El 25/02/2011 12:08, Marco van de Voort escribió:
Have a look at FPC package symbolic. It sounds like roughly the same kind of
soup. (parsing, differentiating, fast eval)
Yes, roughly. Because if I have read it correctly, the package
evaluates things ultimately by calls to the system functions like sin,
arctan, *, etc. This is very different than building on the fly the
minimal object code for evaluating the function, and much slower.
Also, for derivatives, Symbolic builds first the expression that is
the derivative of the initial expression. When the derivative is
called, it evaluates the new function.
Automatic differentiation does not compute the derivative function,
but at each step of the evaluation of the RPN it evaluates the
function and the first (or the function, the first and the second)
derivative of that step. For instance, if the RPN step is "cosine",
the Symbolic approach is to calculate first -sine (and for the second
derivative, to calculate then _again_ the cosine, etc.). But in my
approach, instead of computing the cosine, it is computed the sincos
pair by using the x87 instruction of AMD or Intel. The two values are
used directly for computing the first and second derivatives: thus at
most _one_ call to an expensive FPU instruction.
In my approach, if a call of a RPN step costs n clock cycles, then
the call of the first derivative of that function takes a cost c*n
clock cycles, where c depends only on the function called (that is,
whether it is * or arctan, etc). And the same (with another greater
coefficient) with the second derivative. Suppose that c = 3 for the
operator * and one has to compute the value of x*y*z*w, where
these symbols denote variables or expressions on which the formula
depends. Then Symbolic will compute
x'*y*z*w + x*y'*z*w + x*y*z'*w + x*y*z*w',
that is roughly four times the computation of the function. Automatic
differentiation will take only three (= c) times more (this should be
taken only as a sloppy explanation). Thus, the more complex the
formula the better relative performance.
On the other hand, porting my library to other architectures will be
possible if they have an FPU based on a stack of perhaps more than six
registers, a collection of instructions that move numbers between them
and between them and RAM, and a minimal set of instructions for each
of the basic mathematical functions of Pascal: +,-,*,/,sqr, sqrt,
sin, cos, tan, arctan, abs, exp, log. But I have no idea about those
architectures.
--
montesin at uv dot es
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