Past the cliff: what each method returns
At its defaults it draws past the cliff: what each method returns. Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at 3·10¹⁰ it refuses outright, on a pivot of -6.5·10⁻¹⁷. The sweep and the tree return a usable factorisation of every one of these matrices.
refuse-or-lie is one function in lib/figures/parallel.js —
messages — the count words cannot make, and what the fewest of them costs. 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.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at 3·10¹⁰ it refuses outright, on a pivot of -6.5·10⁻¹⁷. The sweep and the tree return a usable factorisation of every one of these matrices.
n: 8
The arguments are the ones Deciding that a zero has arrived passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at 3·10¹⁰ it refuses outright, on a pivot of -6.5·10⁻¹⁷. The sweep and the tree return a usable factorisation of every one of these matrices.
kappas: [100000000, 10000000000, 30000000000, 1000000000000]
The arguments are the ones The licence is not the boundary passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at 3·10¹⁰ it refuses outright, on a pivot of -6.5·10⁻¹⁷. The sweep and the tree return a usable factorisation of every one of these matrices.
kappas: [100000000, 20000000000, 40000000000, 1000000000000]
The arguments are the ones The licence is not the boundary passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at two of them — 2·10¹⁰, 4·10¹⁰ — it refuses outright, the first on a pivot of -2.7·10⁻¹⁶. The sweep and the tree return a usable factorisation of every one of these matrices.
kappas: [100000000, 30000000000, 300000000000, 1000000000000]
The arguments are the ones The licence is not the boundary passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.37 away from orthogonal at κ = 10⁸ and 1.49 at 10¹² — and at 3·10¹⁰ it refuses outright, on a pivot of -6.5·10⁻¹⁷. The sweep and the tree return a usable factorisation of every one of these matrices.
n: 6, kappas: [100000000, 10000000000, 1000000000000, 10000000000000]
The arguments are the ones The licence is not the boundary passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Loss of orthogonality at four condition numbers past κ²u = 1. Cholesky QR returns a factor whose implied Q is 0.17 away from orthogonal at κ = 10⁸ and 1.49 at 10¹³ — and at two of them — 10¹⁰, 10¹² — it refuses outright, the first on a pivot of -3.6·10⁻¹⁵. The sweep and the tree return a usable factorisation of every one of these matrices.
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.
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.
the sweep still returns something usable at κ = 3·10¹⁰ — checked 3 times
a matrix tall enough for the slices to be tall
and it succeeds on a harder matrix than one it declined
and returns a wholly non-orthogonal factor somewhere else in it
Cholesky QR refuses somewhere in this range
condition numbers past where κ²u passes one
each row block is at least as tall as it is wide
matmul shapes agree
the sweep still returns something usable at κ = 10¹⁰
the sweep still returns something usable at κ = 10¹²
the sweep still returns something usable at κ = 10¹³
the sweep still returns something usable at κ = 10⁸
the sweep still returns something usable at κ = 3·10¹¹
Against the rule
It draws a decomposition and prints its residual. It calls
choleskyQR, tsqrDistributed, householderDistributed,
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.
Deciding that a zero has arrived
The previous tolerances were offers — accept this much error, save this much work. A detection threshold is not an offer, because both directions are failures. One matrix here has three genuinely near-invariant subspaces, and the constant somebody typed decides which of them the recurrence stops at; at eight significand bits the same kind of constant produces a proof of something false.
The answer that depends on the machineThe licence is not the boundary
Cholesky QR is licensed by κ²u ≪ 1, which reaches equality at κ = 9.5·10⁷ in double precision. At 10⁸ the factor it returns is already 0.37 away from orthogonal, and it goes on returning factors as far as 10¹³ — refusing at scattered condition numbers in between, at different ones for eight columns and for six.