Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

zolotarev-shape

One function in the lowrank 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 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 the rational function 6 adi shifts amount to, chosen two ways. |∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 6 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.1187. The upper curve uses 6 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8922 — a factor of 7.52 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

zolotarev-shape 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 rational function 6 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 6 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.1187. The upper curve uses 6 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8922 — a factor of 7.52 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts6κ of the spectrum389geometric max0.12equally spaced max0.89the factor between7.5the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 6 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.1187. The upper curve uses 6 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8922 — a factor of 7.52 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

k: 6

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 rational function 6 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 6 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.1187. The upper curve uses 6 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8922 — a factor of 7.52 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts6κ of the spectrum389geometric max0.12equally spaced max0.89the factor between7.5the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 6 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.1187. The upper curve uses 6 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8922 — a factor of 7.52 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

k: 12

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 rational function 12 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 12 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.009967. The upper curve uses 12 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.766 — a factor of 76.9 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts12κ of the spectrum389geometric max0.01equally spaced max0.77the factor between77the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 12 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.009967. The upper curve uses 12 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.766 — a factor of 76.9 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

k: 2

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 rational function 2 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 2 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.6181. The upper curve uses 2 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.9731 — a factor of 1.57 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts2κ of the spectrum389geometric max0.62equally spaced max0.97the factor between1.6the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 2 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.6181. The upper curve uses 2 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.9731 — a factor of 1.57 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

k: 4

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 rational function 4 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 4 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.2711. The upper curve uses 4 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.9341 — a factor of 3.45 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts4κ of the spectrum389geometric max0.27equally spaced max0.93the factor between3.4the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 4 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.2711. The upper curve uses 4 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.9341 — a factor of 3.45 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

k: 8

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 rational function 8 ADI shifts amount to, chosen two ways|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 8 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.05198. The upper curve uses 8 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8497 — a factor of 16.3 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.10¹10²10³10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹λ, over the spectrum of −A|r(λ)|geometric maxequally spacedgeometricone line of codeshifts8κ of the spectrum389geometric max0.052equally spaced max0.85the factor between16the same k solvesand one choice of where

|∏(λ − pⱼ)/(λ + pⱼ)| over the spectrum of −A for a 30-point discretisation, whose ends are 9.86 and 3834 and whose condition number is 388.8. The lower curve uses 8 geometrically spaced shifts and touches zero at each of them, equioscillating between with a maximum of 0.05198. The upper curve uses 8 equally spaced ones: every shift sits in the top decade of the spectrum, the small end is never approached, and the maximum is 0.8497 — a factor of 16.3 worse, which the iteration pays squared. The two runs are the same algorithm, the same number of solves, and one different line where the shifts are chosen.

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.

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.

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

a number of shifts the picture can hold

a spectrum the interval can be read from

an actuator and a sensor on the grid

the geometric choice is the smaller maximum

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 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.

 The eigenvalue problem that is not linear

Two approximants and one matrix size

A polynomial approximant linearises to nd rows and a rational one to n(m+1), so the fair contest fixes the matrix and varies the basis. On an easy target set the two are indistinguishable and the ordering flips with the noise; on one that reaches a branch point the rational pulls away by two orders.

 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.

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