Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Householder — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 2 · orthogonality
  2. 8×8, κ = 10⁸direction at ε = 10⁻², ‖QᵀQ − I‖1.5·10⁻¹⁵scalar at ε = 10⁻², ‖QᵀQ − I‖0.065direction at ε = 10⁻², residual0.00510⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²relative perturbation εdeparture‖QᵀQ − I‖, scalar‖A − QR‖/‖A‖, scalar‖A − QR‖/‖A‖, direction‖QᵀQ − I‖, directionflat line: a different reflection is still a reflectionsloped line: a non-reflection is not

    One number that has to be right

    Householder's orthogonality was called structural: a reflection is built from a unit vector, so rounding the vector names a different reflection rather than a broken one. Tested by breaking it, the claim is narrower and sharper. Perturb every component of the reflector by a relative 10⁻², and ‖QᵀQ − I‖ stays at 1.5·10⁻¹⁵ while the factorisation moves to 5·10⁻³. Perturb the one stored scalar by the same amount and ‖QᵀQ − I‖ is 6.5·10⁻². The structure is one degree of freedom, and the departure is four times its relative error.

    part 3 · orthogonality
  3. what a block storesnumbers in T, block of 11numbers in T, block of 16136‖QᵀQ − I‖ there3.9·10⁻¹⁵10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶block size‖QᵀQ − I‖124816T perturbed by 10⁻⁶T as computed1, 3, 10, 36, 136 computed numbers in the triangleand a factor of five in what they produce

    A triangle where the scalar was

    Every level-3 QR assembles a block of reflectors into Q = I − Y T Yᵀ, and T is computed by a recurrence whose inputs are its own previous columns. A block of sixteen carries 136 computed numbers where sixteen separate reflections carry sixteen. The orthogonality it produces is 3.9·10⁻¹⁵ against the single reflector's 7.8·10⁻¹⁶ — a factor of five for a hundred and thirty-six times as many things that have to be right.

    part 4 · orthogonality

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