Steps to a relative error of 10⁻⁶, on one 60×12 problem at five condition numbers
At its defaults it draws steps to a relative error of 10⁻⁶, on one 60×12 problem at five condition numbers. Two curves against the condition number on a logarithmic horizontal axis, with four seeds drawn at each. The two recurrences compute the same iterates in exact arithmetic. At κ = 10² they cost 16 and 16 steps; at κ = 10¹⁰ they cost 110 and 209.
two-recurrences is one function in lib/figures/lsqrgmres.js —
the other two krylov regularisers — one sequence, two recurrences, and a weight that stops being a function of σ. 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.
Two curves against the condition number on a logarithmic horizontal axis, with four seeds drawn at each. The two recurrences compute the same iterates in exact arithmetic. At κ = 10² they cost 16 and 16 steps; at κ = 10¹⁰ they cost 110 and 209.
target: 0.000001
The arguments are the ones One sequence and two recurrences passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against the condition number on a logarithmic horizontal axis, with four seeds drawn at each. The two recurrences compute the same iterates in exact arithmetic. At κ = 10² they cost 16 and 16 steps; at κ = 10¹⁰ they cost 110 and 209.
target: 0.001
The arguments are the ones One sequence and two recurrences passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against the condition number on a logarithmic horizontal axis, with four seeds drawn at each. The two recurrences compute the same iterates in exact arithmetic. At κ = 10² they cost 15 and 15 steps; at κ = 10¹⁰ they cost 101 and 159.
target: 0.00001
The arguments are the ones One sequence and two recurrences passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against the condition number on a logarithmic horizontal axis, with four seeds drawn at each. The two recurrences compute the same iterates in exact arithmetic. At κ = 10² they cost 15 and 16 steps; at κ = 10¹⁰ they cost 105 and 194.
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.
21 distinct claims across 4 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 matrix tall enough for the least-squares problem to be one
a width the dense reference is affordable at
an accuracy target above the forward-error floor
and not on an ill-conditioned one
both methods reach 0.001 at κ = 100
both methods reach 0.001 at κ = 10¹⁰
both methods reach 0.001 at κ = 10⁴
both methods reach 0.001 at κ = 10⁶
both methods reach 0.001 at κ = 10⁸
both methods reach 10⁻⁵ at κ = 100
both methods reach 10⁻⁵ at κ = 10¹⁰
both methods reach 10⁻⁵ at κ = 10⁴
both methods reach 10⁻⁵ at κ = 10⁶
both methods reach 10⁻⁵ at κ = 10⁸
both methods reach 10⁻⁶ at κ = 100
both methods reach 10⁻⁶ at κ = 10¹⁰
both methods reach 10⁻⁶ at κ = 10⁴
both methods reach 10⁻⁶ at κ = 10⁶
both methods reach 10⁻⁶ at κ = 10⁸
matmul shapes agree
the two cost the same on a well-conditioned problem
Against the rule
It draws a decomposition and prints its residual. It calls
cgls,
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.