Mathematica 9 is now available
Technology Guide--The Technology Inside Matters
Selective Method Override

Setting a selective method override in a Mathematica function means manually narrowing down how Mathematica should carry out the computation to a particular method or class of algorithms, though not its order or sub-algorithm. This is a halfway point between letting Mathematica automatically select the best option using AAS or manually specifying everything.

For example, ExplicitRungeKutta is a selective method override specifiable in the numerical differential equation solver NDSolve. Choosing it enables all the different orders of Runge-Kutta methods to be utilized under AAS control but prevents the use of other methods, such as Adams or BDF.

Selective Method Override

The chart compares the number of evaluations required to solve a particular ODE with different orders of Runge-Kutta method. Although Mathematica's full automatic algorithm selection produces a near-optimized result, selecting 8th order will in this case be more computationally efficient. However, guessing the best order to use--as required in non-Mathematica, manual-only systems--would typically be highly inefficient.

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