Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Contour eigensolver — where it appears

A method that recovers the eigenvalues inside a region from integrals of the resolvent around its boundary. It works where there is nothing to linearise, and how many eigenvalues it can return is set by the size of the probe block and the number of moments used.

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

Also named here as moment — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Contour integralMomentNonlinear eigenvalue problemNumerical rankQuadratureRankComplex arithmeticCondition numberExact ground truthScaling

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