More Functions in MATLAB

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More Functionsin MATLAB
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Functions that operate on other functions

• A function F() can take another function G() as
  an argument by using a special _at_
  notationF(_at_G,) F() takes G() as an argument
• Matlab documentation refers to these as function
  functions
• The operator _at_ produces what is called a handle
  to the function whose name follows the symbol.
  Although the text discusses 3 other methods for
  passing a function in an argument list to another
  function, the only method acceptable in this
  course is the function handle.

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Functions that operate on other functions

• Let's look at three useful Matlab functions
• fzero() - finds a zero of a function
• fminbnd() - finds a minimum of a function
• quad() - numerically evaluates a definite
  integral over a function

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fzero() - Finds a zero of a function of one
variable (solves Fname(x)0)

• x fzero(_at_Fname,x0)
• Fname a function name
• x0 either a single
• x value starting point or
• xL xR x value range with Fname(xL)being
  opposite in sign from Fname(xR)

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humps(x) - example function
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x fzero(_at_Fname,x0)

• finds x near 1 so humps(x)0
• gtgt fzero(_at_humps,1)
• ans 1.2995
• finds x between 0 and 2 so humps(x) 0
• gtgt fzero(_at_humps,0,2)ans 1.2995

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x2 - 2
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x fzero(_at_Fname,x0)

• First define G(x) in a file G.mfunction y
  G(x)
• y x2 - 2
• Then
• gtgt x fzero(_at_G,0,2)x 1.4142
• gtgt x fzero(_at_G,-2,0)x -1.4142

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fzero() - Additional input variables

• x fzero(_at_Fname,x0,,P1,P2,...)
• P1,P2,...
• These are additional input variables (or
  parameters)
• required by Fname
• after the first parameters
• needed by fzero.

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fzero() - Additional input variables

• x fzero(_at_Fname,x0,,P1,P2,...)
• P1,P2,...
• These are additional input variables (or
  parameters) required by Fname after the first
  parameters needed by fzero.

This is a placeholder. Since there can be two
parameters for fzero, and we only are using one,
we need a null placeholder, (null matrix) to hold
this position.
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Poly4(x) x4 Cx2BxA

• function y Poly4(x,C,B,A)
• function to evaluate x4Cx2BxA
• x can be an array but C,B,A are assumed to be
  scalars
• y x.4Cx.2BxA

This is a sample, user-defined function we will
use to illustrate the ideas of these functions
that pass functions.
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Poly4(x,C,B,A) - with C-3, B0, A0.5
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x fzero(_at_Fname,x0,,P1,P2,...)

• gtgt fzero(_at_Poly4,.5,,-3,0,.5)
• Warning ...
• ans
• 0.4209
• You can do
• warning off
• fzero(...)
• warning on

This warning can be distracting. So
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fzero() - Additional output variables

• x,fval fzero(...)
• x,fval,exitflag fzero(...)
• fval value of fun at solution
• exitflag gt0 if zero is found
• lt0 if zero is not found

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x,fval,exit fzero(...)

• gtgt x,fval,exit fzero(_at_humps,1)
• x
• 1.2995
• fval
• 0
• exit
• 1

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fzero and solving equations

• Given values for A,P, and n, solve the following
  for r

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Rewrite Equation in form f(r,n,A,P)0
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Define Matlab function

• function val intfun(r,n,A,P)
• r is interest rate, n is number of months, A is
  
• amount of loan, P is monthly payment
• Top (r/12) (1r/12) n
• Bot (1r/12) n 1
• val P A Top/Bot

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Use fzero to find one real solution

• Ratefzero(_at_intfun,.1, ,48,21000,500)
• disp('The interest rate that you need is ',
• numstr(Rate),'')
• Command Windowgtgt
• The interest rate you need is 6.7048

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fminbnd() - Finds a local minimum of a function
of one variable

• x fminbnd(_at_Fname,xL,xR)
• Fname a function name
• xL,xR
• range of x to search in

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fminbnd() - Other forms

• Additional output variables
• x,fval fminbnd(...)
• x,fval,exitflag fminbnd(...)
• Additional input variables
• x fminbnd(_at_Fname,xL,xR,,P1,P2,...)

