Using fzero and Applications

Azar Shakoori

UOIT

MATH 2070U

Using fzero and Applications

Using MATLABs fzero

Solving nonlinear equations with parameters

Other solvers

fzero

I MATLAB has practical zero-finding routine fzeroI Can specify initial interval I(0) = [a(0), b(0)] bracketing zero

f = @(x) x-cos(x) % Function to find zeros

I0 = [0.5, 1.0]; % Initial interval

x = fzero( f, I0 )

I Alternatively, fzero accepts a single value x(0) as initialiteratef = @(x) x-cos(x) % Function to find zeros

x0 = 1.0; % Initial iterate

x = fzero( f, x0 )

fzerogui: Illustration of algorithm for fzero

I Graphical interface to fzeroI Can select iterates with mouseI Iterates generated from either

I bisection methodI secant methodI inverse quadratic interpolation

(IQI)I Some examples to try:

I x3 2x 5 on [0, 3]I ln(x+ 2/3) on [0, 1]

Syntax for fzero

x = fzero(fun,x0)

x = fzero(fun,x0,options)

[x,fval] = fzero(...)

[x,fval,exitflag] = fzero(...)

[x,fval,exitflag,output] = fzero(...)

I options is a structure whose fields can specify, displaypreferences, tolerances, maximum number of iterations,etc.

I fzero is a function function: expects function handle funas input

fzero options

I Options may be passed to fzero as a third input argumentI The options are a data structure created by the optimset

commandI options = optimset(par1, val1, par2, val2,)

I parn is the name of the parameter to be setI valn is the value to which to set that parameterI The parameters commonly used with fzero are:I display: when set to iter displays a detailed record of all

the iterationsI tolx: A positive scalar that sets a termination tolerance on

x.

Class activity

Find (any) solutions of the following equation with fzero:

3+ 6t2 = 4t+ 8 (1)

Strategy for equation (1)

I Using an anonymous function in MATLAB define yournonlinear function f

I Use fplot to plot f from t = 2 to t = 6I Use the information from the plot to find out how many

roots are there and in what neighborhood they are located.I Use fzero to find the roots.I Compute the residuals to verify the zeros are correct.

Class activity contd

Find (any) solutions of the following equation with fzero:

x3ex +32= 0 (2)

Strategy for equation (2)

I Again, using anonymous function in MATLAB define yournonlinear function g

I Generate a plot from t = 2 to t = 0. The plot revealsinformation about the roots.

I Use the information from the plot and fzero to find theroots.

Class activity contd

The following MATLAB transcript is used to find the threesmallest positive solutions of the nonlinear equation

x = cot x for = 1.

What is wrong with our solutions?

Nonlinear functions with parameters

I Parametrised families of nonlinear equations are common

e.g., sin x+ ex2/2 = 0 where , R

I Possible approaches to solve parametrised equations1. use extra parameters as global variables in function m-files2. define functions to accept extra parameters & use solvers

that accept variable length lists of inputs (c.f. bisection.m,newton)

3. use anonymous functions to encapsulate functions withfixed parameter values

2 1.5 1 0.5 0 0.5 1 1.5 21

0.5

0

0.5

1

1.5

2

2.5y = sin x+ex

2/2, = 1, = 2

x

y

1 0.5 0 0.5 11.5

1

0.5

0

0.5

1

1.5y = sin x+ex

2/2, = 3, = 1

x

y

Example: using global variablesSolve a sin x+ bex2/2 = 0 when a = 1, b = 2 & when a = 3,b = 1

Example: using functions with extra parameters

Example: using anonymous functionsSolve a sin x+ bex2/2 = 0 when a = 1, b = 2 & when a = 3,b = 1

Other solvers

ROOTS

I Polynomial equations: use roots in MATLABI Based on eigenvalue solver, companion matrixI Requires MATLAB convention for polynomials:

Example

>> p = [1, -2, 0, 3, -5]; % p= x^4-2x^3+3x-5

>> roots(p)

>> ans =

-1.3934

1.8984

0.7475 + 1.1539i

0.7475 - 1.1539i

Using Matlab's fzero Solving nonlinear equations with parametersOther solvers