[Maxima-bugs] [ maxima-Bugs-3123933 ] Incorrect solution to equation

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Bugs item #3123933, was opened at 2010-11-30 23:29
Message generated for change (Comment added) made by crategus
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Category: Lisp Core - Solving equations
Group: None
>Status: Pending
>Resolution: Works For Me
Priority: 5
Private: No
Submitted By: quwerty (doe140)
Assigned to: Nobody/Anonymous (nobody)
Summary: Incorrect solution to equation

Initial Comment:
I have the foolowing code:
c:3e+8$
mu_r1:1$
mu_r2:1$
sigma_1:0$
sigma_2:0$
eps_r1:3.3$
eps_r2:1.6$
h:1e-6$
f:180e+12$
P2:0.01e-3$
eps_0:8.85e-12$
mu_0:4*%pi*1e-7$
eps_a1:eps_r1*eps_0$
eps_a2:eps_r2*eps_0$
mu_a1:mu_0*mu_r1$
mu_a2:mu_0*mu_r2$
omega:2*%pi*f$
/* [wxMaxima: input   end   ] */
/* [wxMaxima: input   start ] */
kill(x, y)$
x:find_root(sqrt(omega^2*h^2*(eps_a1*mu_a1-eps_a2*mu_a2)-x^2)+eps_a2/eps_a1*x*tan(x)=0, x, 0, 5);
y:float(sqrt(omega^2*h^2*(eps_a1*mu_a1-eps_a2*mu_a2)-%^2));
y1:float(-eps_a2/eps_a1*x*tan(x))$

With these limits  y!=y1. If I run 

x:find_root(..., x, 1.7, 2);

I get correct answer. This seems a little strange. May this be of tan(x)'s behaviour at x=%pi/2?

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>Comment By: Dieter Kaiser (crategus)
Date: 2010-12-03 21:03

Message:
At first, it is very helpful to have a small example. I think the function
of interest for this example is

eqn: 0.4848484848484849*x*tan(x)+sqrt(0.7799327999999998*%pi^3-x^2)

One important and documented restriction of the algorithm of find_root is,
that the function has to be continuous over the interval, which is given as
an argument to find_root. If I do a plot of the function for the interval
[0,5] with the command

   plot2d(eqn,[x,0,5],[y,-100,100]) 

I can see that the function is not continuous at two points and that the
function has one root. Furthermore, the root is near the value 1.75.

If I take the function of this example and the interval [1.6,2.0] I get
the desired root:

(%i1) eqn:.4848484848484849*x*tan(x)+sqrt(.7799327999999998*%pi^3-x^2);
(%o1) .4848484848484849*x*tan(x)+sqrt(.7799327999999998*%pi^3-x^2)

This is the root of the equation:

(%i2) result:find_root(eqn,x,1.6,2);
(%o2) 1.753812411159384

Backsubstitution shows that the result is a root within an expected
accuracy

(%i3) float(subst(result,x,eqn));
(%o3) -1.77635683940025e-15

The algorithm of find_root fails in the interval [0,5] because the
function is not continuous over the interval.

I think we do not have a bug. The numerical routine find_root has
documented limitations. Setting the status to pending and the resolution to
"works for me".

Dieter Kaiser

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