Astronomy & Celestial Mechanics
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Gravitation

An error that grows like a random walk

A long integration accumulates two errors with opposite habits. One falls when the step is made smaller and grows in proportion to the time; the other grows when the step is made smaller and accumulates as a square root. Which of the two dominates decides whether a billion-year integration means anything.

Assumes Numerical integration and Chaos.

17 min read 8 figures Scales that defeat intuitionSmall perturbations, large consequences

Everything published about the long-term stability of the solar system rests on numerical integrations that run for billions of years and take billions of steps. The obvious question about such a calculation is what its error is, and the obvious answer — make the step smaller until the answer stops changing — is wrong in a way that is not obvious at all.

Below a certain step size, making the step smaller makes the answer worse. The step at which that happens is not far from the step such integrations actually use, and the reason is that there are two errors rather than one.

A step size below which a smaller step is worse. The error left in a long integration against the step size, for methods of three different orders, with both contributions drawn. The falling lines are truncation error, whose slope on these axes is exactly the order of the method. The rising line is round-off, identical for all three because it is a property of the arithmetic and not of the algorithm: every operation loses a few bits, the losses are independent, and they accumulate as the square root of the number of steps — which is why its slope is −1/2 and why it rises as the step shrinks. Each method's total has a minimum, at a step of 1.0e-6, 5.0e-6, 9.7e-4 for orders 1, 2, 4. Below that minimum every halving of the step costs time and makes the answer worse. That is the practical reason a solar-system integration is not run at an arbitrarily fine step, and it is a reason with nothing to do with computer time.
Fig. 1 The two errors, against the step size, for methods of three different orders. The falling lines are truncation, whose slope on these axes is exactly the order of the method. The rising line is round-off, which is identical for all three because it is a property of the arithmetic rather than of the algorithm. Each total has a minimum, and below that minimum every halving of the step costs time and buys a worse answer.

It is worth putting a number on the scale of the problem before analysing it. An integration of the outer solar system over five billion years at a step of a few hundred days takes of order 101010^{10} steps. Every step performs perhaps a hundred floating-point operations per body, so the calculation performs something like 101310^{13} roundings, each losing about a part in 101610^{16}. If those losses added up in the worst possible way the accumulated error would be a factor of ten larger than the quantity being computed; if they add up in the best possible way — as a random walk — the total is three parts in 101110^{11}. The entire difference between a meaningful result and a meaningless one lies in which of those two happens, and it is decided by details of the arithmetic that no equation in celestial mechanics mentions.

Two errors with opposite signs of slope

Truncation error is what the method gets wrong because it is an approximation. A method of order pp commits an error per step proportional to hp+1h^{p+1}, and over T/hT/h steps that accumulates to something proportional to hph^p. Halving the step divides it by 2p2^p — a clean, predictable gain, and the whole reason higher-order methods exist.

Round-off error is what the arithmetic gets wrong because a real number is stored in a finite number of bits. Every addition of a small increment to a large coordinate loses the bits that fall off the bottom, and the loss is of order the machine epsilon times the magnitude of the number — about 10−1610^{-16} in double precision. That error per step does not depend on the step size at all, so a smaller step means more steps and more losses.

The way those losses accumulate is the crux. If the rounding is unbiased — as likely to go up as down — the losses are independent random increments and their sum is a random walk, growing as the square root of the number of steps. The number of steps is T/hT/h, so the total round-off error grows as h−1/2h^{-1/2}: it rises as the step falls.

The sum of a falling power and a rising one has a minimum, and its position depends on the order of the method. Higher order shifts the minimum to a larger step and lowers it, which is a second and less-discussed reason to use a high-order method: not to take fewer steps, but to be allowed to.

