Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

An exact solve of a measured 8 × 8 system, against the error the measurement put there

One function in the integer library, called 20 times across 6 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 an exact solve of a measured 8 × 8 system, against the error the measurement put there. The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 1.342, which is 1.3·10⁶ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

inexact-data is one function in lib/figures/integer.js — exact arithmetic — no residual to print, and a cost measured in the length of the numbers. 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 1.342, which is 1.3·10⁶ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻⁷10⁻⁵10⁻³10⁻¹10¹relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻⁶answer error1.3amplification1.3·10⁶rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 1.342, which is 1.3·10⁶ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

family: "hilbert", logNoise: -12

The arguments are the ones A fraction recovered from one remainder 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-12 the answer is wrong by 9.988·10⁻⁴, which is 10·10⁸ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻⁷10⁻⁵10⁻³10⁻¹10¹relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻¹²answer error10·10⁻⁴amplification10·10⁸rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-12 the answer is wrong by 9.988·10⁻⁴, which is 10·10⁸ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

family: "random", logNoise: -8

The arguments are the ones An answer with no error in it 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-8 the answer is wrong by 2.049·10⁻⁸, which is 2.05 times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻⁸answer error2·10⁻⁸amplification2rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-8 the answer is wrong by 2.049·10⁻⁸, which is 2.05 times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

family: "random", logNoise: -6

The arguments are the ones An exact answer to a measured problem 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 2.049·10⁻⁶, which is 2.05 times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻⁶answer error2·10⁻⁶amplification2rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-6 the answer is wrong by 2.049·10⁻⁶, which is 2.05 times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

family: "hilbert", logNoise: -14

The arguments are the ones An exact answer to a measured problem 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-14 the answer is wrong by 1.002·10⁻⁵, which is 10⁹ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻⁷10⁻⁵10⁻³10⁻¹10¹relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻¹⁴answer error10⁻⁵amplification10⁹rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-14 the answer is wrong by 1.002·10⁻⁵, which is 10⁹ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

family: "hilbert", logNoise: -8

The arguments are the ones An exact answer to a measured problem 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.

An exact solve of a measured 8 × 8 system, against the error the measurement put thereThe rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-8 the answer is wrong by 0.2705, which is 2.7·10⁷ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻⁷10⁻⁵10⁻³10⁻¹10¹relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻⁸answer error0.27amplification2.7·10⁷rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

The rising curve is the distance from the exact solution of a perturbed system to the exact solution of the unperturbed one, as the perturbation grows from a relative 10⁻¹⁶ to 1. Every point on it was computed with no rounding anywhere: the elimination is over the rationals, the residual of each solve is the zero vector, and the error is entirely the data's. At a measurement accurate to 10^-8 the answer is wrong by 0.2705, which is 2.7·10⁷ times the data's own error — the matrix's condition number, doing the only job left for it to do. The straight line of slope one is the whole finding: removing the arithmetic error changes nothing about this quantity, because this quantity was never about the arithmetic.

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.

7 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.

a family with a known conditioning

a relative error the data was measured to

a system size

each solution entry is the ratio of two determinants, as Cramer's rule says

every Bareiss division is exact

the answer's error is at least the data's own, so exactness has not removed it

the assertion refuses a counterexample

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 214 of 397 generators — 194 print a residual and 20 are exempt with a published reason; 183 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.

Exact arithmetic, and what it costs instead

A fraction recovered from one remainder

A solution over the rationals can be computed modulo a prime power and then recovered — the residue determines the fraction uniquely, but only once the modulus is twice the square of the fraction's longer part. Below that there is no partial credit: the algorithm returns a different fraction with the same residue, and it is a perfectly good one.

 Exact arithmetic, and what it costs instead

An answer with no error in it

An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.

 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.

 Exact arithmetic, and what it costs instead

Every intermediate is a minor

Fraction-free elimination divides by the previous pivot at every step and the division is always exact. Not usually, not for these entries — always, because the number being divided is a determinant with that pivot as a factor, which is a theorem and is checked here against the minors themselves.

 Exact arithmetic, and what it costs instead

How many primes the answer needs

Work modulo a word-sized prime and no intermediate can exceed twenty-six bits, whatever the matrix does. The catch is that the answer must be reassembled from several such computations, and the number of them has to be fixed before the first one runs — by a theorem about how large a determinant can be, not by trying more until it settles.

 Exact arithmetic, and what it costs instead

The answer is longer than the question

An exact solution of an integer system is a vector of fractions, each of them a ratio of two determinants. So the output carries 2n long integers where the input carried n² short ones, and no algorithm can write it down more cheaply — the length of the answer is a floor under every exact solver rather than a property of one.

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