Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Jacobi iteration — where it appears

6 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
0408012016020024010⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹iteration‖e‖_A / ‖e₀‖_Ameasuredκ bound119 steps40×40, spectrum spread evenly in logbound permits 1417

The rate the condition number predicts

Conjugate gradients converge at a rate governed by the square root of the condition number. That is a bound rather than an estimate, it is provable, and it is loose enough that provisioning iterations from it wastes nine out of ten.

iterative · krylov 051015202530354010⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹iteration‖r‖ / ‖b‖plain CGIC(0) CGwhat the preconditioner didκ(A)48κ(L⁻¹AL⁻ᵀ)5.1‖A − LLᵀ‖/‖A‖0.0832D Laplacian, n = 100√κ ratio predicts 3.07×

Changing the condition number on purpose

Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.

iterative · preconditioning 015030045060075090010⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹iteration‖r‖ / ‖b‖JacobiGauss–SeidelSOR ω=1.78closed form vs measuredρ Jacobi, exact0.99ρ measured0.99ρ Gauss–Seidel, exact0.981D Laplacian, n = 24ω optimal = 1.777

A rate that is known in advance

On the model problem, Jacobi contracts by cos(π/(n+1)) per step, Gauss–Seidel by its square, and optimally relaxed SOR by a number given in closed form. Three rates, all known before anything runs, and all measurable against what runs.

iterative · stationary 00.250.50.751-1-0.500.51mode frequency θ / πdamping factor per sweeppredicted±0.333measureda coarse grid seestwo routes to one factorsmoothing factor, scanned0.33smoothing factor, closed form0.33worst mode disagreement4.4·10⁻¹⁶63 interior points, one sweepthe left-hand end is what the coarse grid is for

The error smoothing cannot reach

One weighted Jacobi sweep multiplies every mode of the error by a number, and the number is a sine. Half the modes are cut by three or better, and the other half come back at 0.999 — which is not a failure of the method but the fact the whole of multigrid is built on.

iterative · multigrid 15731pointsfinest gridone unknown — the recursion bottoms out in a divisionthe coarse operator, two ways‖RA_hP − A_2h‖/‖A_2h‖10⁻¹⁸unknowns / finest grid1.7cycle cost, in fine sweeps12each coarse point reaches three fine ones½, 1, ½ — and the restriction is its transpose

The same problem on a coarser grid

Restriction, the coarse operator and interpolation are three matrices with nine distinct entries between them. Two of the three are each other's transpose, and their product with the fine operator is the coarse discretisation exactly — not approximately, entry for entry, at every level.

iterative · multigrid 10¹10¹.³10¹.⁶10¹.⁹⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰¹10².²00.250.50.751grid points nresidual reduction per stepJacobiGauss–SeidelV-cycleV-cycle spread, 8× in size0.003Jacobi at n = 1270.99work exponent, fitted0.079the dashed curve is cos(πh), Jacobi's closed formthe flat line is the whole method

A rate that does not notice the size

The V-cycle reduces the residual by a factor of ten a cycle at fifteen points and at a hundred and twenty-seven. Jacobi on the same four problems goes from 0.981 to 0.9978, climbing towards one. One of those is a constant and the other is an exponent, and that is the whole distinction the field turns on.

iterative · multigrid

Named alongside it

The objects these essays reach for when they reach for this one.

ResidualCondition numberConjugate gradientsModel problemFourier modesKrylov subspacePreconditioningSparsityThe V-cycleCholeskyConvergence factorConvergence rate

All concepts