Effect of Simplify on numeric vs symbolic

• To: mathgroup at smc.vnet.net
• Subject: [mg55971] Effect of Simplify on numeric vs symbolic
• Date: Tue, 12 Apr 2005 05:26:32 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```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.

Here is a simpler example that illustrates the point, also taken from
the same course.  LineGaussRuleInfo returns weights and abcissae of 1D
for exact information if False; if True it returns N[information].
All computations involve nunbers only.

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 *)

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] ];
];

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}]];

Print[PerverseExpression[4,50,True ]];
Print[PerverseExpression[4,50,False]];
Print[PerverseExpression[4,50,False]]; (* Cached *)

```

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