A step size below which a smaller step is worse. The error left in a long integration against the step size, for methods of three different orders, with both contributions drawn. The falling lines are truncation error, whose slope on these axes is exactly the order of the method. The rising line is round-off, identical for all three because it is a property of the arithmetic and not of the algorithm: every operation loses a few bits, the losses are independent, and they accumulate as the square root of the number of steps — which is why its slope is −1/2 and why it rises as the step shrinks. Each method's total has a minimum, at a step of 1.3e-5, 1.6e-3, 3.1e-2 for orders 2, 4, 8. Below that minimum every halving of the step costs time and makes the answer worse. That is the practical reason a solar-system integration is not run at an arbitrarily fine step, and it is a reason with nothing to do with computer time.
Fig. 2 The same competition for a longer integration and higher orders. Ten times as many steps raises the round-off line by a factor of about three, which moves every minimum to the right and up: a longer run wants a coarser step, which is the opposite of intuition and follows directly from the square-root accumulation. The eighth-order method’s optimum sits nearly two decades coarser than the second-order method’s, and its error there is six decades lower.

One more feature of the minimum is worth noticing: it is not sharp. The total error near its optimum varies slowly, because it is the sum of two power laws crossing at a shallow angle, so a step size within a factor of three of the optimum costs very little. That is fortunate, since the optimum depends on the length of the run, on the machine’s precision and on the problem’s dynamic range, none of which is known exactly in advance. The practical procedure is not to find the optimum but to stay comfortably to the right of it, and to check afterwards that the energy error grew as a square root rather than linearly — which is a measurement of whether the arithmetic regime was the intended one.

Brouwer’s law, and why the exponents are what matter

Fixing the step and asking how the error grows in time gives a second and sharper statement, and it was made by Dirk Brouwer in 1937 for exactly this purpose.

An unbiased error in the energy performs a random walk, so it grows as t1/2t^{1/2}. An error in the energy is an error in the mean motion, and an error in the mean motion integrates once more into an error in the position along the orbit. The double integral of a random walk grows as t3/2t^{3/2}.

A biased error behaves differently at both stages. A rounding rule that always goes the same way, or a truncation error that does not average out over an orbit, accumulates linearly: the energy error grows as tt and the position error as t2t^2.

Four exponents, and the difference between them is a gigayear. How the error in a long integration grows with the number of steps, for the two kinds of arithmetic error and for the two quantities that matter. An unbiased error — one as likely to round up as down — accumulates as a random walk, so the energy error grows as the square root of the step count and the position error, which is its double integral, as the three-halves power. A biased error accumulates linearly and gives t and t². The exponents differ by a factor of two in the logarithm, which over the ten decades drawn here is a factor of 2.0e+3 in the final position error. That is why the arithmetic of a long integration is engineered rather than accepted: the choice is not between a fast code and a slow one but between an answer and a plausible-looking number.
Fig. 3 The four growth laws. The difference between the biased and unbiased cases is a factor of two in the exponent, which over the ten decades of steps drawn here is an enormous factor in the final position error. This is the reason the arithmetic of a long integration is engineered rather than accepted: the choice is not between a fast code and a slow one but between an answer and a plausible-looking number.

The practical target that follows is called Brouwer’s law, and a solar-system integrator is said to achieve it when its energy error grows as t1/2t^{1/2} rather than as tt. Achieving it is not automatic. It requires that the truncation error average to zero over an orbit, which is what a symplectic method guarantees, and that the round-off be unbiased, which requires care with the arithmetic.

Four exponents, and the difference between them is a gigayear. How the error in a long integration grows with the number of steps, for the two kinds of arithmetic error and for the two quantities that matter. An unbiased error — one as likely to round up as down — accumulates as a random walk, so the energy error grows as the square root of the step count and the position error, which is its double integral, as the three-halves power. A biased error accumulates linearly and gives t and t². The exponents differ by a factor of two in the logarithm, which over the ten decades drawn here is a factor of 5.0e+3 in the final position error. That is why the arithmetic of a long integration is engineered rather than accepted: the choice is not between a fast code and a slow one but between an answer and a plausible-looking number.
Fig. 4 The same laws with a smaller bias and a longer run. Reducing the bias by a factor of four moves the crossing point out by a factor of sixteen, because the crossing is where a linear growth overtakes a square-root one and that scales as the inverse square of the bias. Two decades of extra integration are bought by one factor of four in the arithmetic — which is why compensated summation and careful rounding are worth the trouble, and why they buy time rather than accuracy.

