121 lines
5.0 KiB
Markdown
121 lines
5.0 KiB
Markdown
---
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title: Numerical method and error
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tags: numerical, error, iteration, conditioning, convergence
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summary: The behaviour of a computed number that no exact answer exists for, and the differences between being wrong, being imprecise, and being unstable.
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---
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# Numerical method and error
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Most quantities in orbital work have no closed form. The period of an
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elliptical orbit under a perturbed field, the argument of perigee after a drag
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history, the instant a satellite enters a cone of view: these are defined by an
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equation that must be solved, not evaluated. Solving it numerically replaces
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one question, which has an answer, with another, which does not.
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## Definition
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Three different things go wrong when a number is computed, and they are worth
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keeping apart:
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- **Truncation error.** The method computes an approximation to the exact
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answer by stopping early or by replacing a function with a simpler one.
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- **Rounding error.** The representation of the number itself is finite. A
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double-precision value holds about fifteen to seventeen significant
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decimal digits; beyond that the digits are noise.
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- **Instability.** A small error in the input, or in an intermediate step,
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grows into a large error in the answer.
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The first two are bounded and can be estimated. The third is a property of the
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problem and cannot be estimated from the arithmetic; it has to be recognised.
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## Mathematics
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### Iteration
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A fixed point is reached by repeated substitution, $x$ becoming `g(x)`. It
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converges when the derivative of $g$ at the root is small in magnitude:
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$$
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\lvert g'(x) \rvert < 1 \quad \text{(convergence)}, \qquad
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\lvert g'(x) \rvert > 1 \quad \text{(divergence)}
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$$
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Kepler's equation, $M = E - e\sin E$, has no closed form solution for $E$. It
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is solved by Newton's method,
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$$
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E_{n+1} = E_n - \frac{E_n - e \sin E_n - M}{1 - e \cos E_n}
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$$
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which converges quadratically near the root when `e` is not close to one. The
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iteration is fast and its rate depends on eccentricity; for a near-parabolic
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orbit the denominator approaches zero and a different formulation is needed.
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A bisection is slower per step but cannot diverge, because it halves an
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interval that is known to bracket the root. Which is preferable depends on
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whether divergence or cost dominates.
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### Conditioning
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The sensitivity of an answer to a perturbation of its input is the
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**condition number**. For a function $f$,
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$$
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\kappa = \frac{\lvert f'(x) \rvert \, \lvert x \rvert}{\lvert f(x) \rvert}
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$$
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A large condition number means a small input error becomes a large output
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error, whatever the method. Near a singular point the condition number
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diverges: computing the radius of an orbit from its semi-major axis and
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eccentricity is well conditioned, while computing the time to perigee from the
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same elements is not, because $\sqrt{1 - e^2}$ appears in the denominator and
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goes to zero as `e` approaches one. The problem is ill conditioned, not the
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arithmetic, and more digits do not help.
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### Truncation in step size
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An integration over a fixed step accumulates a global error proportional to
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the step, and a local error proportional to the step squared. Halving the
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step therefore improves the answer by a factor of two, not four, which is why
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"just halve the step" is an expensive way to buy accuracy.
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For a method of order $p$, the global error falls as the step to the power
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$p$. A fourth order Runge-Kutta gives $h^4$, which is why it is common: the
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cost per step is four function evaluations, and halving the step costs eight
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times as much for a factor of sixteen less error.
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## Unit conventions
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- Tolerance is relative to a scale of the answer, never absolute. An absolute
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tolerance of `1e-9` on a radius of `7000` km is far tighter than the
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representation can hold, and the iteration will not terminate.
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- The scale used for the relative tolerance should be the magnitude of the
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quantity, not the quantity's units, so that switching between kilometres and
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metres does not change the answer.
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## Applicability
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A method that converges is not necessarily accurate. Newton's method on
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Kepler's equation converges to the root of the equation as posed; if the
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equation omits a perturbation the solution is the root of the wrong problem,
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and it will converge confidently.
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Iteration also fails silently when it stops on a loose criterion. Reporting
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the number of iterations rather than only the answer lets a caller see that a
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result was reached in the expected number of steps.
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The three failure modes are separable in practice: rounding shows up as a
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floor on achievable accuracy, truncation as an error that shrinks when the
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method is refined, and instability as an error that grows.
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## Standards
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- Newton, *De analysi per aequationes numero terminas infinitas*, 1669
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- Higham, *Accuracy and Stability of Numerical Algorithms*, 2nd edition, 2002
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- Quarteroni, Sacco, Saleri, *Numerical Mathematics*, 2nd edition, 2007
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## See also
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- [[units-and-dimensions]] for what the numbers mean
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- [[ephemeris-interpolation]] for the error of a sampled table
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- [[higher-order-gravity]] for a model where the series does not converge |