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Making Mathematica Functions Evaluate Rapidly?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg2206] Making Mathematica Functions Evaluate Rapidly?
*From*: siegman at ee.stanford.edu (A. E. Siegman)
*Date*: Mon, 16 Oct 1995 11:53:04 -0400
*Organization*: Stanford University
I frequently want to evaluate (plot, numerically integrate, calculate
numerical moments of) functions which start out complex, e.g., something
like
f[x,y,z] = (1/ f1[x,y,z] ) Exp[ f2[x,y,z] ]
where f1[x,y,z] and f2[x,y,z] may contain various purely real
coefficients, call 'em a,b,c,... which have fixed (predefined) values, as
well as the variables x,y,z , and some explicit I's (the imaginary unit
I), and some standard functions, e.g., Cos[], Sin[], etc.. In other words,
all input values are purely real, and all I's are explicit.
Then, what I really want to do is to generate purely real outputs, either
g1[x,y,z] = Abs[f[x,y,z]]
or
g2[x,y,z] = Abs[f[x,y,z]]^2
in a form that will evaluate as rapidly as possible (on a PowerMac).
What's the best way to do this? Should I define the initial functions
using = or := ? At what stage should I compile (and how)? Can I compile
a function that contains other functions that have already been compiled?
I sometimes seem to get very small complex values coming out of the
compiled versions of g1 or g2 (i.e., the Abs[] functions), even though
I've used _Real on x,y,z in the compilation, which makes me think they may
being evaluated with complex values even though everything is supposedly
real. Do I need to use ComplexExpand somehow on f[x,y,z] to separate it
into Re and Im parts, then square them independently to get Abs[]^2 ?
And finally, what if Pi or Sqrt[2] or ... are also contained in the
functions f1, f2 ? Do I need to do something to force those into
numerical form also?
(My experience is that depending on just how I set up a calculation like
the above I get wildly different speeds -- but there seems to be no
systematic way to know how to get the fastest evaluation.)
Email replies to siegman at ee.stanford.edu appreciated -- thanks.
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