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humps(x)
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x,y,exit fminbnd(...)

• gtgt x,y,exit fminbnd(_at_humps,.4,.8)
• x
• 0.6370
• y
• 11.2528
• exit
• 1

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Poly4(x,C,B,A) - with C-3, B0, A.5
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x fminbnd(_at_Fun, xL,xR,,P1,P2,...)

• gtgt fminbnd(_at_Poly4,1,2,,-3,0,0.5)ans
  1.2248
• gtgt fminbnd(_at_Poly4,-1.5,-0.5,,-3,0,.5)ans
  -1.2247

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Finding a maximum

• How would we find a local maximum? We might
  expect to have a fmaxbnd function, but we do not.
• To find the maximum of a function, you must find
  the minimum of the negative of the function.
• To do so, you'll have to define an M-file which
  is the negative of the desired function.
• The desired maximum value is the negative of the
  minimum value so obtained.

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sin(x)
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Local maximum of sin(x)

• First define Nsin(x) in a file Nsin.m
• function y Nsin(x)
• y -sin(x)
• Then
• gtgt x,y fminbnd(_at_Nsin,0,2pi)
• x
• 1.5708
• y
• -1.0000
• gtgt maxval -y
• maxval
• 1.0000

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Poly4(x,C,B,A) - with C-3, B0, A0.5
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Local maximum of Poly4(x,C,B,A)

• First define Poly4neg in a file Poly4neg.m as
  follows
• function y Poly4neg(x,C,B,A)
• POLY4NEG Negative of the example 4th order
  polynomial
• y -Poly4(x,C,B,A)

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Local maximum of Poly4(x,C,B,A)

• gtgt x,y fminbnd(_at_Poly4neg,-1,1,,-3,0,0.5)
• x
• 0
• y
• -0.5000
• gtgt maxval -y
• maxval
• 0.5000

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quad() - Evaluates the definite integral

• area quad(_at_Fun,xL,xR)
• Fun a function name
• xL,xR
• range of x over which to
• integrate

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quad() - Other forms

• Additional input variables
• area quad(_at_Fun,xL,xR,,,P1,P2,...)
• Important
• Fun must be a function that takes an input vector
  and returns an output vector containing values of
  the function at each element of the input.

There are two parameters for area that we are not
using, so we have two null matrices, which tell
area to use its default values.
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sin(x)
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area quad(_at_Fun,xL,xR)

• gtgt area quad(_at_sin,0,pi)
• area
• 2.0000
• gtgt area quad(_at_sin,0,2pi)
• area
• -1.5389e-016

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Poly4(x,C,B,A) - with C-3, B0, A0.5
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area quad(_at_Fname, xL,xR,,,P1,P2,)

• gtgt quad(_at_Poly4,0.5,1.5,,,-3,0,0.5)
• ans
• -1.2375

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Passing Functions as Arguments to OtherFunctions
in MATLAB

• When we invoke "function functions" such as fzero
  and fminbnd, we pass a "handle" to the function
  whose zero or minimum we want, using the syntax
  _at_Fname, where Fname is the name of an existing
  function.
• You can write your own "function function" which
  expects a function handle as an argument, if you
  remember to use the Matlab function feval
  whenever you need to evaluate the function whose
  handle is being passed.

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Passing Functions as Arguments to OtherFunctions
in MATLAB

• For example, suppose you want to write a function
  which will generate the x and y points for a
  graph
• function x,y Setup(Fname,a,b,npts)
• produces x and y vectors by generating npts,
• evenly spaced values of x
• between a and b and corresponding function
• values y Fname(x). Fname must be a function
  handle
• xlinspace(a,b,npts) produce x values using
  linspace
• to evaluate a function handle, must use feval.
• yfeval(Fname,x) this evaluates Fname into y

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Passing Functions as Arguments to OtherFunctions
in MATLAB

• We could then invoke Setup as
• x,ySetup(_at_sin,0,2pi,200)
• to produce 200 points on the sin curve for a
  plot.
• The first argument to feval is the function
  handle while the remaining arguments are those
  required by the function being evaluated. In the
  example above, we assumed that Fname requires
  only a single argument.