# fzero

## Purpose

Zero of a function of one variable.

## Synopsis

    z = fzero('function',x0)
    z = fzero('function',x0,tol)
    z = fzero('function',x0,tol,trace)

## Description

`fzero('function',x0)` finds a zero of the function `function(x)` that is near `x0`. `fzero` identifies only points where the function actually crosses the x-axis. Points where the function touches the x-axis, but does not cross it, are not considered zeros.

`fzero('function',x0,tol)` returns an answer accurate to within a relative error of `tol`. The default value for `tol` is `eps`.

`z = fzero('function',x0,tol,trace)` displays information at each iteration if `trace` is nonzero.

## Examples

    Calculate pi by finding the zero of the sine function near 3.
    x = fzero('sin',3)

    x =
        3.1416

To find a zero of the function

write an M-file called `f`.`m`.

    function y = f(x)
    y = x.^3-2*x-5;

To find the zero near 2

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

Since this function is actually 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

## Algorithm

`fzero` 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

`fzero` 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 (0,0). Since the function never crosses the x-axis, however, no zero is found. For functions with no valid zeros, `fzero` executes until you terminate it.

## See Also

    eps, fmin, roots

## References

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

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

(c) Copyright 1994 by The MathWorks, Inc.