Summary: Performance of numerical functions decreases with time

*To*: mathgroup <mathgroup at yoda.physics.unc.edu>*Subject*: Summary: Performance of numerical functions decreases with time*From*: Magnus Nordborg <magnus at fisher.stanford.edu>*Date*: Fri, 16 Jul 93 09:40:58 -0700

Thanks to all that responded to this question, including several staff members at WRI. The problem (see original posting below) is caused by a bug in Mathematica 2.1's pseudo-compiler (invoked automatically by FindMinimum) that causes symbols to accumulate over time. Five fixes were suggested: 1.) SetOptions[FindMinimum,Compiled->False] 2.) Use something like: leftoverCompileNames = Names["Compile`*"]; Scan[Unprotect, leftoverCompileNames]; Scan[Clear, leftoverCompileNames]; Scan[Remove, leftoverCompileNames]; every loop. 3.) Compile the function yourself, then use 1.) 4.) Get 2.2 5.) Use C/MathLink I tried 1.) and 5.). It might be of interest that 1.) finished in less than 10 hours, compared to the 20 h I estimated from running 30 iterations with Compiled->True. Using MathLink (and still looping inside Mathematica, but using a Brent's minimizer from NetLib) as in 5.) was about 40 times faster again. Again, thanks to all that responded, -Magnus --- Magnus Nordborg magnus at fisher.stanford.edu (NeXT mail preferred) Department of Biological Sciences Stanford University Stanford, CA 94305-5020 +1 (415) 723-4952 (office) Begin original message: Date: Tue, 13 Jul 93 13:57:47 -0700 From: Magnus Nordborg <magnus at fisher.Stanford.EDU> To: mathgroup <mathgroup at yoda.physics.unc.edu> Subject: Performance of numerical functions decreases with time Hi, I have a "time-critical" question here. I want to find the minimum of a simple function for many different parameters. I define something like: MyFunction[d_,s_,z_,a_,b_,g_,k_] := Module[{fn,res}, fn = - 2*(1 - d)*M^z*(1 - M^g)^k*s - (1 - M^g)^k*(1 - (1 - M^a)^b)*(1 - M^z*s); res = FindMinimum[ fn, {M,0.5}][[2]]; If[ MatchQ[res,{Rule_}], M /. res, (* else *) res = FindMinimum[ fn, {M,0.1}][[2]]; If[ MatchQ[res,{Rule_}], M /. res, (* else *) res = FindMinimum[ fn, {M,0.9}][[2]]; If[ MatchQ[res,{Rule_}], M /. res, (* else *) Null ] ] ] ] This particular pattern tries several starting points, but everything I say below applies equally well to the simplest possible pattern. Now, I am trying to use "Table" to find the minimum for 10^5 different values, say. I did some small number (like 10) and then extrapolated to find that my job would take on the order of 20 hours (this is using Sparc 2 and NeXT). 80 hours later the machines are still crunching. So I did a little test. I timed 35 cases, then redid the same 35 cases, and so on 10 times. To my chagrin, I find that the performance is decreasing over time! why = Table[ Timing[ Table[ {0.1,0.1,0.1,1.5,5,1.5,k, MyFunction [0.1,0.1,0.1,1.5,5,1.5,k]}, {k,1.5,10,0.25}] ], {i,10} ]; Transpose[why][[1]] {23.8333 Second, 26.2833 Second, 28.85 Second, 31.6667 Second, 34.5167 Second, 37.0833 Second, 39.7 Second, 42.6 Second, 45.5 Second, 48.5833 Second} It doesn't take the proverbial rocket scientist to see that if time to completion more than doubles after doing 300 cases, 10^5 will take a bit MORE than 20 h... What on earth is going on here? What is causing this, and is there a work-around? We really need these numbers fast, unfortunately. Time for C...? Grateful for a quick reply, -Magnus --- Magnus Nordborg magnus at fisher.stanford.edu (NeXT mail preferred) Department of Biological Sciences Stanford University Stanford, CA 94305-5020 +1 (415) 723-4952 (office) P.S. I'm using Mathematica 2.1 on NeXT and Sparc.