The funding project is MAC; that is, the ‘MAC’ in ‘MACSYMA’. The codes are of historical interest only—it was only a year later that Risch published his decision procedure. SIN & SAINT are still wonderful bits of software to explore.
The Abstract is a good summary. I am unsure how useful this is today. We now have battle tested numerical libraries that use Runge Kutta and similar to solve ODEs.
Abstract:
SIN and SOLDIER are heuristic programs written in LISP which solve symbolic integration problems. SIN (Symbolic INtegrator) solves inde-finite integration problems at the difficulty approaching those in the larger integral tables. SIN contains several more methods than are used in the previous symbolic integration program SAINT, and solves most of the problems attempted by SAINT in less than one second. SOLDIER (SOLu-tion of Ordinary Differential Equations Routine) solves first order, first degree ordinary differential equations at the level of a good col-lege sophomore and at an average of about five seconds per problem attempted. [...]
In some situations, there is a huge performance advantage to symbolic integration. The symbolic integration is expensive, but once it's done, you can evaluate integrals of different parameterisations of your function in a handful of cycles. With numerical integration, you have to spend however many thousands of cycles it takes for every parameterisation.
If you're evaluating integrals as part of the objective function in an optimisation over a multidimensional space, then you're going to be doing a lot of optimisations, and that saving can add up.
Fair point. The same thing applies to differentiation, where e.g. computing analytic Jacobians can be useful to save precious CPU cycles in embedded control systems.
Even very high-order Runge-Kutta solvers may fail to preserve simple qualitative properties such as having a linear oscillator stay in its phase space. Meaning: simulations of physical systems will typically diverge unless you're very lucky. Might mean a satellite failing to enter stable orbit. Solvers for systems with certain specific phase space properties may be found (for satellites there are particular solvers), but do you always know beforehand what you're dealing with?
This "numerical power will solve every problem" attitude is bound to lead us to disaster. But then, so was the "ORM will solve every problem" attitude, for example. Luckily we have people actively researching symbolic computation. Jeez, without symbolic computation finding very high-order Runge-Kutta methods is exceedingly hard and error-prone.
Symbolic integration requires an accurate model of the system, whereas you can throw numerical methods at most problems and get results that are good enough. If you worry about your satellite falling down you can always try energy-preserving integrators.
I would wager the guess that most PID controllers, arguably the most common application of numerical integration, work just fine with just basic forward Euler integration.
I can't disagree with your reply because it seems we're living in entirely different planets.
It's a good thing that we don't have to develop a full-scale polemic (I was thinking sociologists and economists, but heck look at Taleb the professional polemist) since the world can reward us both for adding value as technologists and engineers.