There is a subtlety in the second step of that argument that is worth spelling out, because it explains why position errors are so much larger than energy errors in practice. An error in the energy is an error in the semi-major axis, and by Kepler’s third law an error in the semi-major axis is an error in the period. A body whose period is wrong by one part in 101110^{11} drifts one part in 101110^{11} of a revolution per revolution, and the drift accumulates: after NN orbits it is NN times as large. That is why the position error carries an extra power of time compared with the energy error, and it is why a beautifully conserved energy is not by itself evidence that a computed position means anything. An integrator can be right about how much and wrong about where, and this is the arithmetic reason as well as the structural one.

The tricks that make the rounding unbiased

Three techniques do most of the work, and each addresses a different way the arithmetic goes wrong.

Compensated summation. The classic loss is adding a small increment to a large accumulator: the low bits of the increment fall off the end and are gone. Kahan’s algorithm keeps a running record of what was lost and adds it back on the next step. It costs three extra floating-point operations per addition and it converts a linear accumulation into a random walk — which is to say it buys a change of exponent rather than a change of coefficient.

Storing the increment rather than the coordinate. A planet’s position is a large number and its change per step is a small one, so representing the state as an initial position plus an accumulated offset keeps the additions between numbers of similar size. The same idea appears in every long-baseline timing problem, and the reason is always the same: precision is relative, so the arithmetic should be done on quantities that are small.

Rounding that does not favour a direction. The default round-to-nearest with ties-to-even is very nearly unbiased, and that is not an accident of the standard; it was chosen for this. Truncation towards zero, which some older hardware did, is biased and destroys Brouwer’s law immediately.

The adaptive step starts 5.7× more accurate and ends 1.3× worse. The envelope of the relative energy error against elapsed revolutions — the worst error within each plotted interval rather than the error at one sample in it — for the same second-order symplectic integrator run two ways on a Kepler orbit at e = 0.5. The flat band is a fixed step of one 200th of a period: its error oscillates once per revolution and the envelope does not grow, because a symplectic method at constant h is the exact solution of a nearby Hamiltonian and is conserving that one. The rising curve is the same method with the step refined where the orbit is fast — h ∝ r^3/2, the free-fall time, varying by a factor of 5 around the orbit — which is the first thing anybody reaches for at a close encounter and which is strictly more accurate step for step: over its first three revolutions it stays a factor of 5.7 below the fixed run. By 240 revolutions it is a factor of 1.3 above it and still climbing. Changing the step changes which Hamiltonian is being conserved, the errors from successive steps stop cancelling, and what is left is a random walk with no bound at all. The practical consequence is that a solar-system integration cannot adapt its step: it either keeps a step short enough for the closest encounter it will ever meet, or it detects the encounter and hands that piece of the trajectory to an entirely different, non-symplectic method for the duration — which is what every long-term integration of the planets actually does.
Fig. 5 The comparison the first figure abstracts, measured on a real orbit: the error against step size for three methods, in the regime where truncation dominates and round-off is invisible. Every line is straight and its slope is the method’s order, which is the check that the implementation is what it claims to be. The round-off floor lies below the bottom of this plot, and finding it requires either a much longer run or a much finer step — which is exactly why it is so often discovered by accident, in the form of a convergence test that stops converging.

None of the three tricks is exotic and all three are routinely omitted. A code written without them is not obviously wrong: it passes short tests, converges under step refinement over the range anybody checks, and conserves energy well over a thousand orbits. It fails only at durations long enough that nobody has an independent answer to compare against, which is precisely the regime the code exists for.

What was actually measured

The results that these techniques exist to support are the long integrations of the solar system, and the checks on them are worth stating because they are the only available evidence that the answers mean anything.

The energy error is monitored and its growth exponent is measured. A good modern integration of the outer solar system over five billion years reports a relative energy error of order 10−1110^{-11} growing as the square root of time. The exponent is fitted from the run itself rather than assumed, and a run that shows linear growth is diagnosed as having a bug in its arithmetic rather than as being slightly less accurate.

