Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

approximant-contest

One function in the ratapprox library, called 10 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 two approximants at equal linearisation size, at a reach of 0.06. ‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.2, 2.9, 8.0, 23.7, 73.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

approximant-contest is one function in lib/figures/ratapprox.js — the problem the solver was given — the approximation committed before the arithmetic. 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.

Two approximants at equal linearisation size, at a reach of 0.06‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.2, 2.9, 8.0, 23.7, 73.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻⁵10⁻³10⁻¹rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.06largest size drawn78rational there2·10⁻⁵polynomial there0.0015the ratio73no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.2, 2.9, 8.0, 23.7, 73.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

reach: 0.06

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.

Two approximants at equal linearisation size, at a reach of 0.06‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.2, 2.9, 8.0, 23.7, 73.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻⁵10⁻³10⁻¹rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.06largest size drawn78rational there2·10⁻⁵polynomial there0.0015the ratio73no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.2, 2.9, 8.0, 23.7, 73.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

reach: 1

The arguments are the ones Two approximants and one matrix size 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.

Two approximants at equal linearisation size, at a reach of 1‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 0.4, 1.4, 4.4, 1.5, 0.0 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻¹⁰10⁻⁸10⁻⁶10⁻⁴rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach1largest size drawn78rational there1.2·10⁻⁸polynomial there5.2·10⁻¹⁰the ratio0.043no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 0.4, 1.4, 4.4, 1.5, 0.0 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

reach: 0.3

The arguments are the ones Two approximants and one matrix size 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.

Two approximants at equal linearisation size, at a reach of 0.3‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 0.9, 2.5, 7.7, 25.6, 89.4 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻⁸10⁻⁶10⁻⁴10⁻²rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.3largest size drawn78rational there8.7·10⁻⁸polynomial there7.8·10⁻⁶the ratio89no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 0.9, 2.5, 7.7, 25.6, 89.4 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

reach: 0.02

The arguments are the ones Two approximants and one matrix size 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.

Two approximants at equal linearisation size, at a reach of 0.02‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.3, 2.6, 6.5, 17.6, 49.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻⁴10⁻²rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.02largest size drawn78rational there1.6·10⁻⁴polynomial there0.0079the ratio49no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.3, 2.6, 6.5, 17.6, 49.2 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

reach: 0.008

The arguments are the ones Two approximants and one matrix size 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.

Two approximants at equal linearisation size, at a reach of 0.008‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.4, 2.3, 5.2, 12.7, 32.7 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.263850627410⁻⁴10⁻²1rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.008largest size drawn78rational there5.4·10⁻⁴polynomial there0.018the ratio33no winner on an easy targetand two orders on a hard one

‖g − r‖ against the size of the matrix the eigensolver is then handed: n(m+1) for a rational approximant with m poles and nd for a polynomial of degree d, so d = m + 1 is the fair comparison. At this target set the ratio of the polynomial's error to the rational's runs 1.4, 2.3, 5.2, 12.7, 32.7 across the five sizes. On a target set that stands clear of the branch point the two are the same measurement and the ordering flips with the noise; on one that reaches it the rational pulls away, because a polynomial converges algebraically against an algebraic singularity. Neither method is better — the question is different, and which question it is was decided by whoever said where to look.

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 branch point to the left of the spectrum

a size the linearisation can afford

a target set that stops short of the branch point

a target set the fits can be made on

two fits at every size

Against the rule

It draws a decomposition and prints its residual. It calls fitRational, fitPolynomial, 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.

 The eigenvalue problem that is not linear

The eigenvalues that are answers to nothing

A rational approximant of degree five turns a six-by-six problem into a thirty-six-by-thirty-six one, and thirty-six numbers come back. Six are the answer. The rest are exact eigenvalues of the approximant, lying where the function it approximates is not a real number at all.

 The eigenvalue problem that is not linear

The problem the solver was actually given

A linearisation is exact — it has the polynomial's eigenvalues, with their multiplicities, and the whole loss is arithmetic. A nonlinear eigenvalue problem does not offer that. Every algorithm replaces the function first, and the term that replacement contributes is committed before any number is rounded and appears in no residual.

 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.

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