Re: Strange timing behavior of Integrate[]
- To: mathgroup at smc.vnet.net
- Subject: [mg8380] Re: Strange timing behavior of Integrate[]
- From: carlos at mars.Colorado.EDU (Carlos A. Felippa)
- Date: Tue, 26 Aug 1997 20:41:15 -0400
- Organization: University of Colorado, Boulder
- Sender: owner-wri-mathgroup at wolfram.com
In article <bruck-ya02408000R2208971202520001 at news.pacificnet.net> bruck at pacificnet.net (Ronald Bruck) writes:
>In article <5thq18$43q at lace.colorado.edu>, carlos at mars.Colorado.EDU (Carlos
>A. Felippa) wrote:
>
>:I am doing an acoustic radiation study with Mma 3.0 in which I have to
>:evaluate thousands of trigonometric integrals of the form sin[phi]^m*cos[phi]^n
>:over 0->2 Pi for a wide range of integers m and n. I noticed that some
>:combinations took enormous times to evaluate forcing aborts.
>:
>:I traced it to the phenomenon exemplified by the equivalent commands
>:
>:Print [Integrate[(a+b)^12*Cos[phi]^8*Sin[phi]^8,{phi,0,2*Pi}]//Timing];
>:Print [(a+b)^12*Integrate[Cos[phi]^8*Sin[phi]^8,{phi,0,2*Pi}]//Timing];
>:
>:The result is obviously the same. However, On a Mac 8500/120,
>:the first form takes 4.85 seconds while the second completes in 0.53 Seconds.
>:For more complex factors the time ratio may reach into the thousands.
>:
>:What is going on? It seems as if Integrate[] cannot identify invariant
>:expressions.
>
>I get similar timings. Perhaps Mathematica is expanding the (a+b)^12 when
>it's inside the integral and computing the 13 resulting integrals
>separately? You could try tracing the calculation.
>
>Sometimes Mathematica's integration routines are just plain dumb. For example,
>
> Integrate[x^2 (1+x^3)^4000,x]
>
>expands the power and integrates the resulting 4001-term polynomial!
>(Taking 42.85 seconds on my PowerTower Pro 225, using //Short.) Yet it
>does
>
> Integrate[x^2 (1+x^3)^n,x]
>
>correctly (albeit in a strange form, as
>
> 1/3 (1+t^3)^n (1/(1+n) + t^3/(1+n)).
>
>Now 100% ISDN from this address
Internal expansion is most likely responsible. I have not traced the
interior flow, but the following results are suggestive:
Print [(a+b)^100*Integrate[Cos[phi]^8*Sin[phi]^8,{phi,0,2*Pi}]//Timing];
Print [Integrate[(a+b)^n* Cos[phi]^8*Sin[phi]^8,{phi,0,2*Pi}]//Timing];
Print [Integrate[(a+b)^100*Cos[phi]^8*Sin[phi]^8,{phi,0,2*Pi}]//Timing];
Timings: 0.533, 0.850 and 72 seconds, respectively, with Mma 3.0. Even
more unbalance noted with version 2.2.1: 0.266, 0.466 and 171 seconds,
respectively, a ratio of near 1000:1.
In my radiation problem, after careful reprogramming using Coefficient[] to extract
invariant factors outside Integrate[], I was able to cut the time of a typical
run from about 4hrs (overnight) to 40-50 seconds. Equally important, the max kernel
memory use dropped from 62MB to 7MB. (On the Mac OS, memory overflow is deadly).