kron-cost
At its defaults it draws multiplications in one product with the 3-dimensional laplacian, assembled against reshaped. Both curves are exact counts rather than estimates. The assembled matrix has n^3 rows, so multiplying by it is n^6 multiplications; applying the 3 one-dimensional factors along their own indices is 3·n^4. The fitted slopes are 6.00 and 4.00 against 6 and 4 exactly. At n = 256 that is 2.81·10¹⁴ against 1.29·10¹⁰, a factor of 2.18·10⁴ — and the ratio is n^2, so it grows with every size rather than settling.
kron-cost is one function in lib/figures/tensor.js —
an index that is a tuple — a matrix that is d small ones, and the inverse that is nearly one. 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.
Both curves are exact counts rather than estimates. The assembled matrix has n^3 rows, so multiplying by it is n^6 multiplications; applying the 3 one-dimensional factors along their own indices is 3·n^4. The fitted slopes are 6.00 and 4.00 against 6 and 4 exactly. At n = 256 that is 2.81·10¹⁴ against 1.29·10¹⁰, a factor of 2.18·10⁴ — and the ratio is n^2, so it grows with every size rather than settling.
d: 3
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.
Both curves are exact counts rather than estimates. The assembled matrix has n^3 rows, so multiplying by it is n^6 multiplications; applying the 3 one-dimensional factors along their own indices is 3·n^4. The fitted slopes are 6.00 and 4.00 against 6 and 4 exactly. At n = 256 that is 2.81·10¹⁴ against 1.29·10¹⁰, a factor of 2.18·10⁴ — and the ratio is n^2, so it grows with every size rather than settling.
d: 2
The arguments are the ones An equation whose unknown is a matrix 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.
Both curves are exact counts rather than estimates. The assembled matrix has n^2 rows, so multiplying by it is n^4 multiplications; applying the 2 one-dimensional factors along their own indices is 2·n^3. The fitted slopes are 4.00 and 3.00 against 4 and 3 exactly. At n = 256 that is 4.29·10⁹ against 3.36·10⁷, a factor of 128 — and the ratio is n^1, so it grows with every size rather than settling.
d: 4
The arguments are the ones An index that is a pair 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.
Both curves are exact counts rather than estimates. The assembled matrix has n^4 rows, so multiplying by it is n^8 multiplications; applying the 4 one-dimensional factors along their own indices is 4·n^5. The fitted slopes are 8.00 and 5.00 against 8 and 5 exactly. At n = 256 that is 1.84·10¹⁹ against 4.4·10¹², a factor of 4.19·10⁶ — and the ratio is n^3, so it grows with every size rather than settling.
d: 6
The arguments are the ones An index that is a pair 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.
Both curves are exact counts rather than estimates. The assembled matrix has n^6 rows, so multiplying by it is n^12 multiplications; applying the 6 one-dimensional factors along their own indices is 6·n^7. The fitted slopes are 12.00 and 7.00 against 12 and 7 exactly. At n = 256 that is 7.92·10²⁸ against 4.32·10¹⁷, a factor of 1.83·10¹¹ — and the ratio is n^5, so it grows with every size rather than settling.
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.
3 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.
a number of indices the counts are meaningful at
and the reshaped one costs n^{d+1}
the assembled product costs n^{2d}
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 146
of 287 generators —
131 print a residual and
15 are exempt with a published reason;
141 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.
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.
When the index is a tupleA solve that is d decompositions
A Kronecker sum is closed under nothing useful — its inverse is not a Kronecker sum and no factorisation of it is one. What it has instead is eigenvectors that are Kronecker products, so a solve with 1,728 unknowns takes one decomposition of a 12 × 12 matrix and nothing else.
Structure, and the solver that cannot see itAn equation whose unknown is a matrix
AX + XB = C is linear in X, so it has a coefficient matrix, and writing it down is the obvious thing to do. At n = 100 that matrix has a hundred million entries for a problem with ten thousand unknowns, and the algorithm everybody uses instead never forms it. Its conditioning is not the eigenvalue gap either, which is the number a reader is invited to consult.
When the index is a tupleAn index that is a pair
A discretisation on a two-dimensional grid of n points a side has n² unknowns and a matrix with n⁴ entries — 10⁸ at n = 100. What that matrix is instead is two Kronecker products of an n × n matrix, which is 2n² numbers, and nothing has been approximated: assembling it was the mistake.
Sparsity, and what elimination costsThe factor is not sparse
A sparse matrix has a factor that is not sparse, and the gap between them is the entire reason iterative methods exist. The entries elimination creates can be counted before any arithmetic runs, from the graph alone.
When the index is a tupleThe format that does not notice the dimension
A Tucker core is r^d numbers, so the format that repaired the definition still cannot go past five indices. Cutting between the indices rather than across them gives d − 1 ranks instead of d, storage linear in the number of indices, and a family whose ranks are two everywhere by an addition formula.
Where the flop count stopped predicting the timeThe order the products are taken in
The sparsity field's first essay says the elimination order decides the memory. This is the same sentence about arithmetic: a contraction of several tensors over shared indices has one value and many evaluation orders, and on the inner product of two trains they differ by a factor of two million.