Computation speeds: numerical vs symbolic

*To*: mathgroup at smc.vnet.net*Subject*: [mg55741] Computation speeds: numerical vs symbolic*From*: AES <siegman at stanford.edu>*Date*: Tue, 5 Apr 2005 03:21:06 -0400 (EDT)*Organization*: Stanford University*Sender*: owner-wri-mathgroup at wolfram.com

No actual question, or at least no immediately useful question in this post. I'm just expressing surprise and/or asking for enlightenment on what seems like a vast disparity in speeds of symbolic vs numerical calculations in the latest version of Mathematica. To be specific, take an expression like v = Exp[ -a (s+y)] ( Exp[2 a y] Cos[s0 + q (s-y)] - Cos[s0 + q (s+y)] ) which happens to have come up in a problem I'm just working on (y and s are the independent variables of interest, dimensionless surrogates for space and time; a, q and s0 are parameters; q happens to be an integer in the end). I then want to evaluate the integral w = Integrate[v^2, {y, 0 , Pi}] (symbolically) and make some plots of w versus s, with values of a, s0 and q being set by rules for each plot. The integration involved here (nothing but products of exponentials and cosines) I would describe as trivial but tedious, that is, I could do it by hand, but it would be a (very) dull job. Yet on my Mac iBook G4 under Mathematica 5.1, it takes 15 seconds to do this particular integration. If I follow the above cell with a second cell putting in the "simplifying" assumption w = Integrate[v^2, {y, 0 , Pi}, Assumptions->Element[q,Integers]] this calculation takes 27 seconds. Oddly, if I immediate re-evaluate either of the cells it only takes 2.7 seconds or 5 seconds, respectively, for the results to come up the second time, but having evaluated the first cell (without the Assumption) doesn't seem to speed up the second cell (with the Assumption), so it's not like some integration rules have gotten loaded and cached in evaluating the first cell and used in evaluating the second cell. So, I don't see how I can use this speedup in any practical way (???). The speed contrast comes when I then make a single DisplayPlot containing Plots of the w function from above and two very similar functions wp and wpp versus s for 10 instances of rules specifying values of a, s0 and q, in other words, thirty decaying-sinusoid-like curves in all, each containing 5 or 6 complete cycles. This plot appears on the screen faster than I can release the Enter key to create it. So, I'm just curious, how can a calculate and plot command like this be so stunningly fast? and the trivially simple symbolic integration above be so slow?

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