Matpic
mathematics, in pictures
Concept

Polynomial approximation

Replacing a function by a polynomial that agrees with it in some stated sense. The sense may be agreement at a point to high order, exact agreement at several points, or the smallest possible worst error across an interval, and a polynomial optimised for one of those loses badly under another.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

analysis · Taylor series What the degree-5 sum costs, and what the bound claims. The error of the degree-5 Taylor polynomial of sin x against x, on a logarithmic scale, with Lagrange's bound drawn above it. The bound exceeds the error by a factor of 1.8 at the right-hand end.

An error with an unknown in it

Taylor's theorem does not say a partial sum is close to anything. It says the error is one more derivative evaluated somewhere nobody can name, and everything the theorem is worth comes from what happens when that somewhere is replaced by the worst case.

analysis · Taylor series 1/(1 + x²), expanded about 1.2. 1/(1 + x²) with Taylor sums of degree 2, 6, 14 about x = 1.2 rather than about zero. The interval they converge on reaches 1.562 either side of the centre.

The centre is a choice

A Taylor series is nearly always written about zero, and nothing about the construction prefers zero. Moving the centre moves the interval the series works on, and moving it repeatedly walks the function into places its first series could never reach.

analysis · Taylor series 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 ln(1 + x) past its radius, with a denominator allowed. ln(1 + x), its Taylor sum of degree 8, and its Padé approximants of order 2 and 4. At x = 3 the Taylor sum is out by 5.95e+2 and the highest-order approximant by 2.97e-4.

A denominator that reaches past the radius

The Taylor series of ln(1 + x) is useless beyond x = 1 however many terms it is given. The same coefficients spent on a numerator and a denominator converge at x = 3, and at x = 100, because a polynomial cannot imitate a singularity and a quotient of two polynomials can.

analysis · Taylor series Where interpolation through 11 nodes magnifies an error. Lebesgue functions of equispaced, Chebyshev and optimal nodes of degree 10; constants 29.900, 2.489, 2.052.

The best nodes have no formula

Interpolating through n points magnifies any error in the data by at most the Lebesgue constant of the points. Chebyshev's points keep it near (2/π)·log n; stretching them to the ends of the interval brings it within two hundredths of the best possible; and the best possible points, characterised in 1978 by having every bump of the error curve the same height, have never been given a formula.

analysis · Taylor series

Named alongside it

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

ConvergenceTaylor seriesRadius of convergenceAnalytic functionApproximationAnalytic continuationComplex numbersDerivativeInterpolationRemainderSingularityBound

All concepts