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Re: NIntegrate-FindRoot acting up in version 5.1

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  • Subject: [mg57004] Re: [mg56943] NIntegrate-FindRoot acting up in version 5.1
  • From: Daniel Lichtblau <danl at>
  • Date: Thu, 12 May 2005 02:33:03 -0400 (EDT)
  • References: <>
  • Sender: owner-wri-mathgroup at

John Roberts wrote:
>  I originally made the input shown below in Mathematica 4.1.1. Version 
> 4.1.1 ran it flawlessly and always produced the correct result from 
> NIntegrate with no warnings or error messages. Now, when I run the same 
> notebook with version 5.1.0 it crashes and gives the "FindRoot: :nlnum" 
> message shown below:
> In1:   len = Sqrt[ (z^2 + (x Cos[ang] + r Sin[b] )^2  + (-r Cos[b] - x 
> Sin[ang])^2 ] ;
> In2:  speed = NIntegrate[ (eqn = FindRoot[ len == c b r / v, {b, 
> 0}, WorkingPrecision->100,
>         AccuracyGoal->80 ] ; beta = b  /.  eqn;  fnax) , {ang,  0,  
> 3Pi/2,  2Pi},  WorkingPrecision->80,  AccuracyGoal->9 ]
> Out2: FindRoot: :nlnum : The function value {0. + Sqrt[0.0172266 + (0. + 
> 0.05 <<1>>)^2 + (-0.125 - 0.05 Sin[<<1>>])^2]
>          is not a list of numbers with dimensions {1} at {b} = {0.}.
> As can be seen from the input shown above, NIntegrate integrates the 
> expression fnax with respect to the angle ang. But fnax is also a 
> function of the initial angle beta or b (beta = b), so each time 
> NIntegrate calculates the value of fnax it must first use FindRoot to 
> find the value of beta that corresponds to the value of ang that it is 
> using. I did not include the expression for fnax here because it is 
> rather large, but there is nothing exotic about fnax, it is just a lot 
> of terms with Sin and Cos functions.
> All of the values z, x, r, c v are input with 120 decimal places of 
> precision or with infinite precision (no decimal point).
> It should also be noted that I checked the FindRoot part alone (without 
> NIntegrate) at various points along the range of integration from ang 
> = 0 to ang = 2 Pi, and FindRoot got the correct value of beta at 
> every point with no warnings or error messages; so the problem appears 
> to be associated with how NIntegrate uses FindRoot in Mathematica 5.1 
> rather than with FindRoot itself.
> Thanks in advance for any help you can give me,
> John R.

I got identical garbage in 4.1 and 5.1. Which is as I would expect, 
because the code is attempting numerical algorithms on nonumerical data. 
Even with values for the parameters noted, it is not clear what fnax 
should be. (Moral: Post the code you actually used.)

Here is a way to do this that might be what you have in mind. It gives 
the same numerical result in both versions.

{c,x,r,v,z} = Table[Random[Integer,{1,100}], {5}];
len = Sqrt[z^2 + (x*Cos[ang] + r*Sin[b])^2  +
   (-r*Cos[b] - x*Sin[ang])^2];
integrand[ang_?NumberQ] := b /. FindRoot[len == c*b*r/v, {b,0},
   WorkingPrecision->100, AccuracyGoal->80]
NIntegrate[integrand[ang], {ang,0,3*Pi/2,2*Pi},
   WorkingPrecision->80,  AccuracyGoal->9]

Depending on parameter values and other considerations you may be able 
to dispense with or at least weaken the precision and accuracy requirements.

Daniel Lichtblau
Wolfram Research

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