Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Nonlinear eigenvalue problem — where it appears

A problem in which the matrix depends on the eigenvalue through something that is not a polynomial. It has no linearisation and generally infinitely many eigenvalues, so the only finite question is how many of them lie inside a given region.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

-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.

polynomial · nonlinear eigenvalue 23456710⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²poles in the approximantresiduals and errorforward error‖T(λ)x‖‖T̃(λ)x‖one extra evaluationagainst the approximant1.5·10⁻¹²against the problem asked1.9·10⁻⁶forward error4.8·10⁻⁵‖g − r‖ there8.5·10⁻⁵the free residual is flatand the answer is not

The problem the solver was actually given

A linearisation is exact — it has the polynomial's eigenvalues, with their multiplicities, and the whole loss is arithmetic. A nonlinear eigenvalue problem does not offer that. Every algorithm replaces the function first, and the term that replacement contributes is committed before any number is rounded and appears in no residual.

polynomial · approximation before linearisation 11.31.61.92.210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10²log₁₀ quadrature pointsdistance from the true counthalf an eigenvaluean integer, eventuallytrue count2at 4 points2at 128 points2finest error1.4·10⁻¹³the integral is an integerand a rounding hides how far it was

The last digit is the cheapest

Every cost curve on this site has the same shape: the first digits are cheap and the last ones are not. One method inverts it. Doubling the work buys twice as many digits as the previous doubling did, so the price of a digit halves every time it is paid.

cost · nonlinear cost 110¹10⁻⁷10⁻⁵10⁻³distance from the branch point, λ + c|g − r|the eigenvaluesdegree 98 poleschosen, not computedreach0.06left end, from the cut0.2rational, worst8.5·10⁻⁴polynomial, worst0.0068linearisation size, both54committed before the solveand invisible to it

An error committed before the arithmetic

Before a nonlinear eigenvalue problem is solved, somebody says where they think the eigenvalues are. That sentence sets the accuracy of everything that follows by five orders, costs nothing to say, and cannot be revised once the approximation built on it is in hand.

polynomial · approximation before linearisation 11.31.61.92.210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10²log₁₀ quadrature pointsdistance from the true counthalf an eigenvaluean integer, eventuallytrue count2at 4 points2at 128 points2finest error1.4·10⁻¹³the integral is an integerand a rounding hides how far it was

Counting what is inside a circle

A trace of a matrix nobody wants to form, integrated around a contour, gives an integer — how many eigenvalues are inside. It converges exponentially, it is estimated with random probes, and the probe block is a ceiling that the answer does not mention.

randomised · trace estimation -10123456789-101λbranch point at −0.4the closed formwhich of these is an answereigenvalues returned36wanted6past the branch point6complex24worst against the closed form7.7·10⁻⁴all of them exactfor a problem nobody asked

The eigenvalues that are answers to nothing

A rational approximant of degree five turns a six-by-six problem into a thirty-six-by-thirty-six one, and thirty-six numbers come back. Six are the answer. The rest are exact eigenvalues of the approximant, lying where the function it approximates is not a real number at all.

polynomial · spurious spectrum 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 integralQuadratureExact ground truthApproximation before linearisationComplex arithmeticCondition numberRankRational approximationArgument principleBranch pointContour eigensolverLinearisation

All concepts