Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Orthogonal invariant — where it appears

A quantity unchanged by multiplying a matrix by orthogonal factors on either side. The singular values are, the sparsity pattern is not, and which properties survive an orthogonal transformation is what decides what an algorithm may do.

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

the mirrorx, length 4.000Hx = (-4.000, 0)v = x − αe₁safe sign: α = −‖x‖, so v is formed from a sum and nothing cancelsunsafe sign: α = +‖x‖ gives ‖v‖ only 35.1% of ‖x‖ + ‖x‖ — the digits go‖HᵀH − I‖5·10⁻¹⁶‖Hx‖ − ‖x‖8.9·10⁻¹⁶second component2.2·10⁻¹⁶built from a unit vectororthogonality is structural

A reflection cannot stop being one

Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.

orthogonality · householder numbers stored, 256 × 256clustered, strong27,008clustered, weak24,064the dense matrix65,536shuffled, strong65,536shuffled, weak118,208the same matrix, twiceκ, clustered24κ, shuffled24‖A‖_F, clustered6140‖A‖_F, shuffled6140shuffled weak ⁄ dense1.8the compressibility is in the numberingand the numbering is not in the matrix

The same matrix, numbered twice

One symmetric permutation. The condition number is 24.3948 either way to eight digits and the Frobenius norm is 6.13996414·10³ either way to twelve. The partition that stored 27,008 numbers now finds no admissible pair anywhere and stores all 65,536, and the format that compresses regardless stores 118,208.

hierarchy · admissibility

Named alongside it

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

AdmissibilityCatastrophic cancellationCluster treeCondition numberElimination orderFill-reducing orderingGaussian eliminationGram–SchmidtHierarchical matrixHouseholder reflectionJacobi's eigenvalue methodMatrix norm

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