Numerical Linear Algebra
where the algebra stops being the arithmetic
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The thread: Exact ground truth — page 3

Essays 49 to 50 of the 50 on this theme, in the same order.
1234567810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, largest firstrelative errorone-sided Jacobizero-shift QRshifted QReigenvalues of BᵀBthe same four routes, reversedσ_min, exactly5.2·10⁻²⁶Jacobi's error on it0.015sweeps, zero shift400sweeps, shifted16a method is not accuratea method on a matrix is Eigenvalues, singular values, rank

Accurate is not a property of a method

A bidiagonal matrix whose every entry is 1 or 4096 has singular values spanning thirty decades. On it, the method recommended for small singular values loses the small one by one and a half per cent, the sweep with the theorem behind it does not converge at all, and the shift the theorem is a warning about gets every value to 5·10⁻¹⁶. Nothing there contradicts the theory.

 -4-2.35371-0.7074290.9388572.585144.231435.8777102468101214shift σν(σ)an answer that is an integershifts2000disagreements0eigenvalues14steps14the marks are a Jacobi decompositionand the staircase never saw one Eigenvalues, singular values, rank

An eigenvalue count that cannot be slightly wrong

Every spectral computation on this site returns floats with errors in them. Counting eigenvalues below a shift by the signs of an unpivoted elimination returns an integer, and an integer cannot be 6.9999999997 — so the answer is exactly right, or wrong by a whole eigenvalue, and where the second happens is a band of measurable width.

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