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Re: Effect of Simplify on numeric vs symbolic
*To*: mathgroup at smc.vnet.net
*Subject*: [mg56027] Re: Effect of Simplify on numeric vs symbolic
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Thu, 14 Apr 2005 08:54:18 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <d3g8vq$t01$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <d3g8vq$t01$1 at smc.vnet.net>, carlos at colorado.edu wrote:
> In my previous example of FEM symbolic vs numeric speed all
> computations were with numbers. No variables were involved.
> The difference (minutes vs seconds) due largely due to Simplify.
I could find no mention of Gauss-Legendre quadrature in the index of
your INTRODUCTION to FINITE ELEMENT METHODS course at
http://caswww.colorado.edu/courses.d/IFEM.d/Home.html ?
> Here is a simpler example that illustrates the point, also taken from
> the same course. LineGaussRuleInfo returns weights and abcissae of 1D
> Gauss-Legendre quadrature rules of 1 to 5 points. Argument numer asks
> for exact information if False; if True it returns N[information].
> All computations involve nunbers only.
The expressions involve arithmetic with complicated radicals (that can
be avoided -- see below). Manipulation of such expression is often very
slow.
> Results of calls (at end of post) on a Mac laptop:
>
> numer rule n Eval time Simplify time
> True 4 50 0.00 sec 0.00 sec
> False 4 50 0.00 sec 78.01 sec (* Note: FullSimplify needed *)
> False 4 50 0.00 sec 0.00 sec (* found result in cache *)
I ran your code (Mac OS X 10.3.8, Mathematica 5.1) and don't get these
timings? The _first_ time I call
PerverseExpression[4, 50, False]
I get the correct answer in 0.16 sec.
> LineGaussRuleInfo[{rule_,numer_},point_]:= Module[
> {g2={-1,1}/Sqrt[3],w3={5/9,8/9,5/9},
> g3={-Sqrt[3/5],0,Sqrt[3/5]},
> w4={(1/2)-Sqrt[5/6]/6, (1/2)+Sqrt[5/6]/6,
> (1/2)+Sqrt[5/6]/6, (1/2)-Sqrt[5/6]/6},
> g4={-Sqrt[(3+2*Sqrt[6/5])/7],-Sqrt[(3-2*Sqrt[6/5])/7],
> Sqrt[(3-2*Sqrt[6/5])/7], Sqrt[(3+2*Sqrt[6/5])/7]},
> g5={-Sqrt[5+2*Sqrt[10/7]],-Sqrt[5-2*Sqrt[10/7]],0,
> Sqrt[5-2*Sqrt[10/7]], Sqrt[5+2*Sqrt[10/7]]}/3,
> w5={322-13*Sqrt[70],322+13*Sqrt[70],512,
> 322+13*Sqrt[70],322-13*Sqrt[70]}/900,
> i=point,p=rule,info={{Null,Null},0}},
> If [p==1, info={0,2}];
> If [p==2, info={g2[[i]],1}];
> If [p==3, info={g3[[i]],w3[[i]]}];
> If [p==4, info={g4[[i]],w4[[i]]}];
> If [p==5, info={g5[[i]],w5[[i]]}];
> If [numer, Return[N[info]], Return[info] ];
> ];
The abscissas and weights for arbitrary n can be computed directly as
follows:
x[n_, m_] := Root[Function[x, LegendreP[n, x]], m]
w[n_, m_] := 2 (1 - x[n, m]^2)/((n + 1)^2 LegendreP[n + 1, x[n, m]]^2)
a direct implementation of equation (10) at
http://mathworld.wolfram.com/Legendre-GaussQuadrature.html
To convert these Root objects to radicals, we use ToRadicals:
TableForm[ToRadicals[Table[x[n, m], {n, 2, 5}, {m, 1, n}]]]
Similarly, the weights reduce to
TableForm[RootReduce[Table[w[n, m], {n, 3, 5}, {m, 1, n}]]]
These results agree, as expected with your gn's and wn's.
However, there really is no need to convert the abscissas and weights
expressions to radicals. In general, especially for large n (in fact,
even for n=6), it is better to work with representations involving Root
objects and use RootReduce to simplify combinations of such objects.
> PerverseExpression[rule_,n_,numer_]:=Module[{xw,tEval,tSymb,x},
> xw=Table[LineGaussRuleInfo[{rule,numer},i],{i,1,rule}];
> {tEval,x}=Timing[Product[(i+2)*xw[[i,1]]^n,{i,1,rule}]];
> {tSymp,x}=Timing[FullSimplify[x]];
> Return[{tEval,tSymp,x}]];
This expression is just
g[n_, p_] := Product[(i + 2) x[n, i]^p, {i, 1, n}]
Computing
RootReduce[g[4, 50]]
takes about 0.4 sec and gives the same result as
Print[PerverseExpression[4,50,False]]
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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