Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Separable expansion — where it appears

An approximation of a two-variable kernel as a sum of products of a function of one variable with a function of the other. Truncated at p terms it is a rank-p matrix written down without any linear algebra, and it converges when the two clusters are far enough apart.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

hierarchy · off diagonal rank 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.

hierarchy · off diagonal rank -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.

hierarchy · off diagonal rank

Named alongside it

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

AdmissibilityKernel matrixNumerical rankOff diagonal rankAsymptotic analysisHierarchical matrixLow-rank approximationSpectral decayCounterexampleDiscretisationFourier modesLower bound

All concepts