fzero - Root of nonlinear function - MATLAB (original) (raw)

Root of nonlinear function

Syntax

Description

[x](#btoc6lj-1-x) = fzero([fun](#btoc6lj-1-fun),[x0](#btoc6lj-1-x0)) tries to find a point x where fun(x) = 0. This solution is where fun(x) changes sign—fzero cannot find a root of a function such as x^2.

example

[x](#btoc6lj-1-x) = fzero([fun](#btoc6lj-1-fun),[x0](#btoc6lj-1-x0),[options](#btoc6lj-1-options)) uses options to modify the solution process.

example

[x](#btoc6lj-1-x) = fzero([problem](#btoc6lj-1-problem)) solves a root-finding problem specified by problem.

example

[[x](#btoc6lj-1-x),[fval](#btoc6lj-1-fval),[exitflag](#btoc6lj-1-exitflag),[output](#btoc6lj-1-output)] = fzero(___) returns fun(x) in the fval output, exitflag encoding the reason fzero stopped, and an output structure containing information on the solution process.

example

Examples

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Calculate π by finding the zero of the sine function near 3.

fun = @sin; % function x0 = 3; % initial point x = fzero(fun,x0)

Find the zero of cosine between 1 and 2.

fun = @cos; % function x0 = [1 2]; % initial interval x = fzero(fun,x0)

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

Find a zero of the function f(x) = x_3 – 2_x – 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)

Since f(x) is a polynomial, you can find the same real zero, and a complex conjugate pair of zeros, using the roots command.

ans = 2.0946
-1.0473 + 1.1359i -1.0473 - 1.1359i

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)

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)

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.

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]

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

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Function to solve, specified as a handle to a scalar-valued function or the name of such a function. fun accepts a scalarx and returns a scalarfun(x).

fzero solves fun(x) = 0. To solve an equation fun(x) = c(x), instead solvefun2(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

Initial value, specified as a real scalar or a 2-element real vector.

• Scalar — fzero begins at x0 and tries to locate a point x1 where fun(x1) has the opposite sign of fun(x0). Then fzero iteratively shrinks the interval where fun changes sign to reach a solution.
• 2-element vector — fzero checks that fun(x0(1)) and fun(x0(2)) have opposite signs, and errors if they do not. It then iteratively shrinks the interval where fun changes sign to reach a solution. An interval x0 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 for solution process, specified as a structure. Create or modify the options structure using optimset.fzero uses these options structure fields.

Example: options = optimset('FunValCheck','on')

Data Types: struct

Root-finding problem, specified as a structure with all of the following fields.

For an example, see Solve Problem Structure.

Data Types: struct

Output Arguments

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Location of root or sign change, returned as a scalar.

Function value at x, returned as a scalar.

Integer encoding the exit condition, meaning the reason fzero stopped its iterations.

Information about root-finding process, returned as a structure. The fields of the structure are:

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 forfzero.

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

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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 for TolX and FunValCheck.
• fzero does not support the fourth output argument, the output structure.

Version History

Introduced before R2006a