Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

two-routes-rank

One function in the kernel library, called 7 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 17 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 three routes to the error of a rank-k approximation of one block, at q = 0.5. The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

two-routes-rank is one function in lib/figures/kernel.js — one block — where a matrix with no zero entries turns out to have five useful columns. 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.

Three routes to the error of a rank-k approximation of one block, at q = 0.5The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹rank of the approximationrelative errorthe boundthe expansion, evaluatedthe decompositiontwo routes, one shapebound at rank 50.063expansion at rank 50.0048decomposition at rank 52.8·10⁻⁹written, decades a column0.36best, decades a column1.7one route used the matrixand the other used four numbers

The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

n: 96

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.

Three routes to the error of a rank-k approximation of one block, at q = 0.5The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹rank of the approximationrelative errorthe boundthe expansion, evaluatedthe decompositiontwo routes, one shapebound at rank 50.063expansion at rank 50.0048decomposition at rank 52.8·10⁻⁹written, decades a column0.36best, decades a column1.7one route used the matrixand the other used four numbers

The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

n: 48

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.

Three routes to the error of a rank-k approximation of one block, at q = 0.5The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.37 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹rank of the approximationrelative errorthe boundthe expansion, evaluatedthe decompositiontwo routes, one shapebound at rank 50.063expansion at rank 50.0048decomposition at rank 52.7·10⁻⁹written, decades a column0.37best, decades a column1.7one route used the matrixand the other used four numbers

The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.37 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

n: 192

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.

Three routes to the error of a rank-k approximation of one block, at q = 0.5The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹rank of the approximationrelative errorthe boundthe expansion, evaluatedthe decompositiontwo routes, one shapebound at rank 50.063expansion at rank 50.0048decomposition at rank 52.8·10⁻⁹written, decades a column0.36best, decades a column1.7one route used the matrixand the other used four numbers

The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

n: 144

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.

Three routes to the error of a rank-k approximation of one block, at q = 0.5The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.0246810121410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹rank of the approximationrelative errorthe boundthe expansion, evaluatedthe decompositiontwo routes, one shapebound at rank 50.063expansion at rank 50.0048decomposition at rank 52.8·10⁻⁹written, decades a column0.36best, decades a column1.7one route used the matrixand the other used four numbers

The upper line is q^(p+1)/(1 − q), which used four numbers about two intervals and no entry of the matrix. The middle line is the truncated expansion actually evaluated — a rank p + 1 matrix written down as a table of powers, with no decomposition anywhere in it — and it falls at -0.36 decades a column, close to the log₁₀ q = -0.30 the bound predicts. The lower line is the decomposition, which is optimal by construction, and it falls at -1.74 — 4.8 times as fast. Two of these curves share no arithmetic. What they agree on is that the error is geometric in the rank; what they disagree on is the base, by a factor in the exponent rather than a constant, and the disagreement is why a partition allocated from the bound is safe and wasteful at the same time.

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.

17 distinct claims across 5 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.

the written expansion at order 0 is inside its own bound — asserted 10 times

a block size the decomposition is affordable at

a block size the dense SVD below is affordable at

a kernel this file defines

a truncation order inside the range this stays conditioned at

and the decomposition at the same rank is at least as good

matmul shapes agree

the kernel the expansion is written for

Against the rule

It draws a decomposition and prints its residual. It calls twoRouteSweep, 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.

 Eigenvalues, singular values, rank

The best approximation there is

The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.

 Least squares, and the road not to take

The projection and the right angle

The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.

 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.

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