Multiplications in a hierarchical solve and in a dense factorisation, and where they cross

At its defaults it draws multiplications in a hierarchical solve and in a dense factorisation, and where they cross. Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.13, 1.93, 1.74 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 128: below it the format is the more expensive way to solve the system, at 1.48 times the dense count, and at n = 512 it is 6.2 times cheaper. Every point returns an answer at a backward error of about 1.4·10⁻¹⁰, so the comparison is between two ways of getting the same thing.

solve-cost is one function in lib/figures/hsolve.js — solving with it — the recursion that assembles nothing, and the backward error that was chosen. 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.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.13, 1.93, 1.74 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 128: below it the format is the more expensive way to solve the system, at 1.48 times the dense count, and at n = 512 it is 6.2 times cheaper. Every point returns an answer at a backward error of about 1.4·10⁻¹⁰, so the comparison is between two ways of getting the same thing.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.13, 1.93, 1.74 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 128: below it the format is the more expensive way to solve the system, at 1.48 times the dense count, and at n = 512 it is 6.2 times cheaper. Every point returns an answer at a backward error of about 1.4·10⁻¹⁰, so the comparison is between two ways of getting the same thing.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.25, 1.95, 1.78 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 128 and n = 256: below it the format is the more expensive way to solve the system, at 2.36 times the dense count, and at n = 512 it is 3.4 times cheaper. Every point returns an answer at a backward error of about 2.4·10⁻¹⁴, so the comparison is between two ways of getting the same thing.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 1.82, 1.80, 1.62 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 64: below it the format is the more expensive way to solve the system, at 0.28 times the dense count, and at n = 512 it is 47.4 times cheaper. Every point returns an answer at a backward error of about 2.6·10⁻⁴, so the comparison is between two ways of getting the same thing.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.06, 1.87, 1.66 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 64: below it the format is the more expensive way to solve the system, at 0.61 times the dense count, and at n = 512 it is 17.5 times cheaper. Every point returns an answer at a backward error of about 1.5·10⁻⁶, so the comparison is between two ways of getting the same thing.

Both counted rather than estimated: the recursion carries a counter and reports what it actually did. The dense factorisation is n³/3 and rises at exactly three per doubling. The hierarchical solve rises at 2.13, 1.88, 1.73 — falling, because the cost is n log²n and its exponent is on its way to one. The two cross between n = 64 and n = 64: below it the format is the more expensive way to solve the system, at 0.92 times the dense count, and at n = 512 it is 10.3 times cheaper. Every point returns an answer at a backward error of about 2.3·10⁻⁸, so the comparison is between two ways of getting the same thing.

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.

9 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.

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