Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Extrapolation — where it appears

Predicting a value outside the data a fit was made from — here, the next member's answer from the answers already computed. The coefficients alternate in sign, so the fit amplifies whatever error the stored answers carry, and the degree worth using is the one whose truncation term is larger than that amplified error.

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

10⁻²10⁻¹10¹10²drift between consecutive membersinner steps over the whole runthe previous answer — a warm startthe parabola through the last threethe line through the last twodegree zero is a choicemembers per factorisation4warm, at drift 0.02160linear, at 0.0212quadratic, at 0.0226the factor13drifts where warm fails2one more term in the extrapolationand a factor of thirteen

A warm start is degree zero

The previous answer used as the next member's starting point costs 160 inner steps over twenty members. The line through the last two answers costs 12 — a factor of thirteen, for three vector operations and no extra storage. The parabola through the last three costs 26, which is worse than the line and better than the point.

sequence · Sequence of solves 10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶050100150200250relative residual tolerance of every solveinner steps over twenty membersprevious answerline, last twoparabola, last threea straight path, twenty members, drift 0.02tolerance 10⁻⁶: previous answer62tolerance 10⁻⁶: line11tolerance 10⁻⁶: parabola17tolerance 10⁻¹⁰: previous answer160tolerance 10⁻¹⁰: line12tolerance 10⁻¹⁰: parabola14tolerance 10⁻¹⁴: previous answer261tolerance 10⁻¹⁴: line17tolerance 10⁻¹⁴: parabola23tighter to the lefteach point is a whole run of twenty

A straight path has nothing for a parabola to fit

The line through the last two answers beat the parabola through the last three on a drifting sequence, and the reason offered was that the parabola amplifies the stored answers' error. Tightening the solve tolerance from 10⁻⁶ to 10⁻¹⁴ should have reversed that, and it does not: the line needs 7 to 18 inner steps over twenty members at every tolerance and the parabola 14 to 23. The sequence's roots move along a straight line, so the line is exact and there is nothing else to fit. Bend the path by a part in ten thousand and the parabola wins below 10⁻⁹, by 105 steps to 75 at 10⁻¹⁴.

sequence · Sequence of solves 010203040run, by bend then tolerancesteps beyond the better fixed degreestraightbend 10⁻⁵bend 10⁻⁴bend 0.001bend 0.01bend 0.1● second difference ■ residual at both ○ wrong fixed degreethirty runs: steps beyond the better degreesecond difference: runs matching the better degree, of 3026second difference: worst excess, steps5residual at both: runs matching the better degree, of 3029residual at both: extra residuals, in steps4.5wrong fixed degree: worst excess, steps35zero is the better degree, known in advanceeach group of five is one bend

The degree the history chooses

A sequence of solves can start each member from the line through its last two answers or the parabola through its last three, and which is better depends on how much its path bends — which a code does not know. Over thirty runs of bend and tolerance, always taking the line costs 456 inner steps more than the better choice; always taking the parabola costs 33. A free rule reading the stored answers closes that to 11. A rule that evaluates the residual at both starts picks the better one on 29 runs of 30, and pays 135 steps for the evaluations.

sequence · Sequence of solves

Named alongside it

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

Cholesky factorisationFlop countNewton iterationStopping criterionWarm startExact ground truthCondition number

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