Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Discretisation — where it appears

The replacement of a continuous problem by a finite one on a mesh. Every quantity on this site that depends on it is worth checking against refinement, because a constant that grows with the mesh is a different claim from one that does not.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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 00.250.50.75100.250.50.751xuexactcentral differencesupwindthe oscillation is exact‖Ax − b‖/‖b‖ for the central answer4.6·10⁻¹⁸values outside [0, 1]16worst excursion0.52the dashed lines are 0 and 1, which the equation guaranteesno solver was involved

The stencil that is not symmetric

Past a cell Péclet number of exactly one — measured by bisection at 1.0000000000000002 — the central-difference solution of a convection–diffusion problem oscillates from point to point and leaves the interval the equation guarantees, at 16 of 31 grid points. It is the exact solution of its own linear system, to 4.6·10⁻¹⁸. No solver was involved.

iterative · convection

Named alongside it

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

AdmissibilityArtificial diffusionAsymptotic analysisConvection diffusionHierarchical matrixKernel matrixLow-rank approximationM matrixModel problemNon symmetric operatorNumerical rankOff diagonal rank

All concepts