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title	tags	summary
Numerical method and error	numerical, error, iteration, conditioning, convergence	The behaviour of a computed number that no exact answer exists for, and the differences between being wrong, being imprecise, and being unstable.

Numerical method and error

Most quantities in orbital work have no closed form. The period of an elliptical orbit under a perturbed field, the argument of perigee after a drag history, the instant a satellite enters a cone of view: these are defined by an equation that must be solved, not evaluated. Solving it numerically replaces one question, which has an answer, with another, which does not.

Definition

Three different things go wrong when a number is computed, and they are worth keeping apart:

Truncation error. The method computes an approximation to the exact answer by stopping early or by replacing a function with a simpler one.
Rounding error. The representation of the number itself is finite. A double-precision value holds about fifteen to seventeen significant decimal digits; beyond that the digits are noise.
Instability. A small error in the input, or in an intermediate step, grows into a large error in the answer.

The first two are bounded and can be estimated. The third is a property of the problem and cannot be estimated from the arithmetic; it has to be recognised.

Mathematics

Iteration

A fixed point is reached by repeated substitution, x becoming g(x). It converges when the derivative of g at the root is small in magnitude:


\lvert g'(x) \rvert < 1 \quad \text{(convergence)}, \qquad
\lvert g'(x) \rvert > 1 \quad \text{(divergence)}

Kepler's equation, M = E - e sin E, has no closed form solution for E. It is solved by Newton's method,


E_{n+1} = E_n - \frac{E_n - e \sin E_n - M}{1 - e \cos E_n}

which converges quadratically near the root when e is not close to one. The iteration is fast and its rate depends on eccentricity; for a near-parabolic orbit the denominator approaches zero and a different formulation is needed.

A bisection is slower per step but cannot diverge, because it halves an interval that is known to bracket the root. Which is preferable depends on whether divergence or cost dominates.

Conditioning

The sensitivity of an answer to a perturbation of its input is the condition number. For a function f,


\kappa = \frac{\lvert f'(x) \rvert \, \lvert x \rvert}{\lvert f(x) \rvert}

A large condition number means a small input error becomes a large output error, whatever the method. Near a singular point the condition number diverges: computing the radius of an orbit from its semi-major axis and eccentricity is well conditioned, while computing the time to perigee from the same elements is not, because sqrt(1 - e^2) appears in the denominator and goes to zero as e approaches one. The problem is ill conditioned, not the arithmetic, and more digits do not help.

Truncation in step size

An integration over a fixed step accumulates a global error proportional to the step, and a local error proportional to the step squared. Halving the step therefore improves the answer by a factor of two, not four, which is why "just halve the step" is an expensive way to buy accuracy.

For a method of order p, the global error falls as the step to the power p. A fourth order Runge-Kutta gives h^4, which is why it is common: the cost per step is four function evaluations, and halving the step costs eight times as much for a factor of sixteen less error.

Unit conventions

Tolerance is relative to a scale of the answer, never absolute. An absolute tolerance of 1e-9 on a radius of 7000 km is far tighter than the representation can hold, and the iteration will not terminate.
The scale used for the relative tolerance should be the magnitude of the quantity, not the quantity's units, so that switching between kilometres and metres does not change the answer.

Applicability

A method that converges is not necessarily accurate. Newton's method on Kepler's equation converges to the root of the equation as posed; if the equation omits a perturbation the solution is the root of the wrong problem, and it will converge confidently.

Iteration also fails silently when it stops on a loose criterion. Reporting the number of iterations rather than only the answer lets a caller see that a result was reached in the expected number of steps.

The three failure modes are separable in practice: rounding shows up as a floor on achievable accuracy, truncation as an error that shrinks when the method is refined, and instability as an error that grows.

Standards

Newton, De analysi per aequationes numero terminas infinitas, 1669
Higham, Accuracy and Stability of Numerical Algorithms, 2nd edition, 2002
Quarteroni, Sacco, Saleri, Numerical Mathematics, 2nd edition, 2007

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