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Concept

Error analysis

Accounting for how far a computed answer is from the exact one, and where the discrepancy came from. It separates what the mathematics guarantees from what the arithmetic delivers, which are routinely different: a construction can be exact on paper and lose every significant digit in floating point.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

How fast an iteration arrives. The distance from the fixed point plotted against the step number on a logarithmic vertical axis, for two ordinary iterations and for Newton's method, whose curve bends downward.

How fast the staircase arrives

The slope at a crossing decides whether an orbit reaches it. The same number decides how fast — and when the slope is zero the arithmetic changes kind, from a fixed factor per step to a doubling of the correct digits.

dynamics · Iteration 11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350.

The points that ruin the fit

A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.

analysis · Taylor series A series that gets better, then worse. The error of Euler's series against the number of terms kept, at x = 0.05 and 0.1 and 0.2. At 0.05 the error falls to 1.1e-8 at 20 terms and then climbs without limit. At 0.1 the error falls to 1.8e-4 at 10 terms and then climbs without limit. At 0.2 the error falls to 1.8e-2 at 5 terms and then climbs without limit.

A series that converges nowhere

Expand Euler's integral in powers of x and the coefficients are the factorials, so the series converges at no x but nought. Stopped at its smallest term it still computes the integral to within about e^(−1/x) — and every term added after that makes the answer worse.

analysis · Taylor series How many steps Newton's method needs on (z − 1)²(z + 1). A square of the complex plane shaded in bands by the number of iterations a root-finding method needs from each starting point, darker meaning slower, with the roots marked.

A double root halves the error instead of squaring it

Near an ordinary root, Newton's method squares its error at every step and a handful of steps reach full precision. Near a double root it only halves the error — twenty steps where five would do, and a ceiling of about eight correct digits that no number of steps can break through. Doubling the step repairs the double root and ruins the simple one.

dynamics · Newton basins The best degree-3 polynomial to eˣ: its error touches its largest size 5 times. The error curve of the best uniform polynomial approximation of degree 3 to eˣ on the interval from −1 to 1. It reaches its maximum size 5.528 × 10⁻³ at 5 points, alternately above and below, at x = -1.000, -0.682, 0.050, 0.732, 1.000.

The error that keeps coming back to its worst

Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.

analysis · Taylor series

Named alongside it

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

ApproximationBoundConvergenceConvergence rateDerivativeInterpolationIterationNewtons methodRadius of convergenceTaylor seriesAnalytic functionAsymptotic series

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