MATLAB: Positive roots of trigonometric equation.

fzerorootstrigonometric

Dear all,
What would be a sound and reliable way for finding the first n positive roots of the following?
 a   = 918 ;
 b   = 47 ; 
 c   = 3e-2 ;
 fun = @(x) x .* tan(x) - (a-x.^2) ./ (b + (a-x.^2)*c) ;
My first idea was to use FZERO from a large number of starting values, keep unique (with some tolerance) positive roots, and eliminate those close (k+1/2)pi .. but there has to be a sounder and more reliable approach (?)
Thank you,
Cedric
PS/EDITs:
  • And eliminate as well roots of the denominator.

Best Answer

The function
     f(x) = x .* tan(x)
is continuous and strictly monotonically increasing in every interval [-pi/2,pi/2]+k*pi while the function
     g(x) = (a-x.^2) ./ (b + (a-x.^2)*c)
is continuous and strictly monotonically decreasing everywhere except at its unique pole xp.
Because f and g have opposite monotonicity in each interval [-pi/2,pi/2]+k*pi not containing xp, and because f(x) ranges from -inf to +inf there, the equation
                    f(x)=g(x)
will have exactly one solution in each of those intervals. These solutions are identically the roots of f(x)-g(x). The anomalous interval [-pi/2,pi/2]+k0*pi containing the pole xp can be handled by splitting it into 2 subintervals to the left and right of xp. Each of those subintervals must have exactly one root by the same monotonicity arguments.
Thus, you need simply loop over the first successive n of these intervals,subdividing the k0-th interval appropriately. In each interval, use fzero to find the unique root there.
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