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2. MATLAB FUNCTIONS
2.1 MATHEMATICAL FUNCTIONS
COMMON MATH FUNCTIONS
The common math functions include functions to compute the absolute value of a value or
the square root of a value or to round a value. We now list these functions, along with brief
descriptions.
Function
Descriptions
abs(x)
Computes the absolute value of x.
sqrt(x)
Computes the square root of x.
round(x)
Rounds x to the nearest integer.
fix(x)
Rounds (or truncates) x to the nearest integer toward 0.
floor(x)
Rounds x to the nearest integer toward -∞.
ceil(x)
Rounds x to the nearest integer toward ∞.
sign(x)
Returns a value of -1 if x is less then 0, a value of 0 if
x equals 0, and a value of 1 otherwise.
rem(x,y)
Returns the reminder of x /y . For example, rem(25,4) is 1,
and rem(100,21) is 16. This function is also called a modulus function.
exp(x)
Computese^x , wheree is the base for natural logarithms, or approximate
ly 2.718282.
log(x)
Computes ln x, the natural logarithm of x to the base e .
log10(x)
Computes log10 x, the common logarithm of x to the base 10.
 round(-2.6)
ans =
-3
 fix(-2.6)
ans =
-2
floor(-2.6)
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ans =
-3
 ceil(-2.6)
ans =
-2
 sign(-2.6)
ans =
-1
 rem(15,2)
ans =
1
 floor(ceil(10.8))
ans =
11
 log10(100) + log10(0.001)
ans =
-1.0000
 abs(-5:5)
ans =
5 4 3 2 1 0 1
 round([0:0.3:2,1:0.75:4])
ans =
0 0 1 1 1 1 2
2
3
4
5
1
2
3
3
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4
TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS
Trigonometric Functions : The trigonometric functions assume that angles are represented in radians. To convert radians to degrees to radians, use the following conversions, whic
h use the fact that 180°= π radians:
angle_degrees=angle_radians*(180/pi);
angle_radians=angle_degrees*(pi/180);
We now list the trigonometric functions with brief descriptions:
Function
Description
Sin(x)
Computes the sine of x, where x is in radians.
cos(x)
Computes the cosine of x, where x is in radians.
Tan(x)
Computes the tangents of x, where x is in radians.
asin(x)
Computes the arcsine or inverse sine of x, where x must be between –1 and 1.
The function returns an angle in radians between -π/2 and π/2.
acos(x)
Computes the arccosine or inverse cosine of x, where x must be between -1
and 1. The function returns an angle in radians between 0 and π.
atan(x)
Computes the arctangent or inverse tangent of x. The function returns an
angle in radians between -π/2 and π/2.
atan2(y,x)
Computes the arctangent or inverse tangent of the value x /y . The function
returns an angle in radians that will be between -π and π, depending on the
signs of x and y.
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Hyperbolic Functions : The hyperbolic functions are functions of the natural exponential
function, e x ; the inverse hyperbolic functions are functions of the natural logarithm function
, ln x . MATLAB includes several hyperbolic functions, as shown in these brief descriptions:
Function
Description
sinh(x)
Computes the hyperbolic sine of x, which is equal to
cosh(x)
tanh(x)
asinh(x)
e x  e x
.
2
e x  e x
Computes the hyperbolic cosine of x, which is equal to
.
2
Computes the hyperbolic tangent of x, which is equal to
sinh x
.
cosh x
Computes the inverse hyperbolic sine of x, which is equal to


ln x  x x 2  1 .
acosh(x)
Computes the inverse hyperbolic cosine of x, which is equal to


ln x  x 2  1 for x greater than or equal to 1.
atanh(x)
Computes the inverse hyperbolic tangent of x, which is equal to
ln
1 x
for x 1 .
1 x
Other hyperbolic and inverse hyperbolic functions can be computed using the following equ
ations:
cosh x
for x  0
sinh x
1
1
sec h x 
, csc h x 
cosh x
sinh x
coth x 
acoth x  ln
x 1
for x  1
x 1
 1  1  x2
asech x  ln 

x


 for 0  x  1


1
1  x2
acsch x  ln  
x
x





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COMPLEX NUMBER FUNCTIONS
Complex numbers are needed to solve many problems in science and engineering. Recall tha
t a complex number has the form a  ib where is 1 , a is the real part of the value, and
b is the imaginary part of the value.
