Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

adi-identity

One function in the lowrank library, called 8 times across 4 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 adi error is the rational function, not merely bounded by it. ‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

adi-identity is one function in lib/figures/lowrank.js — why a gramian can be truncated — a rational function's poles, and the decay they explain. 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.234567891010⁻⁴10⁻²1ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn9ratio, worst0.77spread of the ratio1error at k = 103.9·10⁻⁴its residual1.1·10⁻⁴four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

n: 30

The arguments are the ones An error committed before the arithmetic 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.234567891010⁻⁴10⁻²1ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn9ratio, worst0.77spread of the ratio1error at k = 103.9·10⁻⁴its residual1.1·10⁻⁴four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

n: 16

The arguments are the ones Where to put the poles of a rational function 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 16-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.759 at k = 2 and 0.748 at k = 10, a spread of 1.0384 while the error itself falls from 0.193 to 5.03·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.234567891010⁻⁵10⁻³10⁻¹ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn9ratio, worst0.78spread of the ratio1error at k = 105·10⁻⁵its residual1.6·10⁻⁵four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 16-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.759 at k = 2 and 0.748 at k = 10, a spread of 1.0384 while the error itself falls from 0.193 to 5.03·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

n: 10

The arguments are the ones Why a Gramian can be truncated at all 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 10-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.732 at k = 2 and 0.703 at k = 9, a spread of 1.0415 while the error itself falls from 0.119 to 1.81·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.2345678910⁻⁵10⁻³10⁻¹ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn8ratio, worst0.73spread of the ratio1error at k = 91.8·10⁻⁵its residual6.9·10⁻⁶four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 10-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.732 at k = 2 and 0.703 at k = 9, a spread of 1.0415 while the error itself falls from 0.119 to 1.81·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

n: 22

The arguments are the ones Why a Gramian can be truncated at all 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 22-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.776 at k = 2 and 0.769 at k = 10, a spread of 1.0296 while the error itself falls from 0.248 to 1.61·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.234567891010⁻⁴10⁻²1ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn9ratio, worst0.79spread of the ratio1error at k = 101.6·10⁻⁴its residual4.6·10⁻⁵four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 22-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.776 at k = 2 and 0.769 at k = 10, a spread of 1.0296 while the error itself falls from 0.248 to 1.61·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

n: 40

The arguments are the ones Why a Gramian can be truncated at all 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 ADI error is the rational function, not merely bounded by it‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 40-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.766 at k = 2 and 0.752 at k = 10, a spread of 1.0346 while the error itself falls from 0.336 to 7.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.234567891010⁻⁴10⁻²1ADI steps krelative error, and max|r|²max|r|²the measured errora bound that is an identitysteps drawn9ratio, worst0.78spread of the ratio1error at k = 107.9·10⁻⁴its residual2.1·10⁻⁴four orders of errorand one constant

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 40-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.766 at k = 2 and 0.752 at k = 10, a spread of 1.0346 while the error itself falls from 0.336 to 7.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

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.

8 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 Gramian the closed form can be checked against

a grid fine enough to have modes and coarse enough to draw

a system with a known eigenbasis

an actuator and a sensor on the grid

and the ratio is one number, flat in k

LU is for square matrices

matmul shapes agree

the error is at most the squared factor

Against the rule

It draws a decomposition and prints its residual. It calls adiLowRank, 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 192 of 346 generators — 174 print a residual and 18 are exempt with a published reason; 154 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

An error committed before the arithmetic

Before a nonlinear eigenvalue problem is solved, somebody says where they think the eigenvalues are. That sentence sets the accuracy of everything that follows by five orders, costs nothing to say, and cannot be revised once the approximation built on it is in hand.

 Reduction, and what a model is for

Interpolating at the model’s own poles

One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.

 Reduction, and what a model is for

Where to put the poles of a rational function

Three times in one field the same question arrives from different directions — ADI shifts, rational approximation of a square root, the decay of a Gramian — and it has one answer. Cluster them geometrically towards wherever the function is difficult, and the alternative that looks reasonable costs orders.

 Reduction, and what a model is for

Why a Gramian can be truncated at all

Every method in this field rests on one fact nobody states the reason for — the eigenvalues of a Gramian fall off a cliff. The equation defining it has a rank-one right-hand side and no low-rank structure anywhere — and the answer's decay is a rational approximation problem with a closed-form rate.

The whole library · All essays · What must fail