Every essay — page 6
Methods that were designed apart
Every field here introduced its own knob and its own vocabulary, and the essays that did so were written as though the methods were alternatives. Put four of them on one problem with an answer that is known and they land within a few per cent of each other, because what each one controls is how much of the data to believe and how much is worth believing belongs to the problem. This field is what happens when two methods meet: which parts of a solver can be computed at three significand bits, which cannot be rounded at all, and where the reading that every knob is the same knob stops being true.
The part of a solver that may be rounded
A preconditioner computed and applied with a three-bit significand still returns thirteen correct digits — it costs seventeen extra iterations and nothing else. Round the working arithmetic instead and the step count barely moves while the answer loses exactly the digits the format dropped.
An answer that changes with the seed
A randomised rank-k solve is a truncation computed in a random subspace, and it reaches the same floor as the deterministic ones. What it does not do is return the same answer twice — a factor of 1.84 across four seeds at rank 8, and 1.02 at the rank where the method is best.
The step that stops mattering
Regularise the problem the iteration has built rather than the problem it was given, and the error curve stops turning. The unregularised run ends 1,127 times above its own best; the same run with a penalty inside it ends 1.000000000003 times above.
A parameter chosen on a smaller problem
Inside a hybrid method the regularisation parameter is chosen on a 25×24 problem rather than a 64×64 one. The rule that reads a residual transfers exactly; the rule that reads a trace is biased by exactly two grid steps at twenty-four steps and one at forty, at every noise level from 10% to 0.1%.
Where the flop count stopped predicting the time
Every cost claim in the other eleven fields is a count of arithmetic, and those counts have not decided which of two implementations is faster for about thirty years. A blocked and an unblocked elimination perform the identical multiplications in a different order, return a factorisation identical to the last bit, and move a factor of two different amounts of data. What is counted here is words moved between a fast memory and a slow one, with the size of the fast memory named on every figure — because nothing in the model is optimal until it is.
The same arithmetic at a different price
A blocked and an unblocked elimination perform 72,568 operations each — the same operations, associated differently — choose the same pivots, and return a factorisation identical to the last bit: ‖PA − LU‖/‖A‖ = 4.487946226420872·10⁻¹⁶ in both. One of them moves 41,332 words between fast and slow memory and the other moves 19,476.
A block size is a property of the machine
Three lines of counting say the best block size is √(M/3). Scanned over every integer at five fast memories, the measured optimum is √M − 2 — exactly, at all five. The count has the right scaling and the wrong constant, low by a factor of 1.56, and the wrong form: the answer is affine in √M rather than proportional to it.
A reduction that changes the order
A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.
The 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.
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.
Memory bought with messages
Holding four copies of the data instead of one is supposed to cut a matrix multiplication's communication by √4. Measured on a machine of 64 processors it costs 14% more traffic; at 576 it saves 44%, which is 72% of what the law promises. The memory is exactly four times, and that part is not asymptotic.