The fzero
function attempts
to find a root of one equation with one variable. You can call this
function with either a one-element starting point or a two-element
vector that designates a starting interval. If you give fzero
a
starting point x0
, fzero
first
searches for an interval around this point where the function changes
sign. If the interval is found, fzero
returns a
value near where the function changes sign. If no such interval is
found, fzero
returns NaN
. Alternatively,
if you know two points where the function value differs in sign, you
can specify this starting interval using a two-element vector; fzero
is
guaranteed to narrow down the interval and return a value near a sign
change.
The following sections contain two examples that illustrate
how to find a zero of a function using a starting interval and a starting
point. The examples use the function humps.m
humps.m
,
which is provided with MATLAB^{®}. The following figure shows the
graph of humps
.
x = -1:.01:2; y = humps(x); plot(x,y) xlabel('x'); ylabel('humps(x)') grid on
You can control several aspects of the fzero
function
by setting options. You set options using optimset
.
Options include:
Choosing the amount of display fzero
generates
— see Set Options, Using a Starting Interval, and Using a Starting Point.
Choosing various tolerances that control how fzero
determines
it is at a root — see Set Options.
Choosing a plot function for observing the progress
of fzero
towards a root — see Plot Functions.
Using a custom-programmed output function for observing
the progress of fzero
towards a root —
see Output Functions.
The graph of humps
indicates that the function
is negative at x = -1
and positive at x
= 1
. You can confirm this by calculating humps
at
these two points.
humps(1)
ans = 16
humps(-1)
ans = -5.1378
Consequently, you can use [-1 1]
as a starting
interval for fzero
.
The iterative algorithm for fzero
finds smaller
and smaller subintervals of [-1 1]
.
For each subinterval, the sign of humps
differs
at the two endpoints. As the endpoints of the subintervals get closer
and closer, they converge to zero for humps
.
To show the progress of fzero
at each iteration,
set the Display
option to iter
using
the optimset
function.
options = optimset('Display','iter');
Then call fzero
as follows:
a = fzero(@humps,[-1 1],options)
Func-count x f(x) Procedure 2 -1 -5.13779 initial 3 -0.513876 -4.02235 interpolation 4 -0.513876 -4.02235 bisection 5 -0.473635 -3.83767 interpolation 6 -0.115287 0.414441 bisection 7 -0.115287 0.414441 interpolation 8 -0.132562 -0.0226907 interpolation 9 -0.131666 -0.0011492 interpolation 10 -0.131618 1.88371e-07 interpolation 11 -0.131618 -2.7935e-11 interpolation 12 -0.131618 8.88178e-16 interpolation 13 -0.131618 8.88178e-16 interpolation Zero found in the interval [-1, 1] a = -0.1316
Each value x
represents the best endpoint
so far. The Procedure
column tells you whether
each step of the algorithm uses bisection or interpolation.
You can verify that the function value at a
is
close to zero by entering
humps(a)
ans = 8.8818e-16
Suppose you do not know two points at which the function values
of humps
differ in sign. In that case, you can
choose a scalar x0
as the starting point for fzero
. fzero
first
searches for an interval around this point on which the function changes
sign. If fzero
finds such an interval, it proceeds
with the algorithm described in the previous section. If no such interval
is found, fzero
returns NaN
.
For example, set the starting point to -0.2
,
the Display
option to Iter
,
and call fzero
:
options = optimset('Display','iter'); a = fzero(@humps,-0.2,options)
Search for an interval around -0.2 containing a sign change: Func-count a f(a) b f(b) Procedure 1 -0.2 -1.35385 -0.2 -1.35385 initial interval 3 -0.194343 -1.26077 -0.205657 -1.44411 search 5 -0.192 -1.22137 -0.208 -1.4807 search 7 -0.188686 -1.16477 -0.211314 -1.53167 search 9 -0.184 -1.08293 -0.216 -1.60224 search 11 -0.177373 -0.963455 -0.222627 -1.69911 search 13 -0.168 -0.786636 -0.232 -1.83055 search 15 -0.154745 -0.51962 -0.245255 -2.00602 search 17 -0.136 -0.104165 -0.264 -2.23521 search 18 -0.10949 0.572246 -0.264 -2.23521 search Search for a zero in the interval [-0.10949, -0.264]: Func-count x f(x) Procedure 18 -0.10949 0.572246 initial 19 -0.140984 -0.219277 interpolation 20 -0.132259 -0.0154224 interpolation 21 -0.131617 3.40729e-05 interpolation 22 -0.131618 -6.79505e-08 interpolation 23 -0.131618 -2.98428e-13 interpolation 24 -0.131618 8.88178e-16 interpolation 25 -0.131618 8.88178e-16 interpolation Zero found in the interval [-0.10949, -0.264] a = -0.1316
The endpoints of the current subinterval at each iteration are
listed under the headings a
and b
,
while the corresponding values of humps
at the
endpoints are listed under f(a)
and f(b)
,
respectively.
Note:
The endpoints |
For the first nine steps, the sign of humps
is
negative at both endpoints of the current subinterval, which is shown
in the output. At the tenth step, the sign of humps
is
positive at the endpoint, -0.10949
, but negative
at the endpoint, -0.264
. From this point on, the
algorithm continues to narrow down the interval [-0.10949
-0.264]
, as described in the previous section, until it
reaches the value -0.1316
.