Independent codes are compared. The two long-running programmes that established the chaotic behaviour of the inner planets used different integrators, different step sizes, different arithmetic and different machines, and they agree on the statistical properties of the outcome — the Lyapunov time of about five million years, and the small probability of Mercury’s eccentricity becoming large — while disagreeing completely on any individual trajectory. That disagreement is expected and is the subject of the expiry date on a prediction; the agreement on the statistics is the evidence that the arithmetic is not driving the result.

And the integration is checked against reality where it can be. A solar-system integration run backwards over a few thousand years reproduces recorded eclipses, and over a few million years reproduces the astronomical cycles recorded in sedimentary rock. The second check is the more remarkable: the periodicities of Earth’s orbital eccentricity and obliquity are imprinted in ocean-floor sediments, and matching a computed sequence to a measured one over tens of millions of years is a direct test of the integration’s phase.

Energy error over 240 revolutions, at one step size. The relative error in total energy against revolution number, for three integrators run on the same Kepler orbit at e = 0.5 with the same step of 200 per revolution. The exact energy is a constant, so every curve here is the method rather than the problem. Euler climbs steadily: its energy at the end is 106.5% wrong, and the orbit it draws has spiralled outwards. Runge–Kutta 4 begins 4.7e+2 times more accurate than leapfrog and ends at 2.22e-4, having grown by a factor of 10 across the run: the error is SECULAR. Leapfrog oscillates inside a band and stays there — worst error 2.62e-3, and the second half of the run is no worse than the first, which is measured here rather than claimed. That is the property that decides whether a five-billion-year integration means anything, and it is not accuracy: a symplectic method is the exact solution of a Hamiltonian a step-size away from the intended one, so its energy cannot wander, while a more accurate non-symplectic method has no such constraint and eventually wanders further.
Fig. 6 The measurement all of this is about, made on a test orbit rather than on the solar system: the energy error against time for three methods at the same step size. The symplectic method’s error is bounded and oscillates with the orbital period rather than growing; the two non-symplectic methods drift. Nothing about the round-off is visible at this scale, and that is the honest order of business — the truncation behaviour has to be right before the arithmetic is worth worrying about.

One further check is available and is under-used: run the same integration forwards and then backwards, and see whether it returns to its starting point. A time-reversible method with exact arithmetic would return exactly; the failure to return measures the irreversible part of the error, which is precisely the round-off. It is a strong test because it requires no reference solution and no theory of what the answer should be, and it separates the reversible truncation error — which cancels — from the irreversible arithmetic one, which does not.

Why the order of the method is not the whole story

There is a temptation to read the first figure as an argument for the highest available order, and the temptation should be resisted for a reason that is specific to this subject.

Order, measured off the slope rather than quoted. Position error after one complete revolution against step size, both logarithmic, for the same three integrators. The straight-line slope of each is its convergence order, fitted here over the three finest steps rather than taken from the method's name: euler 0.93, rk4 4.03, leapfrog 2.00, against the 1, 4, 2 the definitions promise. Read alone, this figure says to use Runge–Kutta: on the fitted slopes, halving the step buys 1.9× for euler, 16.3× for rk4, 4.0× for leapfrog. Read beside the energy figure it says the opposite for a long integration, and the two together are the whole of why solar-system work is done with symplectic methods that are, step for step, the worse of the two.
Fig. 7 The convergence orders themselves, measured rather than quoted: the error against step size for three methods, with the slopes read off the drawing. Each method delivers exactly the exponent its construction promises, over the range where truncation dominates. What the plot cannot show is the cost per step, and a method of twice the order that costs four times as much per step is a gain only if the required accuracy is high enough.

A high-order method takes more force evaluations per step, and a force evaluation in an NN-body problem costs N2N^2. So the comparison is between accuracy at fixed work rather than at fixed step, and the crossover depends on the accuracy required. For the accuracies that matter here — parts in 101110^{11} over gigayears — high order wins comfortably, which is why the field settled on high-order symplectic schemes. For a short, coarse integration it does not.

More importantly, the order says nothing about the structure of the error. A high-order non-symplectic method has a smaller energy error than a low-order symplectic one over a few orbits and a larger one over a million, because one drifts and the other does not. Order is about the coefficient; symplecticity is about the exponent; and this essay is about a third thing, which is the arithmetic, and which is also about the exponent.

