fzero
Root of nonlinear function
Syntax
Description
Examples
Root Starting From One Point
Calculate by finding the zero of the sine function near 3
.
fun = @sin; % function x0 = 3; % initial point x = fzero(fun,x0)
x = 3.1416
Root Starting from an Interval
Find the zero of cosine between 1
and 2
.
fun = @cos; % function x0 = [1 2]; % initial interval x = fzero(fun,x0)
x = 1.5708
Note that and differ in sign.
Root of a Function Defined by a File
Find a zero of the function f(x) = x3 – 2x – 5.
First, write a file called f.m
.
function y = f(x)
y = x.^3 - 2*x - 5;
Save f.m
on your MATLAB® path.
Find the zero of f(x)
near 2
.
fun = @f; % function x0 = 2; % initial point z = fzero(fun,x0)
z = 2.0946
Since f(x)
is a polynomial, you can
find the same real zero, and a complex conjugate pair of zeros, using
the roots
command.
roots([1 0 -2 -5])
ans = 2.0946 -1.0473 + 1.1359i -1.0473 - 1.1359i
Root of Function with Extra Parameter
Find the root of a function that has an extra parameter.
myfun = @(x,c) cos(c*x); % parameterized function c = 2; % parameter fun = @(x) myfun(x,c); % function of x alone x = fzero(fun,0.1)
x = 0.7854
Nondefault Options
Plot the solution process by setting some plot functions.
Define the function and initial point.
fun = @(x)sin(cosh(x)); x0 = 1;
Examine the solution process by setting options that include plot functions.
options = optimset('PlotFcns',{@optimplotx,@optimplotfval});
Run fzero
including options
.
x = fzero(fun,x0,options)
x = 1.8115
Solve Problem Structure
Solve a problem that is defined by a problem structure.
Define a structure that encodes a root-finding problem.
problem.objective = @(x)sin(cosh(x)); problem.x0 = 1; problem.solver = 'fzero'; % a required part of the structure problem.options = optimset(@fzero); % default options
Solve the problem.
x = fzero(problem)
x = 1.8115
More Information from Solution
Find the point where exp(-exp(-x)) = x
, and display information about the solution process.
fun = @(x) exp(-exp(-x)) - x; % function x0 = [0 1]; % initial interval options = optimset('Display','iter'); % show iterations [x fval exitflag output] = fzero(fun,x0,options)
Func-count x f(x) Procedure 2 1 -0.307799 initial 3 0.544459 0.0153522 interpolation 4 0.566101 0.00070708 interpolation 5 0.567143 -1.40255e-08 interpolation 6 0.567143 1.50013e-12 interpolation 7 0.567143 0 interpolation Zero found in the interval [0, 1]
x = 0.5671
fval = 0
exitflag = 1
output = struct with fields:
intervaliterations: 0
iterations: 5
funcCount: 7
algorithm: 'bisection, interpolation'
message: 'Zero found in the interval [0, 1]'
fval
= 0 means fun(x) = 0
, as desired.
Input Arguments
fun
— Function to solve
function handle | function name
Function to solve, specified as a handle to a scalar-valued function or
the name of such a function. fun
accepts a scalar
x
and returns a scalar
fun(x)
.
fzero
solves fun(x) = 0
.
To solve an equation fun(x) = c(x)
, instead solve
fun2(x) = fun(x) - c(x) = 0
.
To include extra parameters in your function, see the example Root of Function with Extra Parameter and the section Parameterizing Functions.
Example: 'sin'
Example: @myFunction
Example: @(x)(x-a)^5 - 3*x + a - 1
Data Types: char
| function_handle
| string
x0
— Initial value
scalar | 2-element vector
Initial value, specified as a real scalar or a 2-element real vector.
Scalar —
fzero
begins atx0
and tries to locate a pointx1
wherefun(x1)
has the opposite sign offun(x0)
. Thenfzero
iteratively shrinks the interval wherefun
changes sign to reach a solution.2-element vector —
fzero
checks thatfun(x0(1))
andfun(x0(2))
have opposite signs, and errors if they do not. It then iteratively shrinks the interval wherefun
changes sign to reach a solution. An intervalx0
must be finite; it cannot contain ±Inf
.
Tip
Calling fzero
with an interval (x0
with
two elements) is often faster than calling it with a scalar x0
.
Example: 3
Example: [2,17]
Data Types: double
options
— Options for solution process
structure, typically created using optimset
Options for solution process, specified as a structure. Create or modify
the options
structure using optimset
.
fzero
uses these options
structure
fields.
| Level of display:
|
| Check whether objective function values are valid.
|
| Specify one or more
user-defined functions that an optimization function
calls at each iteration, either as a function handle
or as a cell array of function handles. The default
is none ( |
| Plots various measures of
progress while the algorithm executes. Select from
predefined plots or write your own. Pass a function
name, function handle, or a cell array of function
names or handles. The default is none
(
For information on writing a custom plot function, see Optimization Solver Plot Functions. |
| Termination tolerance on
|
Example: options =
optimset('FunValCheck','on')
Data Types: struct
problem
— Root-finding problem
structure
Root-finding problem, specified as a structure with all of the following fields.
| Objective function |
| Initial point for x ,
real scalar or 2-element vector |
| 'fzero' |
| Options structure, typically created
using optimset |
For an example, see Solve Problem Structure.
Data Types: struct
Output Arguments
x
— Location of root or sign change
real scalar
Location of root or sign change, returned as a scalar.
fval
— Function value at x
real scalar
Function value at x
, returned as a scalar.
exitflag
— Integer encoding the exit condition
integer
Integer encoding the exit condition, meaning the reason fzero
stopped
its iterations.
| Function converged to a solution |
| Algorithm was terminated by the output function or plot function. |
|
|
-4 | Complex function value was encountered while searching for an interval containing a sign change. |
-5 | Algorithm might have converged to a singular point. |
-6 |
|
output
— Information about root-finding process
structure
Information about root-finding process, returned as a structure. The fields of the structure are:
intervaliterations | Number of iterations taken to find an interval containing a root |
iterations | Number of zero-finding iterations |
funcCount | Number of function evaluations |
algorithm |
|
message | Exit message |
Algorithms
The fzero
command
is a function file. The algorithm, created 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 fzero
is
based, is in [2].
Alternative Functionality
App
The Optimize Live Editor task provides a visual interface for
fzero
.
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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
For C/C++ code generation:
The
fun
input argument must be a function handle, and not a structure or character vector.fzero
ignores all options except forTolX
andFunValCheck
.fzero
does not support the fourth output argument, the output structure.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006a
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