Generator

Partial sums of sin x

A generator in the analysis library, called 37 times across 6 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

taylor is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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.

ln(1 + x) past its radius, with a denominator allowed

The poles of the approximants, lined up along the cut

The same coefficients, spent two ways, at x = 3

A series that gets better, then worse

Sums of a series that converges nowhere

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the [2/2] approximant reproduces coefficient 0 of the series exactly ×144
with 2 terms the sum is within the first omitted term at x = 0.013 ×120
with 1 terms at x = 0.05 the error is below the first term left out ×90
the polynomial passes through node 1 ×21
the size of coefficient 1 of 1/(1 − 2x) is what the exact value says ×20
the size of coefficient 1 of 1/√(1 − 4x) is what the exact value says ×20
the size of coefficient 1 of Euler's series is what the exact value says ×20
Laguerre node 1 of 4 is a root ×12
pole 1 of [3/4] is minus one over a Gauss–Laguerre node ×12
pole 1 of [4/4] is minus one over a Gauss node ×12
the [2/2] approximant exists with a denominator starting at one ×12
the degree-1 sum matches to order 1 at the expansion point ×7
the degree-2 sum about 1.2 matches the function to order 2 there ×6
as x shrinks, the point settles at 1/(n+2) for degree 1 ×4
the integral Euler's series belongs to agrees two ways at x = 0.05 ×4
the [2/2] denominator does not vanish on the drawn interval ×3
the best error at x = 0.05 has the size √(2π/x)·e^(−1/x) ×3
the best number of terms at x = 0.05 is within two of where the terms are smallest ×3
the degree-2 sum about -1.4 matches the function to order 2 there ×3
well past the best at x = 0.05, every term added makes the error larger ×3
at x = 0.5, 4 terms lose to 2 ×2
between 3 and 41 nodes ×2
between one and three orders, each between 1 and 12 ×2
close to nought, 4 terms beat 2 ×2
a degree between 1 and 20 ×1
a function that is already a quotient of polynomials is reproduced exactly ×1
a higher-order approximant is at least as close at the probe point ×1
an amplification factor is never below one, since the nodes reproduce themselves ×1
and by the largest count drawn it is orders of magnitude worse ×1
and by the last coefficient they are within ten per cent of it ×1
and by the last they are past k/e, so no radius but nought is possible ×1
and every approximant is closer than the last ×1
and it dwarfs anything in the middle of the interval ×1
and the terms are smallest near 1/x ×1
between 6 and 40 terms ×1
between one and five degrees, each between 1 and 8 ×1
between one and five series ×1
between one and four degrees ×1
between one and four singularities, each a point of the plane ×1
between one and four term counts, each between 1 and 20 ×1
between one and six centres ×1
between one and three values of x, each between 0.04 and 1 ×1
between three and fourteen node counts, each between 3 and 25 ×1
Chebyshev nodes keep the error small across the whole interval ×1
each drawn pole is a root of the denominator ×1
each new centre lies inside the previous disc, so its coefficients are known there ×1
each node is the projection of its angle ×1
equally spaced nodes never amplify less than Chebyshev ones ×1
every pole lies on the cut, at or beyond −1 ×1
every pole lies on the negative real axis ×1
every term count is a whole number between 1 and 40 ×1
inside the new interval more terms help ×1
inside the radius, more terms help ×1
ln(1+x) is centred where it is defined ×1
ln(1+x) is expanded inside its own radius ×1
no centre sits on a singularity ×1
no division by nought ×1
no rational has a denominator of nought ×1
no singularity lies strictly inside a disc ×1
outside it they hurt ×1
outside it, more terms hurt ×1
past the radius every Taylor error is larger than the last ×1
past the radius, the Taylor sum of higher degree is the worse one ×1
the bound is a bound ×1
the centre is inside the drawn domain ×1
the Chebyshev end gaps are several times smaller than the middle one ×1
the drawn domain is an interval ×1
the equally spaced constant grows with every step ×1
the error never exceeds the bound Lagrange's form puts on it ×1
the evenly spaced nodes have one gap throughout ×1
the function has a closed form to compare against ×1
the function has a closed form to draw ×1
the function is one the figure knows ×1
the function is one whose derivatives the figure knows ×1
the interval drawn is inside the function's radius ×1
the interval drawn is not empty ×1
the interval is between 0.4 and 8 wide ×1
the largest order is between 3 and 12 ×1
the last coefficient is between 8 and 200 ×1
the mode draws the functions whose derivative is monotone ×1
the mode of the taylor family is one of sums, radius, remainder, xi, centre, discs, interp, lebesgue, nodes, pade, poles, paderate, root, asymptotic, asymsums ×1
the nodes are equally spaced or Chebyshev ×1
the numerator's degree is a whole number close to the denominator's ×1
the point is one where the function is defined ×1
the point moves steadily in one direction as x grows ×1
the pole picture is drawn for ln(1 + x), 1/(1 + x²), eˣ or Euler's series ×1
the polynomial has the degree it claims ×1
the probe point is inside the domain ×1
the radius is the distance to the nearest singularity ×1
the roots for 1/(1 − 2x) close on one over the distance to the singularity ×1
the roots for 1/√(1 − 4x) close on one over the distance to the singularity ×1
the roots for Euler's series grow at every step ×1
the series is one of log, runge, exp, geom2, central, lacunary, euler ×1
the smallest gap is at one of the ends ×1
the span is between 0.2 and 1.5 ×1
the tail of the series is the difference between the function and its partial sum ×1
the Taylor degree is between 1 and 40 ×1
the unknown point lies strictly between nought and x ×1
the window reaches past −1 and past 0.5 ×1
with equally spaced nodes the worst error is out near the ends ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

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

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

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

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

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

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.

The whole library · What the figures prove