Implied orthogonality of four factorisations of a 256×8 matrix
At its defaults it draws implied orthogonality of four factorisations of a 256×8 matrix. Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.99 and reaches 0.373; the same routine run twice has a slope of 0.96 and reaches 3.75·10⁻⁹, which is where the Householder sweep and the reduction tree are.
second-pass is one function in lib/figures/comm.js —
two ways of buying something back — a second pass, and memory spent on messages. 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.
Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.99 and reaches 0.373; the same routine run twice has a slope of 0.96 and reaches 3.75·10⁻⁹, which is where the Householder sweep and the reduction tree are.
n: 8
The arguments are the ones Doing it twice passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.99 and reaches 0.373; the same routine run twice has a slope of 0.96 and reaches 3.75·10⁻⁹, which is where the Householder sweep and the reduction tree are.
n: 6
The arguments are the ones Doing it twice passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.96 and reaches 0.169; the same routine run twice has a slope of 0.79 and reaches 2.32·10⁻¹⁰, which is where the Householder sweep and the reduction tree are.
n: 16
The arguments are the ones Doing it twice passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.97 and reaches 0.354; the same routine run twice has a slope of 0.91 and reaches 2.08·10⁻⁹, which is where the Householder sweep and the reduction tree are.
n: 12
The arguments are the ones Doing it twice passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves against the condition number, both axes logarithmic. Cholesky QR's implied orthogonality has a fitted slope of 1.98 and reaches 0.762; the same routine run twice has a slope of 0.95 and reaches 1.79·10⁻⁹, which is where the Householder sweep and the reduction tree are.
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 5 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.
both Cholesky runs return an answer at κ = 100 — checked 2 times
a matrix tall enough for the factorisation to be tall-skinny
a width the dense reference is affordable at
and two cost κ
both Cholesky runs return an answer at κ = 10⁴
both Cholesky runs return an answer at κ = 10⁵
both Cholesky runs return an answer at κ = 10⁶
both Cholesky runs return an answer at κ = 10⁷
both Cholesky runs return an answer at κ = 10⁸
condition numbers below the cliff, where all four return an answer
each row block is at least as tall as it is wide
matmul shapes agree
one pass costs κ²
with the pair matching the sweep at the worst conditioning drawn
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
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.
Doing it twice
Cholesky QR squares the condition number — a fitted slope of 1.95 in κ against the Householder sweep's 1.00. Run the identical routine a second time on the Q it returned and the slope is 0.93, the orthogonality is at or below the sweep's at every κ, and the price is one more all-reduce.
Where the flop count stopped predicting the timeThe message and the word
Three factorisations of one matrix on sixteen processors: 48 communication rounds, 4, and 4. The words sent are 1,170, 1,170 and 2,160 — so the method with the fewest rounds sends the most words, and the count that separates the three is the one no operation count can see.