How far |rₙₙ| sits above σₘᵢₙ on Kahan's matrix, against the size and the parameter
At its defaults it draws how far |rₙₙ| sits above σₘᵢₙ on kahan's matrix, against the size and the parameter. 3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 2.3·10⁸ at n = 50, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
kahan-gap is one function in lib/figures/rrqr.js —
the cheap rank — a greedy pivot rule, and the matrix it has nothing to choose on. 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.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 2.3·10⁸ at n = 50, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
maxN: 50
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.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 2.3·10⁸ at n = 50, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
maxN: 64
The arguments are the ones The cheap rank and what it cannot see passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 6.8·10¹⁰ at n = 64, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
maxN: 20
The arguments are the ones The cheap rank and what it cannot see passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 1214 at n = 20, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
maxN: 30
The arguments are the ones The cheap rank and what it cannot see passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 7·10⁴ at n = 30, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
maxN: 40
The arguments are the ones The cheap rank and what it cannot see passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
3 curves of |rₙₙ| ÷ σₘᵢₙ against n, one per Kahan parameter. Every curve rises without turning over, reaching 4·10⁶ at n = 40, c = 0.5. Column pivoting makes no interchange at any point on any of them, so the failure is not a poor choice — there is nothing to choose.
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.
25 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.
a larger Kahan matrix is worse at c = 0.2, n = 20 — checked 15 times
and a larger c is worse at c = 0.35 — checked 2 times
a column-norm rule this file defines
a Kahan parameter strictly inside (0, 1)
a largest size the SVD can afford
a size the decay has room in
at least two Kahan parameters inside (0, 1)
matmul shapes agree
reaching orders of magnitude inside the sizes drawn
with no interchange at any size
Against the rule
It draws a decomposition and prints its residual. It calls
kahanVerdicts,
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.
Eigenvalues, singular values, rankThe cheap rank and what it cannot see
Almost nobody computes singular values to decide a rank. The standard substitute is QR with column pivoting, read off the diagonal of R — and there is a triangular matrix on which the greedy rule makes no interchange at all, has no better column available at any step, and reports a matrix eight orders of magnitude further from singular than it is.