Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Trace estimation — where it appears

Computing the trace of a matrix that is never formed, by averaging quadratic forms against random probes. The variance depends on the distribution the probes are drawn from rather than on the matrix, which makes that choice a decision rather than a detail.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

1112131415193111.365129.731148.096166.461probes takenrunning estimate of the tracenormal±1one probe, no errorthe exact trace99±1 variance, this matrix0±1 variance, rotated57normal variance545the same spectrum in a general basiscosts the ±1 probe its whole advantage

Counting what cannot be looked at

The trace is n additions and one of the most expensive quantities in the subject to estimate, because the matrices whose trace is wanted are never stored. Hutchinson's estimator is unbiased with one line of algebra — and its variance depends on which random vector is used, by a factor that is a property of the matrix, and on a diagonal matrix one choice is exact from the first probe and the other is not.

randomised · trace estimation 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.

Contour integralNonlinear eigenvalue problemQuadratureArgument principleArithmetic costCancellationComplex arithmeticConvergence rateGeneralised cross validationHutchinson's estimatorInfluence matrixKrylov subspace

All concepts