The condition number a sketched preconditioner leaves, against the condition number it was given
At its defaults it draws the condition number a sketched preconditioner leaves, against the condition number it was given. Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 4n-row sketch, is 2.2284 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
precondition-kappa is one function in lib/figures/precond.js —
a sketch as a preconditioner — the same randomness, spent on the cost instead. 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 κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 4n-row sketch, is 2.2284 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
width: 4
The arguments are the ones The sketch that is not the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 4n-row sketch, is 2.2284 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
width: 2
The arguments are the ones The sketch that is not the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 2n-row sketch, is 6.1286 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
width: 3
The arguments are the ones The sketch that is not the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 3n-row sketch, is 2.7861 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
width: 10
The arguments are the ones The sketch that is not the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 10n-row sketch, is 1.6616 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
n: 16
The arguments are the ones The sketch that is not the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves against κ(A), both axes logarithmic. The matrix's own condition number climbs the diagonal from 100 to 10¹⁰; κ(AR⁻¹), where R comes from a QR of a 4n-row sketch, is 2.6304 at every one of them — the same number to ten digits, not a similar one. The reason is two lines of algebra: with G = SU the preconditioned singular values are those of (GᵀG)⁻¹, which has no spectrum of A in it at all.
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.
17 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 number of columns the sweep can afford
a sketch wider than the problem and narrower than the data
and is the SAME number at every one of them, not merely a similar one
enough rows for the sketch to be a sketch
matmul shapes agree
the sketch's R factor is nonsingular at κ = 100
the sketch's R factor is nonsingular at κ = 10¹⁰
the sketch's R factor is nonsingular at κ = 10⁴
the sketch's R factor is nonsingular at κ = 10⁶
the sketch's R factor is nonsingular at κ = 10⁸
while κ(A) moves by orders of magnitude
κ(AR⁻¹) is a small constant at κ(A) = 100
κ(AR⁻¹) is a small constant at κ(A) = 10¹⁰
κ(AR⁻¹) is a small constant at κ(A) = 10⁴
κ(AR⁻¹) is a small constant at κ(A) = 10⁶
κ(AR⁻¹) is a small constant at κ(A) = 10⁸
κ(AR⁻¹) measured and κ(SU) computed from the sketch alone agree
Against the rule
It draws a decomposition and prints its residual. It calls
qrHouseholder, sketchPreconditioner,
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.
The sketch that is not the answer
Sketch-and-solve throws away the original problem and keeps the small one's answer, which is why its answer moves with the seed. Use the same sketch as a preconditioner instead and the condition number the iteration sees is the same number at every κ from a hundred to ten billion — identically the same, to nine digits, because the spectrum cancels out of it.
Two errors, and whose fault they areThe zero you are allowed to write
A deflation criterion sets a subdiagonal entry to zero because it is small. A drop tolerance discards an entry of a factor because it is small. A truncation discards a singular value because it is small. Three fields, three vocabularies, no shared arithmetic — and plotted as work saved against error accepted, one curve.