Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Moment — where it appears

An integral of the resolvent around a contour weighted by a power of the variable. Higher moments cost one multiplication each at points where the expensive solve has already been done, and each one multiplies how many eigenvalues can be recovered.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

polynomial · Nonlinear eigenvalue 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.

polynomial · Nonlinear eigenvalue -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.

polynomial · Nonlinear eigenvalue

Named alongside it

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

Contour eigensolverContour integralNonlinear eigenvalue problemNumerical rankQuadratureRankCondition numberExact ground truthComplex arithmeticRandom probeScaling

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