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 2 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

Named alongside it

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

Analytic function Approximation Convergence Convergence rate Derivative Fixed point Interpolation Iteration Linear convergence Multiplier Newtons method Polynomial approximation

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