The singular values of a sum of two rank-4 blocks, and the 4 a truncation has to discard
At its defaults it draws the singular values of a sum of two rank-4 blocks, and the 4 a truncation has to discard. A rank-4 block times a vector is a rank-4 block times a vector. A rank-4 block times a rank-4 block is a rank-4 block. A rank-4 block PLUS a rank-4 block is a rank-8 block, exactly, and the 8 bars here are why: the sum of two 4-dimensional spaces is generally 8-dimensional, and none of the 8 singular values is small. Truncating back to 4 costs 61.9 per cent of the block. Below the 8th the values are the unit roundoff, which is the check that the doubling is exact rather than approximate. Every product, every factorisation and every Schur complement inside this format is a chain of these, and there is nothing else to do: without the truncation the ranks double at every level and the format is dense by the bottom.
sum-doubles is one function in lib/figures/recompress.js —
arithmetic in the format — the one operation it is not closed under, and what a hundred roundings cost. 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.
A rank-4 block times a vector is a rank-4 block times a vector. A rank-4 block times a rank-4 block is a rank-4 block. A rank-4 block PLUS a rank-4 block is a rank-8 block, exactly, and the 8 bars here are why: the sum of two 4-dimensional spaces is generally 8-dimensional, and none of the 8 singular values is small. Truncating back to 4 costs 61.9 per cent of the block. Below the 8th the values are the unit roundoff, which is the check that the doubling is exact rather than approximate. Every product, every factorisation and every Schur complement inside this format is a chain of these, and there is nothing else to do: without the truncation the ranks double at every level and the format is dense by the bottom.
k: 6
The arguments are the ones The count that is not the budget passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rank-6 block times a vector is a rank-6 block times a vector. A rank-6 block times a rank-6 block is a rank-6 block. A rank-6 block PLUS a rank-6 block is a rank-12 block, exactly, and the 12 bars here are why: the sum of two 6-dimensional spaces is generally 12-dimensional, and none of the 12 singular values is small. Truncating back to 6 costs 57.8 per cent of the block. Below the 12th the values are the unit roundoff, which is the check that the doubling is exact rather than approximate. Every product, every factorisation and every Schur complement inside this format is a chain of these, and there is nothing else to do: without the truncation the ranks double at every level and the format is dense by the bottom.
k: 4
The arguments are the ones The orthogonality that cannot be diagonal passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rank-4 block times a vector is a rank-4 block times a vector. A rank-4 block times a rank-4 block is a rank-4 block. A rank-4 block PLUS a rank-4 block is a rank-8 block, exactly, and the 8 bars here are why: the sum of two 4-dimensional spaces is generally 8-dimensional, and none of the 8 singular values is small. Truncating back to 4 costs 61.9 per cent of the block. Below the 8th the values are the unit roundoff, which is the check that the doubling is exact rather than approximate. Every product, every factorisation and every Schur complement inside this format is a chain of these, and there is nothing else to do: without the truncation the ranks double at every level and the format is dense by the bottom.
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.
4 distinct claims across 3 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 rank the figure has room to draw twice
concatenating the factors doubles the rank
matmul shapes agree
the truncation done through the factors is the truncation of the matrix
Against the rule
It draws a decomposition and prints its residual. It calls
svd, truncate,
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 count that is not the budget
A Cholesky performed inside a low-rank format truncates 0, 2, 10, 34 and 98 times as the leaf falls from 128 to 8, and those five integers are the same at every accuracy from 10⁻¹² to 10⁻². Across all ten decades the factorisation's residual stays below the representation's own error at a ratio between 0.81 and 1.00 — with two entries that read 1.83 and 1.78, and neither of them is accumulation.
When the index is a tupleThe orthogonality that cannot be diagonal
A matrix decomposition hands over orthonormal factors and a diagonal middle at once. For three indices the two come apart, and there is no arrangement that has both — so the question stops being which decomposition to use and becomes which of the two properties the computation needs.
Neither sparse nor denseThe 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.