MATLAB Function Reference

fzero

Find zero of a function of one variable

Syntax

• x = fzero(fun,x0)
  x = fzero(fun,x0,options)
  [x,fval] = fzero(...)
  [x,fval,exitflag] = fzero(...)
  [x,fval,exitflag,output] = fzero(...)
  

Description

x = fzero(fun,x0) tries to find a zero of fun near x0, if x0 is a scalar. fun is a function handle. See Function Handles in the MATLAB Programming documentation for more information. The value x returned by fzero is near a point where fun changes sign, or NaN if the search fails. In this case, the search terminates when the search interval is expanded until an Inf, NaN, or complex value is found.

Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function fun, if necessary.

If x0 is a vector of length two, fzero assumes x0 is an interval where the sign of fun(x0(1)) differs from the sign of fun(x0(2)). An error occurs if this is not true. Calling fzero with such an interval guarantees fzero will return a value near a point where fun changes sign.

x = fzero(fun,x0,options) minimizes with the optimization parameters specified in the structure options. You can define these parameters using the optimset function. fzero uses these options structure fields:

Display	Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'final' displays just the final output; 'notify' (default) displays output only if the function does not converge.
FunValCheck	Check whether objective function values are valid. 'on' displays a warning when the objective function returns a value that is complex or NaN. 'off' (the default) displays no warning.
OutputFcn	Specify a user-defined function that the optimization function calls at each iteration.
TolX	Termination tolerance on x

[x,fval] = fzero(...) returns the value of the objective function fun at the solution x.

[x,fval,exitflag] = fzero(...) returns a value exitflag that describes the exit condition of fzero:

1	Function converged to a solution x.
-1	Algorithm was terminated by the output function.
-3	NaN or Inf function value was encountered during search for an interval containing a sign change.
-4	Complex function value was encountered during search for an interval containing a sign change.
-5	fzero might have converged to a singular point.

[x,fval,exitflag,output] = fzero(...) returns a structure output that contains information about the optimization:

output.algorithm	Algorithm used
output.funcCount	Number of function evaluations
output.intervaliterations	Number of iterations taken to find an interval
output.iterations	Number of zero-finding iterations
output.message	Exit message

Note    For the purposes of this command, zeros are considered to be points where the function actually crosses, not just touches, the x-axis.

Arguments

fun is the function whose zero is to be computed. It accepts a vector x and returns a scalar f, the objective function evaluated at x. The function fun can be specified as a function handle for an M-file function

• x = fzero(@myfun,x0);
  

where myfun is an M-file function such as

• function f = myfun(x)
  f = ...            % Compute function value at x
  

or as a function handle for an anonymous function:

• x = fzero(@(x)sin(x*x),x0);
  

Other arguments are described in the syntax descriptions above.

Examples

Example 1. Calculate by finding the zero of the sine function near 3.

• x = fzero(@sin,3)
  x =
      3.1416
  

Example 2. To find the zero of cosine between 1 and 2

• x = fzero(@cos,[1 2])
  x =
  

        1.5708

Note that cos(1) and cos(2) differ in sign.

Example 3. To find a zero of the function

write an anonymous function f:

• f = @(x)x.^3-2*x-5;
  

Then find the zero near 2:

• z = fzero(f,2)
  z =
      2.0946
  

Because this function is a polynomial, the statement roots([1 0 -2 -5]) finds the same real zero, and a complex conjugate pair of zeros.

•     2.0946
     -1.0473 + 1.1359i
     -1.0473 - 1.1359i
  

If fun is parameterized, you can use anonymous functions to capture the problem-dependent parameters. For example, suppose you want to minimize the objective function myfun defined by the following M-file function.

• function f = myfun(x,a)
  f = cos(a*x);
  

Note that myfun has an extra parameter a, so you cannot pass it directly to fzero. To optimize for a specific value of a, such as a = 2.

1. Assign the value to a.

• a = 2; % define parameter first
  

2. Call fzero with a one-argument anonymous function that captures that value of a and calls myfun with two arguments:

• x = fzero(@(x) myfun(x,a),0.1)
  

Algorithm

The fzero command is an M-file. The algorithm, which was originated by T. Dekker, uses a combination of bisection, secant, and inverse quadratic interpolation methods. An Algol 60 version, with some improvements, is given in [1]. A Fortran version, upon which the fzero M-file is based, is in [2].

Limitations

The fzero command finds a point where the function changes sign. If the function is continuous, this is also a point where the function has a value near zero. If the function is not continuous, fzero may return values that are discontinuous points instead of zeros. For example, fzero(@tan,1) returns 1.5708, a discontinuous point in tan.

Furthermore, the fzero command defines a zero as a point where the function crosses the x-axis. Points where the function touches, but does not cross, the x-axis are not valid zeros. For example, y = x.^2 is a parabola that touches the x-axis at 0. Because the function never crosses the x-axis, however, no zero is found. For functions with no valid zeros, fzero executes until Inf, NaN, or a complex value is detected.

See Also

roots, fminbnd, optimset, function_handle (@), anonymous functions

References

[1]  Brent, R., Algorithms for Minimization Without Derivatives, Prentice-Hall, 1973.

[2]  Forsythe, G. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1976.

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