Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

partition-ranks

One function in the hodlr library, called 15 times across 8 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 15 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 largest rank in each partition, and what refusing to compress a touching pair costs. The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

partition-ranks is one function in lib/figures/hodlr.js — the partition — a rule that reads four numbers a pair and no entry of the matrix. 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.567891003691215log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers6.8·10⁴weak, numbers6.1·10⁴strong ⁄ weak1.1the better partitionis the more expensive one

The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

logEps: -8

The arguments are the ones A block nobody can call sparse 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.567891003691215log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers6.8·10⁴weak, numbers6.1·10⁴strong ⁄ weak1.1the better partitionis the more expensive one

The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

logEps: -4

The arguments are the ones A rank that is a number of digits 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 3 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 5, 6, 7, 7, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 49,152 numbers against 38,912 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.56789100246810log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers4.9·10⁴weak, numbers3.9·10⁴strong ⁄ weak1.3the better partitionis the more expensive one

The strong rule's worst block is rank 3 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 5, 6, 7, 7, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 49,152 numbers against 38,912 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

logEps: -12

The arguments are the ones The size the rank does not notice 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 7 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 12, 14, 16, 18, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 86,784 numbers against 79,872 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.5678910048121620log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers8.7·10⁴weak, numbers8·10⁴strong ⁄ weak1.1the better partitionis the more expensive one

The strong rule's worst block is rank 7 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 12, 14, 16, 18, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 86,784 numbers against 79,872 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

logEps: -2

The arguments are the ones The test that costs what it saves 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 2 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 3, 3, 4, 4, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 39,744 numbers against 25,600 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.567891002468log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers4·10⁴weak, numbers2.6·10⁴strong ⁄ weak1.6the better partitionis the more expensive one

The strong rule's worst block is rank 2 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 3, 3, 4, 4, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 39,744 numbers against 25,600 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

logEps: -10

The arguments are the ones The test that costs what it saves 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 largest rank in each partition, and what refusing to compress a touching pair costsThe strong rule's worst block is rank 6 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 11, 12, 14, 16, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 77,376 numbers against 71,680 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.56789100369121518log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers7.7·10⁴weak, numbers7.2·10⁴strong ⁄ weak1.1the better partitionis the more expensive one

The strong rule's worst block is rank 6 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 11, 12, 14, 16, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 77,376 numbers against 71,680 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

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.

15 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.

and the strong rule never stores less at n = 64, which is the half nobody expects — asserted 4 times

a partition rule this file defines

a power of two, so the bisection is exact at every level

a shift inside the range the matrix stays positive definite and the geometry stays the geometry

a size the dense reference below is affordable at

an accuracy the blocks have singular values across

an accuracy, not a rank

an admissibility constant inside the range the expansion converges over

and strictly more at the largest size, where the partitions have room to differ

and the weak rule's climbs, because it compresses the touching halves

matmul shapes agree

the strong rule's worst rank is one number across the sweep

Against the rule

It draws a decomposition and prints its residual. It calls storageAgainstSize, 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 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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.

Neither sparse nor dense

A block nobody can call sparse

A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.

 Neither sparse nor dense

A rank that is a number of digits

Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.

 Neither sparse nor dense

The kernel with nothing to compress

Hold the geometry fixed at q = ½, fix the wavelength, and scale the picture up by sixteen. A smooth kernel needs six columns at every scale. An oscillatory one needs twelve, sixteen, twenty-two, thirty-three, fifty-three, and there is no scale at which it stops.

 Neither sparse nor dense

The rounding that was not the problem

A rank-k block plus a rank-k block is a rank-2k block, exactly, so every arithmetic in this format truncates after every addition. A Cholesky performed inside it does ninety-eight of those and its residual is 1.14·10⁻⁹ against a representation error of 1.40·10⁻⁹ — the roundings cost nothing measurable.

 Neither sparse nor dense

The size the rank does not notice

Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.

 Neither sparse nor dense

The test that costs what it saves

The partition that refuses to compress a touching pair keeps every rank at five while the other lets them climb from nine to thirteen. It also stores more numbers at every size measured — 67,968 against 61,440 at n = 512 — and which of those two facts matters is a question about how large the problem is going to get.

 Where the flop count stopped predicting the time

Where the format starts paying

A hierarchical solve costs 1.48 times a dense factorisation at 64 unknowns and 0.16 times it at 512. The crossover is between 64 and 128, it walks right when the accuracy is tightened, and the exponent between consecutive sizes is 2.13, 1.93, 1.74 — falling towards one and never arriving.

 Neither sparse nor dense

Which pairs are allowed to be small

A hierarchical representation is a partition of the matrix into blocks, and the rule that produces it reads four numbers per pair of index clusters and not one entry of the matrix. On a 256-square it yields 112 blocks, 66 of them stored as two thin factors, none of rank above five.

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