Every one of them loses the phase. The distance between the integrated position and the exact Kepler position, against revolution number, for the same three integrators and the same orbit as the energy figure. The unit is the semi-major axis. All three grow without bound — euler reaches 2.49e+2, rk4 reaches 1.47e-1, leapfrog reaches 1.30e+0 — and the symplectic method has no advantage here at all. This is the half of the story a "conserves energy" summary throws away: leapfrog's guarantee is about the shape of the orbit, not about where on it the body is. A five-billion-year integration of the solar system is therefore evidence about whether Mercury stays and no evidence whatever about where Mercury will be, and the two statements are made by the same run.
Fig. 8 The other half of the accounting, and the one that decides whether a prediction is usable: the error in the position along the orbit, which is what somebody wanting to know where a body will be actually cares about. It grows faster than the energy error by one power of time for the reason given above, and the gap between the two curves widens throughout. A code that reports only its energy conservation is reporting the more flattering of the two numbers.

There is a further reason order alone is a poor guide, and it is about what the integration is for. A solar-system integration is not trying to follow one trajectory accurately; it is trying to sample the distribution of outcomes. What matters for that is whether the computed trajectory is a typical member of the family of true trajectories consistent with the uncertainties — a property called shadowing — rather than whether it is close to any particular one. A symplectic method with well-behaved arithmetic produces trajectories that shadow; a method with a secular energy drift produces trajectories that systematically leave the family, because a drifting energy is a drifting semi-major axis and the resonance structure a body encounters depends on where it is in the semi-major axis. The failure is not that the answer is imprecise. It is that the answer is a sample from the wrong distribution, and the resonances the body drifts through are the mechanism.

Where the picture stops

Three limits stand out, and the second is the one that recurs in practice.

The step cannot always be fixed. Everything above assumes a constant step, and a constant step is what makes a symplectic method symplectic. A close encounter requires a smaller step, and adapting the step destroys the property that made the method trustworthy. The standard resolutions — switching integrators near an encounter, or regularising the coordinates so that no small step is needed — are both ways of avoiding an adaptive step rather than of making one safe.

Chaos sets a horizon that no arithmetic reaches. With a Lyapunov time of five million years, an error of 10−1610^{-16} in the initial conditions is amplified to order unity in about two hundred million years. Beyond that no integration predicts a trajectory, however perfect its arithmetic, and the object of the calculation changes from a prediction to a statistical statement. Improving the arithmetic pushes the horizon out logarithmically, which is to say hardly at all.

And the model is not the solar system. An integration includes the eight planets, usually Pluto, sometimes the largest asteroids, a relativistic correction and a tidal term. The masses of the asteroids are uncertain at the per cent level and their number is large, and that uncertainty dominates the error budget of a modern ephemeris over centuries — far above anything in this essay. Perfect arithmetic on an imperfect model is a familiar way of being precisely wrong.

One more deserves its own sentence because it is the one that has grown in importance. Modern hardware is fast at operations that are not bit-reproducible: vectorised instructions reorder summations, fused multiply-add changes rounding, and a parallel reduction gives different results depending on how many threads ran. Each of those is harmless for an ordinary calculation and each of them can silently reintroduce a bias into a summation that was carefully arranged not to have one. The result is that a code that achieved Brouwer’s law on one machine may not on another, with no change to its source, and the only way to find out is to measure the growth exponent on the machine that will do the run.

Why an exponent is worth more than a coefficient

The recurring shape here is worth naming, because it appears in every long calculation.

An error’s coefficient is improved by working harder: a finer step, more digits, a better method. An error’s exponent is improved only by changing what kind of error it is. Converting a linear accumulation into a random walk is worth more than any amount of extra precision, because it changes how the error scales with the length of the run, and the run is the thing that is getting longer.

That is why Brouwer’s law is stated as a law about growth rather than about size. A code that achieves it can be run ten times longer for a factor of three in error; a code that does not gains a factor of ten. Over the durations these calculations now reach, that difference is the difference between a result and a rounding artefact.

