Numerical Linear Algebra
where the algebra stops being the arithmetic
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Ladders — page 3

A field says what an essay is about. A ladder says what else there is to say about it — the distinct arguments that stand against one idea, from the one that introduces it to the one that assumes all the others.
1234567891010⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1indexsingular valuecutoff, σ₁ · 10⁻¹⁰numerical rank 10gap 8.2·10⁶an opiniontrue rank 410×10, built with 4 nonzero valuesrank is a decision

Rank is a decision

A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.

1 rung · spectra 3000000120000-21000000-0.5-1.500001.5-0.5000000-2T = ZᵀAZthe highlighted boxes each hold one conjugate pair, and no real rotation removes themthe form, and that it is one‖A − ZTZᵀ‖/‖A‖1.8·10⁻¹⁵‖ZᵀZ − I‖2.5·10⁻¹⁵worst eigenvalue error2.7·10⁻¹⁵surviving subdiagonal26×6, spectrum chosen before the matrix was builtquasi-triangular is as far as the reals go

The form a real matrix can reach

A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.

1 rung · spectra 10²10².³10².⁵⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁶10⁻¹10⁻⁰.⁵1rows in the sketchworst relative distortiondimension 64dimension 2565 seeds per point, band is best to worstthe dimension does not appear

The dimension does not appear

A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.

1 rung · randomised 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.

1 rung · iterative 10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁸10⁻⁶magnitudespacing to the next numberthe smallest normalgradualflush to zerosmallest normal6.1·10⁻⁵smallest subnormal6·10⁻⁸octaves of subnormals10pairs that lie under FTZ10the spacing stops halving and stays putwhich is what makes x − y = 0 mean x = y

The numbers below the smallest one

Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.

1 rung · arithmetic 10¹10²10³10⁴10⁵10⁶10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹number of terms addedrelative error against the exact sumin orderin a treecompensatedbinary32 · terms are 1/icompensated: 3·10⁻⁸

The order they are added in

Addition is associative in the algebra and is not associative in the arithmetic. The same million numbers, added in a different order, give answers that differ in the third significant figure — and the fix is not a wider float, it is a different order.

1 rung · arithmetic 12345678910⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1rank k of the approximation‖A − Aₖ‖measured, 2-normσₖ₊₁, from theorymeasured, Frobeniusthe first two agreeto 4.3·10⁻⁹worst |‖A−Aₖ‖₂ − σₖ₊₁| / σₖ₊₁4.3·10⁻⁹worst Frobenius discrepancy4.3·10⁻⁹κ = 10⁹; 30 random rank-3 matrices, none closerthe error is σₖ₊₁

The best approximation there is

The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.

1 rung · spectra 10¹10²10²size ncondition numberlimit 81measureda limit, as a fraction of itselfreached at n = 1280.99still to go0.011κ at n = 8, as a fraction0.52every point is below the line and none of them is on itthe limit is not a value

A limit the matrix never reaches

Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.

1 rung · structure

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