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Re: Re: Solving a numerical integration
*To*: mathgroup at smc.vnet.net
*Subject*: [mg5502] Re: [mg5476] Re: [mg5440] Solving a numerical integration
*From*: Allan Hayes <hay at haystack.demon.co.uk>
*Date*: Sat, 14 Dec 1996 19:26:06 -0500
*Sender*: owner-wri-mathgroup at wolfram.com
fransm at win.tue.nl (Frans Martens)
in [mg5476] Re: [mg5440] Solving a numerical integration
Gave the attached solution.
This can be speeded up (and maybe simplified ?) by using NDSolve
instead of NIntegrate - see below.
We could to automate the search for an approximate zero
economically by gradually extending the solution and looking for a
change of sign.
I compare the times of analous parts of the two solutions.
New solution using NDSolve:
f[x_]:= Sqrt[1+x +x^2] -3
soln = y/.First[NDSolve[{y'[x]==f[x], y[0]==0},y,{x,0,6}]];//Timing
{0.133333 Second, Null}
Plot[soln[x],{x,0,6}];//Timing
{0.2 Second, Null}
FindRoot[soln[x]==0,{x, 4.8}, Jacobian -> f[x]]//Timing
{0.0166667 Second, {x -> 4.66231}}
Frans' Solution using NIntegrate:
Clear[int]
int[x_]:= NIntegrate[f[t],{t,0,x},
AccuracyGoal -> 6];//Timing
{0.0166667 Second, Null}
Plot[int[x],{x,0,6}];//Timing
{3.8 Second, Null}
FindRoot[int[x]==0,{x,4.8}, Jacobian -> f[x]]//Timing
{0.783333 Second, {x -> 4.66231}}
Allan Hayes
hay at haystack.demon.cc.uk
http://www.haystack.demon.co.uk
******************
Frans' posting
Pere Llosas wrote:
> I would like to solve an equation of this kind
>
> NIntegrate[f[n],{n,0,x}]==0 (f[x_]=Sqrt[1+x+x^2...)
> where x is the searched value, and f cannot be integrated
> analytically.
>
> NSolve[NIntegrate[f[n],{n,0,x}]==0,x], tries to evaluate
> NIntegrate[f[n],{n,0,x}] before assignin a numerical value to x and
> returns an error.
>
> How could this calculation be done without having to write a
> program that searches the root?
The equation NIntegrate[f[n],{n,0,x}]==0 has root x = 0 and the
NSolve function tries to compute the inverse of the function x |->
NIntegrate[f[n],{n,0,x}] .
The function FindRoot is more suitable. Here is an example with a
second root in the neigbourhood of x = 4.8 .
In[25]:=
Clear[int,f]
int[x_]:=NIntegrate[f[t],{t,0,x},
AccuracyGoal -> 6];
f[x_]:=Sqrt[1+x+x^2]-3;
In[28]:=
FindRoot[int[x]==0,{x,4.8},
Jacobian -> f[x]]
Out[28]=
{x -> 4.66231}
>>>>> The whole above without messages <<<<<<<<<
There are two precautions:
1)The option AccuracyGoal in NIntegrate is set to 6 because the
integral int[x] equals zero for the root x.
2)FindRoot uses the method of Newton-Raphson and setting the option
Jacobian prevents the symbolic computation of int'[x] . Note that
int'[x] equals f[x].
You must have a global idea of the roots of the original equation.
Frans Martens
Eindhoven
The Netherlands
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