Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Two approximants at equal linearisation size, at a reach of 0.06

One function in the ratapprox library, called 6 times across 1 essay. 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 Two approximants and one matrix size passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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 nobody placed would be a picture no essay asked for and no claim was ever checked 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: 0.5

The arguments are the ones Two approximants and one matrix size passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two approximants at equal linearisation size, at a reach of 0.5‖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 0.7, 2.1, 6.6, 22.7, 27.2 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.263850627410⁻⁸10⁻⁶10⁻⁴10⁻²rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.5largest size drawn78rational there1.4·10⁻⁸polynomial there3.7·10⁻⁷the ratio27no 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 0.7, 2.1, 6.6, 22.7, 27.2 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: 0.1

The arguments are the ones Two approximants and one matrix size passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two approximants at equal linearisation size, at a reach of 0.1‖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 1.1, 2.9, 8.3, 25.8, 83.0 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.263850627410⁻⁶10⁻⁴10⁻²rows in the linearisation‖g − r‖ on the target setpolynomialrationalequal cost, two basesreach0.1largest size drawn78rational there5.4·10⁻⁶polynomial there4.5·10⁻⁴the ratio83no 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 1.1, 2.9, 8.3, 25.8, 83.0 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: 0.02

The arguments are the ones Two approximants and one matrix size passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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.

What it checked while drawing

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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

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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