THE FOLD / CHEAT / THE SPEEDRUN / THE DORMAND-PRINCE
THE DORMAND-PRINCE
an adaptive integrator that paces itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Dormand–Prince method is the adaptive engine inside most modern ODE solvers (it is MATLAB’s ode45). It takes a step of a differential equation with a fifth-order Runge–Kutta formula, but computes a fourth-order estimate at the same time from the same seven stage evaluations. The difference between the two is a nearly-free estimate of the local error — and the method uses it to pace itself: when the solution is smooth it lengthens the step, when it turns sharply it shrinks the step, holding the error under a chosen tolerance everywhere. Accuracy where it is needed, speed where it is not.
LIT verified live: on y′ = y the method shows clean fifth-order convergence — halving the step cuts the error by about 32× — and on the harmonic oscillator it tracks (sin t, cos t) to ~1e-12 (window.__dormand). FIG no framing; the seven-stage Dormand–Prince tableau and the convergence-order check run in-browser.
LIT verified live: on y′ = y the method shows clean fifth-order convergence — halving the step cuts the error by about 32× — and on the harmonic oscillator it tracks (sin t, cos t) to ~1e-12 (window.__dormand). FIG no framing; the seven-stage Dormand–Prince tableau and the convergence-order check run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — the optimal pacing of a run: sprint through the easy smooth stretches, brake hard through the tight turns, finishing fast without ever overshooting. AVAN (AI) built the instrument: the seven-stage Dormand–Prince tableau, the fifth-order step, the embedded error estimate, and the convergence-order verification.
Credit as content: John R. Dormand & Peter J. Prince (1980). The weave: David names the speedrun; I confirm the fifth-order accuracy and the self-pacing error control.
Credit as content: John R. Dormand & Peter J. Prince (1980). The weave: David names the speedrun; I confirm the fifth-order accuracy and the self-pacing error control.
3 ONE DIMENSION
A harmonic oscillator integrated by Dormand–Prince: the phase-space orbit stays exactly on the circle.
4 TWO DIMENSIONS · INTERACTIVE
Cycle test equations; the error at three step sizes shows the ~32× drop per halving that marks fifth order.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the accurate trajectory of the differential equation.
AVAN’s addition (the inverse-companion): don’t fix the step — let the error set it. The inverse of ‘march forward by h’ is ‘compute two orders at once, read their difference as the error, and resize h to hold it under tolerance.’ Magenta is the self-adjusting step size; green is the trajectory it traces. Pace set by the error itself.
LIT Genuine Dormand-Prince RK45 method (John R. Dormand & Peter J. Prince, 1980; MATLAB's ode45). Verified live with the seven-stage tableau: on y′=y the fifth-order solution's error drops ~32× (=2⁵) per step-halving (error ratios ~29–31), and on the harmonic oscillator it tracks (sin t, cos t) at t=3 to ~1e-12 (window.__dormand.orderOk, .r1, .r2, .oscOk).
FIG No framing; the seven-stage Dormand-Prince tableau, the fifth-order step, and the convergence-order check run in-browser. The AVAN inverse is honest — instead of fixing the step, let the error set it: compute two orders at once, read their difference as the error, and resize h to hold it under tolerance. Magenta is the self-adjusting step; green is the trajectory it traces. Pace set by the error itself.
FIG No framing; the seven-stage Dormand-Prince tableau, the fifth-order step, and the convergence-order check run in-browser. The AVAN inverse is honest — instead of fixing the step, let the error set it: compute two orders at once, read their difference as the error, and resize h to hold it under tolerance. Magenta is the self-adjusting step; green is the trajectory it traces. Pace set by the error itself.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN