RE: Transcendental equation

[Allen.Windhorn]

Guido,

> -----Original Message-----
> From: address@hidden
> 
> I implemented your solution and it is almost correct. This is the
> result:
> 
> a =   0.20276    3.83709    7.01853   10.17550   13.32524
> 16.47188
> 19.61691   22.76099   25.90447   29.04754
> 
> The problem is that I can only find the odd roots (the first, the
> third, the fifth, and so on), infact between each couple of values
> saved in the vector /a/ there is another value that it is not
> saved. I tried changing your solution (*i*pi+pi/4*) in something
> more refined like *i*pi/2+pi/4* and this is the result:
> 
> a =   0.20276    0.20276    3.83709    3.83710    7.01853
> 7.01853
> 10.17550   10.17551   13.32524   13.32526
> 
> I still find the same values than before but this time each one is
> found twice, which clearly is not what I want.
> What am I missing?

Look at the graph of the function.  You are finding all the roots.
Between each pair of roots is a singularity which is not being
found, but this is not the same as a root, since the function
"passes through" infinity, not zero.  It looks like a root on the
graph, only because the plot function is working on discrete
points and doesn't know about infinity.

Regards,
Allen

> Here is my full code:
> 
> D=0.05
> h=100
> k=60.5
> Lc=D/4
> Bi=h*Lc/k
> N=10;
> 
> # Function plot in the interval between x=0 and x=20 to have an
> # idea of the roots' values 
> x=[0.0001:0.001:20];
> y=x.*(besselj(1,x))./(besselj(0, x)).-Bi;
> # [Define f here, then y = f(x), or f(x, Bi)]
> plot(x, y, '-r')
> axis([0 20 -1 1])
> 
> a=[1:N];
> for i=0:(N-1)
>       f=@(x)(x*(besselj(1, x))/(besselj(0, x))-Bi); 
>     # [don't need to define every time, just once]
>       a(i+1)=fsolve(f, (i*pi+pi/4));
> end