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Finding the roots!

  1. Dec 12, 2012 #1
    How would you find the roots of:
    b - tan (b) = 0

    please do not that i have to plot the graphs of y=b and y=tan b and then i should find the solution. I want to know how to do it the other way.
    thank you
  2. jcsd
  3. Dec 12, 2012 #2


    Staff: Mentor

    Another way, besides a graphical solution, is to use a numerical approximation technique such as Newton's Method (also known as Newton-Raphson). If you want to find out more, you can do a web search, which should generate lots of hits.
  4. Dec 12, 2012 #3


    User Avatar
    Homework Helper

    There are an infinite number. You can use an iterative method like Newton's mathod. There are also some asymptotic expansions, the large values of x are approximately pi/2+n pi for some large n.
  5. Dec 14, 2012 #4
    For some iteration initial estimates, consider the following:

    c_m = \frac{2\,m + 1}{2} \, \pi

    Then let the initial estimate [itex]b^{(0)}[/itex] be given by
    b^{(0)} = c_m - u

    Three possible initial estimates, in increasing accuracy, can be given by the three separate values of u as follows

    [tex]u_2 = \frac{1}{c_m}[/tex]
    [tex]u_3 = \frac{1}{c_m} + \frac{1}{c_m^3}[/tex]
    [tex]u_5 = \frac{1}{c_m} + \frac{2}{3}\, \frac{1}{c_m^3}[/tex]
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