MATLAB includes several functions that are specific to complex numbers and their conversi
ons:
Function
conj(x)
Description
Computes the complex conjugate of the complex number x , Thus, if x i
s equal to a  ib , then conj(x) will be equal to a  ib .
real(x)
Computes the real portion of the complex number x .
imag(x)
Computes the imaginary portion of the complex number x .
abs(x)
Computes the absolute value or magnitude of the complex number x .
angle(x)
Computes the angle using the value of atan2(imag(x),real(x)); thus, the a
ngle value is between -π and π.
Polar Plots. Data values are sometimes represented by complex values, which can be considered to be an angle and a magnitude. For example, if we are measuring light intensity around a light source, we might represent the information with an angle from a fixed axis and a
magnitude that represents the intensity. To plot complex data, we may want to use a
polar plot, as opposed to plotting the magnitude and phase information separately.
The MATLAB command for generating a polar plot of the vectors theta and r is the following:
Function
polar(theta,r)
Description
Generates a polar of the angles theta (in radians) versus the magnitude r
.
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POLYNOMIAL FUNCTIONS
Polynomial Evaluation. There are several ways to evaluate a polynomial for a set of values
using MATLAB. To illustrate, consider the following polynomial over the interval[0,5]:
g ( x)  3x 4  0.5 x3  x  5.2
Function
polyval(a,x)
Description
Evaluates a polynomial with coefficients a for the values in x, The result
is a matrix the same size as x.
Polynomials can be also be evaluated using the polyval functions:
The following polyval reference will generate
and plot 210 points of the polynomial over the
desired interval:
 x=0:5/200:5;
 a=[3,-0.5,0.1,-5.2];
 g=polyval(a,x);
 plot(x,g),title('Polynomial Function')
 hold on; set(gcf,'color','w')
Polynomial Operations. If we assume that the coefficients of two polynomials are stored in
vectors a and b , we can the perform polynomial computations using a and b .
Multiplying two polynomials is more complicated than adding or subtracting two polynomials because a number of terms are generated and combined. Similarly, dividing two polynomials is a tedious process because we must multiply and subtract polynomials. MATLAB
contains functions to perform polynomial multiplication and division:
To illustrate the use of the conv and deconv functions for polynomial multiplication and division, consider the following polynomial product:
g ( x)  (3x3  5 x 2  6 x  2)( x5  3x 4  x 2  2.5)
We can multiply these polynomials using the conv function as shown below:
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Function
Description
conv(a,b)
Computes a coefficient vector that contains the coefficients of the
product of polynomials represented by the coefficients in a and b.
The vectors a and b do not have to be the same size.
[q,r]=deconv(n,d)
Returns two vectors. The first vector contains the coefficients of the
quotient and the second vector contains the coefficients of the remainder polynomial.
 a = [3, -5, 6, -2];
 b = [1, 3, 0, -1, 0, 2.5];
 g = conv(a,b)
g=
Columns 1 through 7
3.0000 4.0000 -9.0000 13.0000 -1.0000 1.5000 -10.5000
Columns 8 through 9
15.0000 -5.0000
g represents the following lowing polynomial:
g ( x)  3x8  4 x7  9 x6  13x5  x 4  1.5x3  10.5x 2  15x  5
We can illustrate polynomial division using the previous polynomials:
h( x ) 
3x8  4 x 7  9 x 6  13x5  x 4  1.5 x3  10.5 x 2  15 x  5
x5  3x 4  x 2  2.5
This polynomial division is specified by these commands:
 g = [3,4,-9,13,-1,1.5,-10.5,15,-5];
 b = [1,3,0,-1,0,2.5];
 [g,r]=deconv(g,b)
g=
3 -5 6 -2
r=
0 0 0 0 0
0
0
0
0
The quotient polynomial is 3 x3  5 x 2  6 x  2 , and the remainder vector contains zeros.
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Roots Of Polynomials. The solution to many engineering problems involve finding the root
s of the an equation of the form
y  f ( x)
where the roots are the values of x for which y is equal to 0.
·The MATLAB function for determining the roots of a polynomial is the roots function:
·If we have the roots of a polynomials and want to determine the coefficients of the polynomial when all the linear terms are multiplied, we can use the poly function:
Function
Description
roots(a)
Determines the roots of the polynomial represented by the coefficient vect
or a.
poly(r)
Determines the coefficients of the polynomial whose roots are contained in
the vector r.
The roots function returns a column vector containing the roots of the polynomial; the
number of roots is equal to the degree of the polynomial. To illustrate the use of this function
, assume that we want to determine the roots of this polynomial:
f ( x)  x3  2 x 2  3x  10
The commands to compute and print the roots of this polynomial are
 p = [1,-2,-3,10];
 r = roots(p)
r=
2.0000 + 1.0000i
2.0000 - 1.0000i
-2.0000
polyval(p,r)
ans =
1.0e-013 *
-0.0355 + 0.1377i
-0.0355 - 0.1377i
0.0711
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We can verify that these values are roots by evaluating the polynomial at the roots and
observing that polynomial values are essentially 0:
The output of the function is a row vector containing the polynomial coefficients.
