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Recursive algorithms--often espoused by designers of computer languages--are a mixed blessing when it comes to practical computational algorithms needed in the physical sciences, as we shall discover in this column by discussing several examples from algebraic and numerical computations. I became particularly interested in two of the repetition patterns for mathematics and computing--iteration and recursion--while devising algorithms for one of my current projects, the Atlas for Computing Mathematical Functions. This handbook and accompanying Mathematica and C programs comprise a visual (700 graphics) and computational (200 functions) tour of special functions, such as those in Abramowitz and Stegun, Spanier and Oldham, or Erdelyi et al. Here I describe first the usual understanding of the distinction between iteration and recursion, then I show some failures and successes of recursive algorithms for symbolic and numerical computations.
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