The same logic governs the analytic side of the subject: a perturbation series that is asymptotic rather than convergent improves for a while and then gets worse forever, and knowing which regime a calculation is in matters far more than knowing its current error. Both halves of celestial mechanics are, in the end, about the shape of an error rather than its size.

End on what the whole apparatus is defending. The claim these integrations support is a claim about a real physical system: that the inner solar system is marginally unstable, that Mercury’s eccentricity performs a random walk driven by a resonance with Jupiter, and that there is a small but non-zero probability of a catastrophic outcome within the Sun’s remaining lifetime. That is a statement about the world, and it is supported entirely by arithmetic — there is no closed solution to appeal to and no observation that can be made on the relevant timescale. When the whole of the evidence for a physical claim is a number that came out of a machine, the properties of the machine’s arithmetic stop being a technical detail and become part of the argument.

Where the ladder goes next

Later rungs on this ladder start with regularisation — the change of variables that removes the singularity at a close encounter and lets a fixed step survive one, which is the transformation that turns a collision into a harmonic oscillator. The rung after that is the statistical question these integrations exist to answer: what it means to compute a probability from a system with one realisation, and how many integrations are enough.

Where this goes next

Energy error over 240 revolutions, at one step size. The relative error in total energy against revolution number, for three integrators run on the same Kepler orbit at e = 0.5 with the same step of 200 per revolution. The exact energy is a constant, so every curve here is the method rather than the problem. Euler climbs steadily: its energy at the end is 106.5% wrong, and the orbit it draws has spiralled outwards. Runge–Kutta 4 begins 4.7e+2 times more accurate than leapfrog and ends at 2.22e-4, having grown by a factor of 10 across the run: the error is SECULAR. Leapfrog oscillates inside a band and stays there — worst error 2.62e-3, and the second half of the run is no worse than the first, which is measured here rather than claimed. That is the property that decides whether a five-billion-year integration means anything, and it is not accuracy: a symplectic method is the exact solution of a Hamiltonian a step-size away from the intended one, so its energy cannot wander, while a more accurate non-symplectic method has no such constraint and eventually wanders further. Wrong about where, and right about how much Runge–Kutta is the more accurate method and loses energy steadily; leapfrog is cruder and its energy error never leaves a band. Over five billion years only one of those properties survives — and neither method knows where the planet is. The adaptive step starts 5.7× more accurate and ends 7.5× worse. The envelope of the relative energy error against elapsed revolutions — the worst error within each plotted interval rather than the error at one sample in it — for the same second-order symplectic integrator run two ways on a Kepler orbit at e = 0.5. The flat band is a fixed step of one 200th of a period: its error oscillates once per revolution and the envelope does not grow, because a symplectic method at constant h is the exact solution of a nearby Hamiltonian and is conserving that one. The rising curve is the same method with the step refined where the orbit is fast — h ∝ r^3/2, the free-fall time, varying by a factor of 5 around the orbit — which is the first thing anybody reaches for at a close encounter and which is strictly more accurate step for step: over its first three revolutions it stays a factor of 5.7 below the fixed run. By 1500 revolutions it is a factor of 7.5 above it and still climbing. Changing the step changes which Hamiltonian is being conserved, the errors from successive steps stop cancelling, and what is left is a random walk with no bound at all. The practical consequence is that a solar-system integration cannot adapt its step: it either keeps a step short enough for the closest encounter it will ever meet, or it detects the encounter and hands that piece of the trajectory to an entirely different, non-symplectic method for the duration — which is what every long-term integration of the planets actually does. A step that must not be adapted A symplectic integrator's bounded energy error is a property of a fixed step. It is conserving a Hamiltonian a step-size away from the intended one, and changing the step changes which Hamiltonian — so refining the step at a close encounter, which is the first thing anybody does, destroys the only property the method was chosen for. A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get. A prediction with an expiry date The inner solar system's Lyapunov time is about five million years, so a centimetre of error becomes an orbit in a hundred million. The ephemeris dies while the system survives — because the elements stay bounded when the phase does not, and only one of those is what stability means.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Brouwers lawCompensated summationEnergy driftError propagationFloating-pointRandom walkRound-off errorStep sizeSymplectic integratorTruncation error