Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Contour integral — where it appears

An integral taken around a closed curve in the complex plane and evaluated by quadrature at points on it. For an analytic integrand the trapezoidal rule converges geometrically rather than algebraically, which is what makes contour methods for eigenvalues practical.

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

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

-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 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 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

Named alongside it

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

Nonlinear eigenvalue problemQuadratureArgument principleComplex arithmeticTrace estimationArithmetic costConvergence rateExact ground truthKrylov subspaceLambert wMatrix polynomialNewton iteration

All concepts