Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Nonlinear eigenvalue — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. -2-101234-5-3-1135real partimaginary part4 insidea countable spectruminside the contour4drawn12existing∞worst branch residual1.6·10⁻¹⁵there is no last eigenvalueso the question has to change

    A problem with infinitely many eigenvalues

    Let the matrix depend on λ through something that is not a polynomial and three things stop being true at once. There is no linearisation, there is no characteristic polynomial, and "compute the spectrum" is not a request that can be granted — the only finite question is how many eigenvalues are inside this circle.

    part 1 · polynomial
  2. 123456024681012moments Krank of the block Hankel12 eigenvalues insidethe ceiling K·ℓthe ranka ceiling with a knob on itprobes2eigenvalues inside12rank at K = 12rank at K = 612solves, at every K512one ceiling per probeand K of them per moment

    A ceiling with a knob on it

    A contour method returns at most as many eigenvalues as its probe block has columns, and the object that comes back does not distinguish that from having found everything. One line of the derivation multiplies the ceiling by a number the caller chooses, and it costs no extra solves at all.

    part 2 · polynomial
  3. 123456110²10⁴10⁶moments Kκ of the block Hankelintegrated in zin (z − c)/ρone division per pointradius6unscaled, K = 62.3·10⁶scaled, K = 61750the factor between1291the ceiling rises by Kand so did the conditioning

    The conditioning that rises with the ceiling

    Higher moments multiply a contour method's ceiling by K and grade its block Hankel over ρ to the 2K, so the two knobs are the same knob. One division per quadrature point separates them, and the measurement of what it is worth grows from twenty to twenty thousand.

    part 3 · polynomial
  4. -6-4-20246-6-4-20246real partimaginary partan infinite spectrum, finitely askedinside the contour12probes4moments used3values returned24worst against the closed form6.2·10⁻¹⁵infinitely many eigenvaluesand a question with an answer

    Where a contour's budget should go

    A contour method's ceiling is the number of probes times the number of moments, and the moments are free in solves while the probes are not. Four ways of reaching one ceiling come out four orders apart, the ordering is not monotone, and what separates the best two is not the usual draw but the unlucky one.

    part 4 · polynomial

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