Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

order-stability

One function in the graphwalk library, called 12 times across 6 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 14 of 14 places survive the arithmetic and 3 survive one missing link. The top 15 of a PageRank ranking on an eighty-page link graph, and how many of its 14 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 14 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 11. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 6.28·10⁻⁵ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

order-stability is one function in lib/figures/graphwalk.js — the walk and the ranking — a rate that is not the published one, a positivity the arithmetic loses, and places nothing settles. 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.

14 of 14 places survive the arithmetic and 3 survive one missing linkThe top 15 of a PageRank ranking on an eighty-page link graph, and how many of its 14 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 14 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 11. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 6.28·10⁻⁵ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 14the arithmetic14 of 14one missing link3 of 14α from 0.80 to 0.9011 of 14entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons14smallest gap6.3·10⁻⁵largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 15 of a PageRank ranking on an eighty-page link graph, and how many of its 14 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 14 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 11. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 6.28·10⁻⁵ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

k: 15

The arguments are the ones A chain with no stationary vector 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.

14 of 14 places survive the arithmetic and 3 survive one missing linkThe top 15 of a PageRank ranking on an eighty-page link graph, and how many of its 14 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 14 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 11. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 6.28·10⁻⁵ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 14the arithmetic14 of 14one missing link3 of 14α from 0.80 to 0.9011 of 14entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons14smallest gap6.3·10⁻⁵largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 15 of a PageRank ranking on an eighty-page link graph, and how many of its 14 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 14 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 11. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 6.28·10⁻⁵ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

k: 24

The arguments are the ones A chain with no stationary vector 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.

23 of 23 places survive the arithmetic and 4 survive one missing linkThe top 24 of a PageRank ranking on an eighty-page link graph, and how many of its 23 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 23 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 15. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 4. The gaps between consecutive entries run from 3.78·10⁻⁶ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 23the arithmetic23 of 23one missing link4 of 23α from 0.80 to 0.9015 of 23entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons23smallest gap3.8·10⁻⁶largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 24 of a PageRank ranking on an eighty-page link graph, and how many of its 23 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 23 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 15. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 4. The gaps between consecutive entries run from 3.78·10⁻⁶ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

k: 6

The arguments are the ones A ranking whose order is not determined 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.

5 of 5 places survive the arithmetic and 2 survive one missing linkThe top 6 of a PageRank ranking on an eighty-page link graph, and how many of its 5 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 5 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 4. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 2. The gaps between consecutive entries run from 1.21·10⁻⁴ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 5the arithmetic5 of 5one missing link2 of 5α from 0.80 to 0.904 of 5entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons5smallest gap1.2·10⁻⁴largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 6 of a PageRank ranking on an eighty-page link graph, and how many of its 5 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 5 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 4. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 2. The gaps between consecutive entries run from 1.21·10⁻⁴ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

k: 10

The arguments are the ones A ranking whose order is not determined 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.

9 of 9 places survive the arithmetic and 3 survive one missing linkThe top 10 of a PageRank ranking on an eighty-page link graph, and how many of its 9 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 9 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 8. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 1.21·10⁻⁴ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 9the arithmetic9 of 9one missing link3 of 9α from 0.80 to 0.908 of 9entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons9smallest gap1.2·10⁻⁴largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 10 of a PageRank ranking on an eighty-page link graph, and how many of its 9 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 9 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 8. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 1.21·10⁻⁴ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

k: 20

The arguments are the ones A ranking whose order is not determined 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.

19 of 19 places survive the arithmetic and 3 survive one missing linkThe top 20 of a PageRank ranking on an eighty-page link graph, and how many of its 19 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 19 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 13. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 3.78·10⁻⁶ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.places settled, of 19the arithmetic19 of 19one missing link3 of 19α from 0.80 to 0.9013 of 19entries move by 4.05·10⁻¹¹entries move by 0.00443entries move by 0.00123three kinds of doubtcomparisons19smallest gap3.8·10⁻⁶largest gap0.0024arithmetic band4.1·10⁻¹¹one-link band0.0044α band0.0012the arithmetic settles nearly all of themthe data settles three

The top 20 of a PageRank ranking on an eighty-page link graph, and how many of its 19 adjacent comparisons keep their order under three kinds of doubt. Perturbing the teleport vector at the rounding level moves every entry by at most 4.05·10⁻¹¹ and settles 19 of them. Moving α across the range everybody uses moves entries by 0.00123 and settles 13. Removing a single arc — one link that might or might not have been crawled — moves entries by 0.00443 and settles 3. The gaps between consecutive entries run from 3.78·10⁻⁶ to 0.00241, so the ordering is decided by the data at a scale ten orders above anything the arithmetic does. This is the one field on the site where the computation is not the weakest part of the answer.

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.

7 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 leaderboard length worth reading

a teleportation parameter strictly inside (0, 1)

an arc inside the vertex set

no arc given twice

no self-link

one comparison for each adjacent pair

the arithmetic settles at least as many places as the parameter does

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 214 of 382 generators — 194 print a residual and 20 are exempt with a published reason; 168 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 matrix that is a graph

A chain with no stationary vector

A page with no outgoing links loses forty per cent of the walker's probability in six hundred steps. A directed cycle never converges at all. And on a graph whose links only run one way, the entire rank of half the vertices is exactly one minus the teleportation parameter.

 The matrix that is a graph

A ranking that is an eigenvector

PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.

 The matrix that is a graph

A ranking whose order is not determined

Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.

 The matrix that is a graph

An eigenvector that must not change sign

Perron's theorem says the leading eigenvector of a connected nonnegative matrix is strictly positive. On a clique with a long tail, four of its thirty-six entries come back negative — and beside them is the one two-sided bound on this site that is proved rather than estimated.

 The matrix that is a graph

The rate is the second eigenvalue

A walk forgets where it started at a rate the graph's second eigenvalue names exactly. Across three orders of magnitude in the step count the prediction is five per cent high — and the published rate for PageRank is right for a reason nobody states, which is that a link graph is in pieces.

 The matrix that is a graph

The spectrum is not the graph

Two graphs on six vertices with the same Laplacian characteristic polynomial — as integer polynomials, not to fourteen digits. One contains a triangle; the other is bipartite. Every method in this field that reads only the spectrum is answering about the class.

The whole library · All essays · What must fail