For example, we can compute the coefficients of the polynomial with roots -1,1, 3 with the
following statement:
a( x)  ( x  1)( x  1)( x  3)
 a = poly([-1,1,3])
a=
1 -3 -1 3
The polynomial is x3  3x 2  x  3
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2.2 FUNCTIONS OF TWO VARIABLES
FUNCTION EVALUATION.
Recall that evaluating a function of one variable, such as f ( x ) , involves computing a vector
of x values, and then computing a corresponding vector of function values as illustrated in
these statements:
Two-Dimensional Grid : The meshgrid function generates the two matrices that define the
underlying grid of for a two dimensional function.
Function
[x_grid,y_grid]=
meshgrid(x,y)
Description
Generates two matrices of size m times n , based on values in the vecto
rs x and y which contain m values and n values, respectively.
The x ＿grid matrix contains the values of x repeated in each now,
and y＿grid matrix contains the values of y repeated in each column.
Thus, to generate the two matrices described on the previous page, we could use the following statements:
 x = -2:2;
 y = -1:2;
 [x_grid,y_grid] = meshgrid(x,y)
x_grid =
-2 -1 0 1 2
-2 -1 0 1 2
-2 -1 0 1 2
-2 -1 0 1 2
y_grid =
-1 -1 -1 -1 -1
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
Once the underlying grid matrices have been defined, we can then compute the corresponding function values. To illustrate, assume that we want to evaluate the following function for
the underlying grid defined above:
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z  f ( x, y ) 
1
1  x2  y 2
The corresponding function values can be computed and stored in a matrix z with four rows
five columns with these statements:
 z = 1./(1+x_grid.^2 + y_grid.^2)
z=
0.1667 0.3333 0.5000 0.3333
0.2000 0.5000 1.0000 0.5000
0.1667 0.3333 0.5000 0.3333
0.1111 0.1667 0.2000 0.1667
0.1667
0.2000
0.1667
0.1111
Since the operations are element-by-element, the value in the z matrix with subscripts(1,1)
is computed using the values in x ＿ grid (1,1) and y ＿ grid (1,1) , and so on.
Mesh Plot : There are several ways to plot a three-dimensional surface with MATLAB.
In this discussion we present two types of plots-a mesh plot and a surface plot. A mesh plot
has an open mesh as shown in Figure1, and a surface plot has a shaded mesh as Figure2.
Several variations of the MATLAB commands for generating these plots are now described:
The statements that generated the plots in Figure1 and 2 are these:
 x = -2:0.1:2;
 y = -1:0.1:2;
 [x_grid,y_grid] = meshgrid(x,y);
 z = 1./(1+x_grid.^2 + y_grid.^2);
 subplot(2,1,1), mesh(x_grid,y_grid,z),...
 title('Mesh Plot'),xlabel('x'),...
 ylabel('y'),zlabel('z'),pause
 subplot(2,1,2), surf(x_grid,y_grid,z),...
 title('Surface Plot'),xlabel('x'),...
 ylabel('y'),zlabel('z')
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Function
Mesh(x_pts,y_pts,z)
Description
Generates an open mesh plot of the surface defined by the matrix
z . The arguments x＿pts and y＿pts can be vectors defining
the ranges of values of the x -and y -coordinates, or they can be
matrices defining the underlying grid of x -and y -coordinates.
surf(x_pts,y_pts,z)
Generates a shaded mesh plot of the surface defined by the
matrix z . The arguments x＿pts and y＿pts can be vectors
defining the ranges of values of the x -and y -coordinates, or they
can be matrices defining the underlying grid of x -and y –coordinates.
After you become comfortable using the mesh and surf commands, use the help function to
learn additional options for these command and about the view function (to specify the view
point) and the colormap function (to specify color scaling).
