Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Off-diagonal rank — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, in orderσ ⁄ σ₁eight digitsan independent draw per entry1 ⁄ r, falling to the floora cliff, and a control1/r rank at 10⁻⁸5log r rank at 10⁻⁸5noise rank at 10⁻⁸96σ₂ ⁄ σ₁0.024σ₆ ⁄ σ₁2.8·10⁻⁹the block has full rankand five useful columns

    A block nobody can call sparse

    A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.

    part 1 · hierarchy
  2. 0369121501122334455digits asked for, −log₁₀ εcolumns keptwhat the geometry promiseswhat the matrix costsa rank is a number of digitscolumns a decade0.55bound, a decade3.3rank at 10⁻⁸5bound at 10⁻⁸28q0.5the shape is rightand the constant is not

    A rank that is a number of digits

    Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.

    part 2 · hierarchy
  3. 45678903691215log₂ of the points a sidecolumns above 10⁻⁸two intervals that touchtwo that do notthe measurement that is a null resultadmissible, n = 325admissible, n = 2565touching, n = 329touching, n = 25613stored ⁄ dense at largest0.039the rank belongs to the geometryand not to the sampling

    The size the rank does not notice

    Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.

    part 3 · hierarchy
  4. -1012301020304050log₂ of the segment lengthcolumns above 10⁻⁸cos(κr) ⁄ r at κ = 401 ⁄ r, same pointsthe case with no answerq, held fixed0.51/r at L = 0.561/r at L = 86cos(κr)/r at L = 0.512cos(κr)/r at L = 853the geometry did not moveand the rank did

    The kernel with nothing to compress

    Hold the geometry fixed at q = ½, fix the wavelength, and scale the picture up by sixteen. A smooth kernel needs six columns at every scale. An oscillatory one needs twelve, sixteen, twenty-two, thirty-three, fifty-three, and there is no scale at which it stops.

    part 4 · hierarchy
  5. source segment length btarget length a⅛¼½124⅛¼½1247889888991011108912131313910131618188111318222681013182633columns at 10⁻⁸square block, side 433⅛ against 48square block, side ⅛7κ = 40, separation ratio ½ throughoutthe smaller side sets the rank

    The smaller cluster sets the rank

    An oscillatory kernel block between two equal clusters needs a rank that grows without limit as they grow. Make the clusters unequal at the same separation ratio and the rank stops following the larger one: a target an eighth long against a source of four needs 8 columns where the square block of side four needs 33, and a target a third long against a source 130 wavelengths long needs 11. What decides the rank is the product of the two lengths over their distance — the Fresnel number, the count optics gives for the waves two apertures can exchange.

    part 5 · hierarchy

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