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Re: question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132531] Re: question
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Tue, 8 Apr 2014 03:38:25 -0400 (EDT)
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  • References: <20140405054854.566716A57@smc.vnet.net>

This does not succeed in evaluating your result; however, may be of use in
the future.

Generally you should use NumericQ rather than NumberQ to restrict arguments
for numerical operations. Unlike NumericQ, NumberQ evaluates to False for
symbolic numbers (i.e., not explicit numbers), for example,

NumericQ/@{3.14159,2.71828,1.61803,Pi,E,GoldenRatio}

{True,True,True,True,True,True}

NumberQ/@{3.14159,2.71828,1.61803,Pi,E,GoldenRatio}

{True,True,True,False,False,False}

Your integrals have significant convergence problems. Catastrophic loss of
precision causes one or more of them to return unevaluated and tau not to
be defined. Even with greater working precision, none of the methods for
NIntegrate converge.

In general, to help maintain precision, use exact numbers whenever possible
(use Rationalize if necessary or more convenient) and Simplify expressions
prior to substituting numerical values.

Clear[beta1,beta2,q1,q2,a,b,c,F1,F2,factor1,factor2,g,y]

workingPrecision=30;
meth="GaussKronrodRule";

r1=1;
r2=2;

beta2[t_]={r1*Cos[2*Pi*t],r2*Sin[2*Pi*t]};
beta1[t_]=beta2[t+48/10*t^2*(t-1)^2];

q2[t_]=beta2'[t]/Sqrt[Norm[beta2'[t]]]//
Simplify[#,Element[t,Reals]]&;

q1[s_]=Assuming[{Element[s,Reals]},
beta1'[s]/Sqrt[Norm[beta1'[s]]]//
FunctionExpand//Simplify];

a[t_,z_]=2*q1[t].q2'[z]//Simplify;
b[t_,z_]=q1[t].q2[z]//Simplify;
c[t_,z_]=2*q1'[t].q2[z]//Simplify;

F1[t_,z_]=c[t,z]/b[t,z]//Simplify;
F2[t_,z_]=a[t,z]/b[t,z]//Simplify;

factor1[s_?NumericQ,z_?NumericQ]:=
Exp[-NIntegrate[F2[s,u],{u,0,z},
WorkingPrecision->workingPrecision,
Method->meth]]

factor2[s_?NumericQ,z_?NumericQ]:=
Exp[NIntegrate[F2[s,u],{u,0,z},
WorkingPrecision->workingPrecision,
Method->meth]]

g[s_?NumericQ,z_?NumericQ]:=
NIntegrate[factor2[s,tau]*F1[s,tau],{tau,0,z},
WorkingPrecision->workingPrecision,
Method->meth]

y[s_?NumericQ,z_?NumericQ]:=
factor1[s,z]*g[s,z]

y[2/10,3/10]

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy
after 9 recursive bisections in u near {u} =
{0.175779977684480217211152986508}. NIntegrate obtained
84.5079246037040969376387922437`30. and 84.8861114104337942809878854747`30.
for the integral and error estimates. >>

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy
after 9 recursive bisections in tau near {tau} =
{0.237889352684480217211152986508}. NIntegrate obtained
-1.54198247933449999324805867741*10^36 and
1.54024583236196812134238868827`30.*^36 for the integral and error
estimates. >>

-0.3067284829323721017658899422


Bob Hanlon



On Sat, Apr 5, 2014 at 1:48 AM, Hagwood, Charles R <charles.hagwood at nist.gov
> wrote:

>
> I am so sick of Mathematica.  It no longer seems to be a  package to do
> applied work, but more for the university types.  I have spent several
> hours using several combinations
> of ?NumberQ  in the following code, but still I get an error.  Last
> weekend I spend several hours using FunctionExpand to get results I can
> read.
>
> Any help appreciated.
>
>
>
> r1=1;
> r2=2;
> beta2[t_]:={r1*Cos[2*Pi*t],r2*Sin[2*Pi*t]}
>
> beta1[t_]:=beta2[t+4.8*t^2*(t-1)^2]
>
> q2[t_]:=
> FunctionExpand[beta2'[t]/Sqrt[Norm[beta2'[t]]],Assumptions->t\[Element]
> Reals && beta2'\[Element]Vectors[2,Reals]]
>
> q1[s_]:=FunctionExpand[
> beta1'[s]/Sqrt[Norm[beta1'[s]]],Assumptions->s\[Element] Reals&&
> beta2'\[Element]Vectors[2,Reals]]
>
>
> a[t_, z_] := 2*q1[t].q2'[z] // FunctionExpand
> b[t_, z_] := q1[t].q2[z] // FunctionExpand
> c[t_, z_] := 2*q1'[t].q2[z] // FunctionExpand
>
>
> F1[t_, z_] := c[t, z]/b[t, z]
> F2[t_, z_] := a[t, z]/b[t, z]
>
>
>
> factor1[s_, z_] := Exp[-NIntegrate[F2[s, u], {u, 0, z}]]
>
> factor2[s_, z_] := Exp[NIntegrate[F2[s, u], {u, 0, z}]]
>
> g[s_, z_?NumberQ] :=  NIntegrate[factor2[s, tau]*F1[s, tau], {tau, 0, z}]
>
> y[s_, z_] := factor1[s, z]*g[s, z]
>
> y[.2, .3]
>
>
> I get the error
>
> NIntegrate::nlim: _u_ = _tau_ is not a valid limit of integration
>
>
>




  • References:
    • question
      • From: "Hagwood, Charles R" <charles.hagwood@nist.gov>
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