While scanning through some papers about the Runge-Kutta method, I stumbled upon a 1993 paper entitled “The search for the Holy Grail, Or: Predictor-corrector methods for solving ODEIVPs,” written by Kevin Burrage, a professor of Computational Mathematics at the University of Queensland in Australia.  Sure, it’s pushing 15 years old now, but with a title like that, I couldn’t help but look!

Burrage explains in the preamble to his paper:

The title of this paper and the quote [from “La Queste del Saint Graal”] given  above in some sense reflects my attitude to the continuing search for efficient scientific algorithms, and, in particular, those for the numerical solution of ordinary differential  equations, in a parallel environment.  The “Quest of the Holy Grail” was a vast and complex romantic prose cycle developed in the twelfth and thirteenth centuries and is based on the legends of Arthurian England. . . . [U]nder later authors, and, in particular, Malory it was formalized into a symbol for the Eucharistic vessel.  Initially, the Arthurian knights would set out on their quests full of passionate hope but because of spiritual inadequacies their search, in all but one case, ended in unmitigated failure. . . . [I]n many ways, the prose cycle offers an appropriate symbolism in the search for efficient parallel methods for ordinary differential equations.

The symbolism, actually, does seem rather apt.  At the time, there had been “a number of articles in the literature, some based on prediction-correction techniques, which appear[ed] to offer the apparent impossibility–namely massive parallelism.  However, usually it is found on further reflection that there may be faulty mathematical reasoning . . . or that the methods work only on a very restricted class of problems” (126).  So massive parallelism is like the Holy Grail of scientific computing, and the failures in mathematical reasoning are like the knights’ spiritual inadequacies.

Burrage presents a brief survey of predictor-corrector methods and shows that “such methods are unlikely to be effective for solving nonstiff problems in a parallel environment” (127).  Indeed, the Mathworld entry on Predictor-Corrector methods notes that even in 1992, people were claiming that “predictor-corrector methods [had] been largely supplanted.”

Looking briefly through recent publications about parallelized ODE solvers, I saw a number of iterative Runge-Kutta formulations similar to the kind we talked about in class, with very pragmatic approaches focused to optimize spatial locality or otherwise develop fast solutions for real machines, e.g. “Optimizing Locality and Scalability of Embedded Runge-Kutta Solvers Using Block-Based Pipelining.”

Speaking as a would-be computer scientist, I have to wonder if the knights would have been better off looking for a Grail with locality-optimized semi-holiness, but I guess that’s a question best left to the likes of Malory.