What's new
Essays arrive in groups rather than one at a time, and a group usually opens up a subject the collection had not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.
15 August 2026
12 essays onmethods that were designed apart, structure, and the solver that cannot see it, where the flop count stopped predicting the time, eigenvalues, singular values, rank, the arithmetic underneath and iterating, instead of factorising
A parameter that counts steps
The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.
Methods that were designed apartFour knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.
Methods that were designed apartThe 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.
Methods that were designed apartAn 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.
Structure, and the solver that cannot see itThe circulant that cannot be indefinite
The previous essay found a preconditioner taking 117 steps against an unpreconditioned 59, because its smallest eigenvalue was −0.173. Average the two diagonals instead of choosing between them and the count is 7, 8, 9, 10, 10 across a factor of sixteen in size.
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.
Structure, and the solver that cannot see itTwo dimensions, and the cluster that thins
The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.
Eigenvalues, singular values, rankRestarting is a filter
A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.
Eigenvalues, singular values, rankAn eigenvalue one vector cannot see
A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.
The arithmetic underneathProving the answer is in the box
Every other method here computes a number and estimates how wrong it is. This one returns a verdict: there is exactly one solution in this box, or there is none, or — the honest third outcome — nothing can be said. Two of the three are proofs about infinitely many points from finitely many operations.
Iterating, instead of factorisingThe diffusion that makes the answer exact
Upwinding adds h/2 of artificial diffusion. Central differencing adds none. Add ε·ξ·Pe with ξ = coth(Pe) − 1/Pe and the computed solution is the exact one at every grid point, to 2.4·10⁻¹⁷ — at every Péclet number, on the problem it was derived from and on no other.
Iterating, instead of factorisingAggregating what the matrix calls strong
The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.
Before that
Everything published earlier, newest first. Titles only — the cards are on the full listing.
14 August 2026
12 essays onstructure, and the solver that cannot see it, where the flop count stopped predicting the time, regularisation, and the answer that is chosen, eigenvalues, singular values, rank, the arithmetic underneath and iterating, instead of factorising
- The matrix that is one row — structure, and the solver that cannot see it
- The same arithmetic at a different price — where the flop count stopped predicting the time
- When the answer is a choice — regularisation, and the answer that is chosen
- A block size is a property of the machine — where the flop count stopped predicting the time
- A limit the matrix never reaches — structure, and the solver that cannot see it
- Where the answer stops being in the data — regularisation, and the answer that is chosen
- A preconditioner that changes sign — structure, and the solver that cannot see it
- A reduction that changes the order — where the flop count stopped predicting the time
- Choosing without knowing — regularisation, and the answer that is chosen
- An eigenvalue that arrives twice — eigenvalues, singular values, rank
- A bound that is proved — the arithmetic underneath
- The stencil that is not symmetric — iterating, instead of factorising
12 August 2026
12 essays onthe arithmetic underneath, iterating, instead of factorising and eigenvalues, singular values, rank
- Eight bits, and a format that breaks the rules — the arithmetic underneath
- The coarse problem is a different problem — iterating, instead of factorising
- The gap decides the eigenvector — eigenvalues, singular values, rank
- A direction the smoother cannot see — iterating, instead of factorising
- The direction the error leans — the arithmetic underneath
- The plane survives what its vectors do not — eigenvalues, singular values, rank
- A coin flip that fixes the average — the arithmetic underneath
- Smoothing a whole line at once — iterating, instead of factorising
- Coarsening in one direction only — iterating, instead of factorising
- One exponent for thirty-two numbers — the arithmetic underneath
- The coarse grid the matrix chooses — iterating, instead of factorising
- A hierarchy with no grid behind it — iterating, instead of factorising
10 August 2026
12 essays onsparsity, and what elimination costs, iterating, instead of factorising, eigenvalues, singular values, rank and the arithmetic underneath
- Structure and stability stop being separable — sparsity, and what elimination costs
- A threshold between fill and growth — sparsity, and what elimination costs
- The error smoothing cannot reach — iterating, instead of factorising
- The form a real matrix can reach — eigenvalues, singular values, rank
- The other half of a format — the arithmetic underneath
- What the symbolic phase can only bound — sparsity, and what elimination costs
- A norm that overflows before it is a norm — the arithmetic underneath
- The same problem on a coarser grid — iterating, instead of factorising
- Two shifts that are never formed — eigenvalues, singular values, rank
- A condition number for one eigenvalue — eigenvalues, singular values, rank
- A rate that does not notice the size — iterating, instead of factorising
- The numbers below the smallest one — the arithmetic underneath
6–8 August 2026
34 essays onrandomised, and the guarantee that changes kind, elimination, and the swap, orthogonality, measured, eigenvalues, singular values, rank, two errors, and whose fault they are, sparsity, and what elimination costs, least squares, and the road not to take, iterating, instead of factorising and the arithmetic underneath
- A bound that holds with probability — randomised, and the guarantee that changes kind
- Elimination is a sequence of choices — elimination, and the swap
- Orthogonal is a number — orthogonality, measured
- Symmetry is worth more than precision — eigenvalues, singular values, rank
- The exact answer to a nearby problem — two errors, and whose fault they are
- The factor is not sparse — sparsity, and what elimination costs
- The projection and the right angle — least squares, and the road not to take
- The rate the condition number predicts — iterating, instead of factorising
- What a float can hold — the arithmetic underneath
- A small residual is not a small error — two errors, and whose fault they are
- An orthogonalisation nobody calls one — iterating, instead of factorising
- Cancellation takes the answer, not a digit — the arithmetic underneath
- Rank is a decision — eigenvalues, singular values, rank
- The dimension does not appear — randomised, and the guarantee that changes kind
- The order decides the memory — sparsity, and what elimination costs
- The road that squares the problem — least squares, and the road not to take
- The swap that is not optional — elimination, and the swap
- Two Gram–Schmidts — orthogonality, measured
- A reflection cannot stop being one — orthogonality, measured
- Changing the condition number on purpose — iterating, instead of factorising
- Randomisation does not create structure — randomised, and the guarantee that changes kind
- The best approximation there is — eigenvalues, singular values, rank
- The bound that is never attained — elimination, and the swap
- The condition number is an amplifier — two errors, and whose fault they are
- The order they are added in — the arithmetic underneath
- The valley with no bottom — least squares, and the road not to take
- Two ends of the same arrow — sparsity, and what elimination costs
- An answer that is known — two errors, and whose fault they are
- Buying the accuracy back — the arithmetic underneath
- The algorithm the libraries actually run — eigenvalues, singular values, rank
- The spectrum that predicts nothing — iterating, instead of factorising
- A rate that is known in advance — iterating, instead of factorising
- The form that makes it affordable — eigenvalues, singular values, rank
- Where the hardware went — the arithmetic underneath