Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Greens function — where it appears

The inverse of a differential operator, read as the response at one point to a source at another. A discretised operator's inverse is its discrete counterpart, and away from the diagonal it is smooth, which is what makes blocks of it far from the diagonal compressible.

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

Also named here as separator — the same set of essays touches all of them, so they are one junction rather than several.

0481216202402468101214unknowns on the separatorcolumns above 10⁻⁸the same matrix, renumberedin the separator's own orderthe ordering the geometry hands overseparator 73separator 236renumbered, largest11the block, largest11share of the square stored0.52the fill is totaland it is not independent

The fill that is not independent

Eliminate both halves of a grid and what is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged. Its off-diagonal block is 11 by 12 and six columns describe it to eight digits. Renumber the separator and the same block needs all eleven.

sparsity · Fill 02468101210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹singular value, in orderσ ⁄ σ₁eight digitsrenumberedin the separator's orderthe same cliff, from the other endσ₂ ⁄ σ₁0.062σ₄ ⁄ σ₁1.3·10⁻⁴σ₆ ⁄ σ₁1.5·10⁻⁸renumbered σ₄ ⁄ σ₁0.63rank at 10⁻⁸6the same entriesin two orders

The cliff behind the count

The fill's rank is an integer between three and six across every separator two dense half-eliminations can afford, and this field has already recorded that a handful of such integers cannot carry a law. The singular values underneath are real numbers. They say the cliff's first step is 23.0 at a separator of eleven, 19.1 at fifteen and 16.2 at twenty-three — and that a control with no differential operator behind it gives 14,672.

sparsity · Fill

Named alongside it

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

Kernel matrixNested dissectionNumerical rankOff-diagonal rankSeparatorAnisotropyDiagonal dominanceElimination orderFill-inFill-reducing orderingHierarchical matrixLow-rank approximation

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