Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Stopping criterion — where it appears

The test an iterative method applies to decide it has finished, computed from quantities the method can see rather than from the error.

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

015304560759010512010⁻²10⁻¹110¹steprelative sizeleast error: 20discrepancy stop: 7errorresidualthe knob is an integerleast error, at step20error there0.14error at step 1206the residual falls at every stepthe error turns and keeps rising

A parameter that counts steps

The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.

combination · iterative regularisation 21018263442500102030405060708090100110120130140150160170180190200210220230240250260270280290300310320330340significand bitsiterationspreconditioner roundedarithmetic roundedno preconditionersame bits, different casualtyerror, 3-bit preconditioner8.8·10⁻¹³error, 3-bit arithmetic0.16‖A − LLᵀ‖/‖A‖ of the factor0.083a direction may be roundeda measurement may not

The part of a solver that may be rounded

A preconditioner computed and applied with a three-bit significand still returns thirteen correct digits — it costs seventeen extra iterations and nothing else. Round the working arithmetic instead and the step count barely moves while the answer loses exactly the digits the format dropped.

combination · mixed precision

Named alongside it

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

Conjugate gradientsDiscrepancy principleFilter factorsForward errorHalf precisionIncomplete choleskyIterative regularisationKrylov subspaceMixed precisionPreconditioningReorthogonalisationResidual

All concepts