Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

forward-predicted

One function in the hsolve library, called 6 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 9 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 the forward error of a hierarchical solve, and κ times the backward error that was chosen for it. Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

forward-predicted is one function in lib/figures/hsolve.js — solving with it — the recursion that assembles nothing, and the backward error that was chosen. 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.

The forward error of a hierarchical solve, and κ times the backward error that was chosen for itForward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹‖A − A_H‖ ⁄ ‖A‖, chosen‖x − x*‖ ⁄ ‖x*‖κ × the chosen backward errorthe error in the answerboth factors known firstκ24chosen at 10⁻⁸1.2·10⁻⁹predicted forward3·10⁻⁸measured forward1.6·10⁻⁹bound ⁄ measured19the amplifier, used forwardsfor once

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

n: 256

The arguments are the ones An accuracy that is a backward error 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.

The forward error of a hierarchical solve, and κ times the backward error that was chosen for itForward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹‖A − A_H‖ ⁄ ‖A‖, chosen‖x − x*‖ ⁄ ‖x*‖κ × the chosen backward errorthe error in the answerboth factors known firstκ24chosen at 10⁻⁸1.2·10⁻⁹predicted forward3·10⁻⁸measured forward1.6·10⁻⁹bound ⁄ measured19the amplifier, used forwardsfor once

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 24.39 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.981 and sits 14× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

n: 128

The arguments are the ones An accuracy that is a backward error 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.

The forward error of a hierarchical solve, and κ times the backward error that was chosen for itForward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 20.88 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.971 and sits 16× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹‖A − A_H‖ ⁄ ‖A‖, chosen‖x − x*‖ ⁄ ‖x*‖κ × the chosen backward errorthe error in the answerboth factors known firstκ21chosen at 10⁻⁸1.4·10⁻⁹predicted forward2.9·10⁻⁸measured forward1.6·10⁻⁹bound ⁄ measured26the amplifier, used forwardsfor once

Forward error ⪅ condition number × backward error is the identity this collection is organised around, and it is normally used after the fact: the algorithm ran, somebody measured what it did, and the condition number explains the difference. Here both factors are known in advance — κ = 20.88 is a property of the problem, and the backward error was set at the top of the program — so the upper line is a prediction rather than an account. The measured error tracks it at a slope of 0.971 and sits 16× below it throughout, which is the usual looseness of a worst-case bound applied to one right-hand side and not a failure of the prediction. What the figure licenses is the sentence a code needs: decide how many digits the answer requires, divide by κ, and compress to that.

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.

9 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.

a size the dense reference is affordable at

and it tracks the chosen backward error

enough accuracies above the floor to fit anything

the forward error is inside the amplifier's prediction at ε = 0.01

the forward error is inside the amplifier's prediction at ε = 10⁻¹⁰

the forward error is inside the amplifier's prediction at ε = 10⁻¹²

the forward error is inside the amplifier's prediction at ε = 10⁻⁴

the forward error is inside the amplifier's prediction at ε = 10⁻⁶

the forward error is inside the amplifier's prediction at ε = 10⁻⁸

Against the rule

It draws a decomposition and prints its residual. It calls accuracySweep, 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 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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

An accuracy that is a backward error

Every backward error on this site is something an algorithm produced and somebody then measured. This one is a line in the program. Solving with a compressed matrix gives a residual that is the compression's own error, at a slope of 1.000 over ten decades, so the knob that sets the storage sets the backward error directly.

 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.

 Two errors, and whose fault they are

The 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.

 Structure, and the solver that cannot see it

The condition number of the model

Describe a 40×40 Toeplitz matrix by its 1,600 entries and its condition number is 78,800. Describe it by the one number it actually contains and the condition number is 61.9. The three decades in between are not an approximation or a bound — they are what κ has been over-stating, and the drop is not where the linear algebra is.

 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.

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