Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

linearisation-choice

One function in the qepbe library, called 15 times across 4 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 13 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 six ways of computing one spectrum, at a change of units of 106. Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 106, the best is first/leading at 5.473·10⁻⁶ and the worst second/trailing at 2.234·10⁻⁴, a spread of 40.81. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

linearisation-choice is one function in lib/figures/qepbe.js — the price of linearising — a residual that stays flat while the answer loses eleven orders. 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.

Six ways of computing one spectrum, at a change of units of 106Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 106, the best is first/leading at 5.473·10⁻⁶ and the worst second/trailing at 2.234·10⁻⁴, a spread of 40.81. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading5.47·10⁻⁶first · trailing4.62·10⁻⁵second · leading4.27·10⁻⁵second · trailing2.23·10⁻⁴symmetric · leading5.47·10⁻⁶symmetric · trailing4.05·10⁻⁵all six are the same algebrabest route5.5·10⁻⁶worst route2.2·10⁻⁴spread across the six41condition of the linearisation8.3·10¹²the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 106, the best is first/leading at 5.473·10⁻⁶ and the worst second/trailing at 2.234·10⁻⁴, a spread of 40.81. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

logGamma: 6

The arguments are the ones A backward-stable answer to a problem nobody asked 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.

Six ways of computing one spectrum, at a change of units of 106Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 106, the best is first/leading at 5.473·10⁻⁶ and the worst second/trailing at 2.234·10⁻⁴, a spread of 40.81. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading5.47·10⁻⁶first · trailing4.62·10⁻⁵second · leading4.27·10⁻⁵second · trailing2.23·10⁻⁴symmetric · leading5.47·10⁻⁶symmetric · trailing4.05·10⁻⁵all six are the same algebrabest route5.5·10⁻⁶worst route2.2·10⁻⁴spread across the six41condition of the linearisation8.3·10¹²the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 106, the best is first/leading at 5.473·10⁻⁶ and the worst second/trailing at 2.234·10⁻⁴, a spread of 40.81. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

logGamma: 0

The arguments are the ones A backward-stable answer to a problem nobody asked 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.

Six ways of computing one spectrum, at a change of units of 100Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 100, the best is first/trailing at 4.383·10⁻¹⁵ and the worst first/leading at 7.489·10⁻¹⁴, a spread of 17.08. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading7.49·10⁻¹⁴first · trailing4.38·10⁻¹⁵second · leading2.28·10⁻¹⁴second · trailing8.86·10⁻¹⁵symmetric · leading7.49·10⁻¹⁴symmetric · trailing4.71·10⁻¹⁴all six are the same algebrabest route4.4·10⁻¹⁵worst route7.5·10⁻¹⁴spread across the six17condition of the linearisation452the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 100, the best is first/trailing at 4.383·10⁻¹⁵ and the worst first/leading at 7.489·10⁻¹⁴, a spread of 17.08. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

logGamma: 4

The arguments are the ones A backward-stable answer to a problem nobody asked 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.

Six ways of computing one spectrum, at a change of units of 104Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 104, the best is first/trailing at 2.638·10⁻⁹ and the worst second/trailing at 1.549·10⁻⁸, a spread of 5.872. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading8.57·10⁻⁹first · trailing2.64·10⁻⁹second · leading8.61·10⁻⁹second · trailing1.55·10⁻⁸symmetric · leading8.57·10⁻⁹symmetric · trailing5.87·10⁻⁹all six are the same algebrabest route2.6·10⁻⁹worst route1.5·10⁻⁸spread across the six5.9condition of the linearisation8.3·10⁸the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 104, the best is first/trailing at 2.638·10⁻⁹ and the worst second/trailing at 1.549·10⁻⁸, a spread of 5.872. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

logGamma: 2

The arguments are the ones A matrix that depends on its own eigenvalue 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.

Six ways of computing one spectrum, at a change of units of 102Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 102, the best is first/trailing at 4.159·10⁻¹³ and the worst first/leading at 5.18·10⁻¹², a spread of 12.46. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading5.18·10⁻¹²first · trailing4.16·10⁻¹³second · leading1.2·10⁻¹²second · trailing6.2·10⁻¹³symmetric · leading5.18·10⁻¹²symmetric · trailing7.12·10⁻¹³all six are the same algebrabest route4.2·10⁻¹³worst route5.2·10⁻¹²spread across the six12condition of the linearisation8.3·10⁴the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 102, the best is first/trailing at 4.159·10⁻¹³ and the worst first/leading at 5.18·10⁻¹², a spread of 12.46. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

logGamma: 8

The arguments are the ones Six routes to one spectrum 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.

Six ways of computing one spectrum, at a change of units of 108Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 108, the best is first/leading at 0.001258 and the worst second/trailing at 1.545, a spread of 1229. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.first · leading0.00126first · trailing0.491second · leading0.813second · trailing1.55symmetric · leading0.00126symmetric · trailing0.402all six are the same algebrabest route0.0013worst route1.5spread across the six1229condition of the linearisation8.3·10¹⁶the spectra agreeand the arithmetic does not

Three linearisations — the first companion form, the second, and the symmetric member of the DL(Q) family — each reduced to a standard eigenvalue problem in two ways: by inverting the leading coefficient, whose eigenvalues are λ, and by inverting the trailing one, whose eigenvalues are 1/λ. All six have exactly the eigenvalues of the quadratic in exact arithmetic. Measured against the closed form at γ = 108, the best is first/leading at 0.001258 and the worst second/trailing at 1.545, a spread of 1229. Under the leading reduction the symmetric form and the first companion form are THE SAME MATRIX — −A₁⁻¹A₀ is [[−M⁻¹C, −M⁻¹K], [I, 0]] for both — so the distinction between them is one the inversion discards, and it survives only under the trailing one.

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.

13 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 chain long enough to have a spectrum and short enough to draw

a change of units the sweep can afford

a linearisation has as many eigenvalues as it has rows

a positive change of units

a positive mass

a reduction this file knows

an overdamped chain, whose spectrum is entirely real

damping that removes energy rather than adding it

every computed eigenvalue is matched to an unused exact one

LU is for square matrices

matmul shapes agree

six routes that all return a spectrum

two spectra of the same size

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 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 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 eigenvalue problem that is not linear

A backward-stable answer to a problem nobody asked

One quadratic eigenvalue problem, in nine systems of units, with a change of variable that is exact in both directions. The residual the solver prints stays at the rounding level at every stop. The answer loses eleven orders of magnitude, and the two facts are consistent.

 The eigenvalue problem that is not linear

A matrix that depends on its own eigenvalue

A damped structure does not produce Ax = λx. It produces (λ²M + λC + K)x = 0, where the matrix whose null vector is wanted is a function of the number being solved for — so there is nothing to factorise, an n × n problem has 2n answers, and the eigenvectors cannot be a basis.

 The eigenvalue problem that is not linear

Six routes to one spectrum

Three linearisations of one quadratic, each reduced to a standard eigenvalue problem two ways. All six have exactly the same eigenvalues in exact arithmetic. On a well-scaled problem they differ by noise; on a badly scaled one by a factor of forty; and two of the six are the same matrix.

 The eigenvalue problem that is not linear

The scaling that buys ten orders

Two lines computed from three norms, a change of variable that is exact in both directions, and the whole of the loss the previous essay measured comes back — flat, at every stop, because after scaling every stop is the same problem.

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