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10.1.1: fzero() Examples and Exercises

  • Page ID
    87912
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      fzero() is a "function of a function", because it needs a "handle" to a function that defines the equation whose root it will find. fzero() can be use either to find a zero of a single functions and or to find the intersection point of 2 functions.

      Using fzero() to find the root of a single function

      The way it works is as follows: It finds an interval containing the initial point. It uses nearby points to approximate derivatives and estimate where the zero is. It then re-evaluates the function at this new x value. It repeats this procedure until it finds a point where the function is zero or very nearly zero.

      Example \(\PageIndex{1}\) Find a root of a 5th-order polynomial:

      We will find a root of this 5th-order polynomial:

      y = x.^5 - 4*x.^2 -10*x + 1;

      Method 1: A straightforward way to do this is to create a function file for this:

      % fpoly5.m:

      function y = fpoly5(x)

      y = x.^5 - 4*x.^2 -10*x + 1;

      end

      A separate m-file script (fzero_poly5.m) is created to visualize this function and to estimate the root.

      %% A. Open a new figure and plot the function
      x = -2:0.05:2.2;

      y = fpoly5(x);

      figure

      plot(x,y)

      grid on;

      %% B. the value of x near x=2.1 that makes y = 0

      x_solution = fzero(@fpoly5, 2.1) % @fpoly5 is a function "handle" to the file fpoly5.m

      % fzero() evaluates the function fpoly5(x) multiple times, until it converges to a root. (A root is a value that makes the function = 0)

      % x_solution = 2.0517

      % Verify that this is a solution:

      yb = fpoly5(x_solution)

      % 7.1054e-15

      hold on;

      plot(x_solution,yb,'o')

      ----

      Method 2: Another way is to create a local sub-function at the bottom of the m-file script. This method currently works in MATLAB, but not Octave.

      Solution

      fzero_poly5.png

      .

      Example \(\PageIndex{2}\) fzero_y_fun1

      %% A. Create this function

      function y1 = y_fun1(x)
      y1 = log(x)./x.^2 -0.1;
      end

      %% B. (1 pt) Create an x vector from 0.2 to 2.0
      % Choose an increment for x, such that the plot will be smooth--that is, no visible straight lines.
      x = 1: 0.1 :6;
      % (1 pt) Evaluate the equation using your x vector
      y1 = y_fun1(x);

      %% C. (1 pt) Open figure and plot this function.
      % A plot of the function lets us estimate a root.

      figure(1)
      plot(x,y1)
      grid on;
      title('fzero1root\_fcnfile\_example.m CSmith')
      hold on;
      plot(x,zeros(size(x)),'r') % Plot a line for y = 0
      %% D. (4 pts) Use the fzero() function to find a root near x = 4

      Solution

      x_solution = fzero(@y_fun1, 4)
      % @y_fun1 = "function handle" to y_fun1.m
      % The function y_fun1(x) must have a single input variable.
      % fzero chooses the values of x for computing the function
      % x_solution = 3.5656

      %% E. fzero() can also be used to find a root between 2 values
      x_solution = fzero(@y_fun1, [3,5])

      .

      Exercise \(\PageIndex{3}\) fzero for y = log10(x) + 0.44;

      %% Create a function m-file that computes y = log10(x) + 0.44;
      % Set
      x = 0 : 0.02 : 1;

      % Compute y for this x vector using your function.
      % Open a figure and plot(x, y)
      % Turn the grid on

      % Use fzero() to find a root near 0.4 of this function

      Answer

      Add texts here. Do not delete this text first.

      .

       


      This page titled 10.1.1: fzero() Examples and Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.