.6·10¹³κ · η = 3.6·10⁻⁴, and the measured forward error is 2.7·10⁻⁴.The algorithm is not at fault. The problem is.H10, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

n: 14

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

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H14b = A·(1, 2, …, 14)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 5·10⁻¹⁸the answer you wantedx = (1, 2, …, 14), exactlythe answer you gotx̂, wrong by 19 relativebackward error 5·10⁻¹⁸forward error 19κ = 2.7·10¹⁷κ · η = 1.4, and the measured forward error is 19.The algorithm is not at fault. The problem is.H14, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

n: 18

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

Backward error, forward error and the condition number, with measured valuesTwo boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.the problem you posedA = H18b = A·(1, 2, …, 18)the problem it answered exactlyA + δA, b + δb‖δ‖ / ‖A‖ = 3.9·10⁻¹⁸the answer you wantedx = (1, 2, …, 18), exactlythe answer you gotx̂, wrong by 35 relativebackward error 3.9·10⁻¹⁸forward error 35κ = 3.5·10¹⁸κ · η = 13, and the measured forward error is 35.The algorithm is not at fault. The problem is.H18, LU with partial pivotingresidual and error differ

Two boxes at the top — the problem posed and the nearby problem the algorithm answered exactly — and two answers below them, with the distances between all four labelled by numbers from a Hilbert solve.

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.

5 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 the answer is wrong anyway

LU is for square matrices

the algorithm did its job

the forward error is within κ·η

the residual is nowhere near the error

Against the rule

It calls a factoriser without drawing a factorisation (solve), so the rule is written down as not applying, with the reason: the factorisation is a step towards the solution; the figure is the error identity

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 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.

Two errors, and whose fault they are

A condition number scaling cannot move

Skeel's componentwise condition number is invariant under any row scaling — exactly, before any norm is taken, because two diagonal factors cancel entry by entry. It is never larger than the normwise one and can be arbitrarily smaller, and the ratio between them is a diagnostic for which kind of ill-conditioning a matrix has.

 Two errors, and whose fault they are

A 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.

 Two errors, and whose fault they are

An 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.

 Exact arithmetic, and what it costs instead

An exact answer to a measured problem

The residual is the zero vector, nothing was rounded at any step, and the answer is wrong in its first digit. Data accurate to fourteen places, an exact solve of the system it defines, and an error of 10⁻⁵ — because conditioning was never a statement about arithmetic and removing the arithmetic error removes none of it.

 Two errors, and whose fault they are

The 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.

 Two errors, and whose fault they are

The fifth author

Four authors of a wrong answer have been named on this site and each is a statement about one computation. The fifth is not: it is what separates two computations that are both correct, it is a backward error of measurable size, and no residual, bound or condition number contains it.

 Two errors, and whose fault they are

Three errors and one number

This site's identity has two factors and a division of blame between them. Two fields have now added a third party and a fourth, and only one of the four is a property of anything — the others are decisions, made before the arithmetic, reported by nothing.

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