CONTOUR AND 3-D PLOTS
Contour Plots. A Contour map is essentially an elevation map that contains a group of lines
that connect-equal elevations. We can think of a line that connects points of equal elevation
as a slice of the countryside at that elevation. If we have a map with many lines showing diff
erent elevation, we can determine mountains and valleys from it. A contour map is generated
form 3-D elevation data, and can be generated by MATLAB using matrices that define the r
ange of x - and y -coordinates and the elevation (or z coordinate) data. Two variations of the
contour command and a related variation of the mesh command are described below:
The following commands generate the plots shown in Figure 3 and 4, assuming that the matr
ices x, y, x＿grid , y＿grid , and z are defined using the previews MATLAB statements:
 subplot(2,1,1),contour(x,y,z),...
 title('Contour Plot'),xlabel('x'),...
 ylabel('y'),grid,pause
 subplot(2,1,2),meshc(x_grid,y_grid,z),...
 title('Mesh/Contour Plot'),xlabel('x'),...
 ylabel('y'),zlabel('z')
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Labeling Contours : Each contour level has a value associated with it. clabel uses these val
ues to display labels for 2-D contour lines. The contour matrix contains the values clabel use
s for the labels. This matrix is returned by contour. contour3, and contourf.
clabel optionally returns the handles of the Text objects used as label. You can then use thes
e handles to set the properties of the label string.
Function
Description
Clabel
Generates labels using the contour matrix and displays the labels
in the current Figure.
contour(x,y,z)
Generates a contour plot of the surface defined by the matrix z .
The arguments x and y are vectors defining the ranges of values
of the x -and y -coordinates. The number of contour lines and
their values are chosen automatically.
contour(x,y,z,v)
Generates a contour plot of the surface defined by the matrix z .
The arguments x and y are vectors defining the ranges of values
of the x -and y -coordinates. The vector v defines the values to
use for the contour lines.
contour3
Display 3-D isolines generated form values given by a matrix z .
contourf
Displays a 2-D contour plot and fills the area between the isoline
s with a solid color.
meshc(x_pts,y_pts,z)
Generates an open mesh plot of the surface defined by the
matrix z . The arguments x＿pts and y＿pts are vectors defining
the ranges of values of the x -and y -coordinates, or they can be
matrices defining the underlying grid of x -and y -coordinates. In
addition, a contour plot is generated below the mesh plot.
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 z = peaks;
 [c,h] = contour(z,10);
 clabel(c,h);
 title('Contour Labeled Using-clabel(c,h)')
The 'manual' option enables you
to add labels by selecting the contour you want to label with the
mouse.
You can also use this option to
label only those contours you select interactively.
For example,
clabel(c,h,'manual')
displays a crosshair cursor when your cursor is inside the Figure. Pressing any mouse button
labels the contour line closest to the center of the crosshair.
Filled Contours : contourf displays a two-dimensional contour plot and fills the areas between contour lines. Use the caxis function to control the mapping of contour to color.
For example, this filled contour plot of the peaks data uses caxis to map the fill colors into
the center of the colormap.
 z = peaks;
 [c,h] = contourf(z,10);
 caxis([-20 20])
 title('Filled Contour Plot Using-contourf(z,10)')
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Pcolor Pseudocolor (Checkerboard) Plot.
pcolor(c) is a pseudocolor or "checkerboard" plot of matrix c.
Function
Description
pcolor(c)
The values of the elements of c specify the color in each cell of the plot.
In the default shading mode, 'faceted', each cell has a constant color and
the last row and column of c are not used. With shading('interp'), each
cell has color resulting from bilinear interpolation of the color at its four
vertices and all elements of c are used. The smallest and largest elements
of c are assigned the first and last colors given in the color table; colors
for the remainder of the elements in c are determined by table-lookup
within the remainder of the color table.
pcolor(x,y,c)
where x and y are vectors or matrices, makes a pseudocolor plot on the
grid defined by x and y.
pcolor is really a surf with its view set to directly above.
pcolor returns a handle to a surface object.
 z = peaks;
 pcolor(z);
 z = peaks;
 pcolor(z);
 title('pcolor(z)')
 shading interp
 title('pcolor(z)-shading interp')
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Colorbar Display Color Bar (Color Scale).
Function
Description
colorbar('vert')
This Function appends a vertical color scale to the current axis.
colorbar('horiz')
This Function appends a horizontal color scale.
colorbar(h) places the colorbar in the axes h. The colorbar will be horizontal if the axes h
width > height (in pixels).
colorbar without arguments either adds a new vertical color scale or updates an existing
colorbar.
h = colorbar(...) returns a handle to the colorbar axis.
 z = peaks;
 z = peaks;
 pcolor(z);
 shading interp
 pcolor(z);
 shading interp
 colorbar('vert')
 title('colorbar(vert)')
 colorbar('horiz')
 title('colorbar(horiz)')
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EXAMPLE : CONTOUR, PCOLOR
MATLAB Program File : 'exam3.m'
fid1=fopen('exam.dat','r');
fid2=fopen('exam.out','w');
ni=fscanf(fid1,'%g',[1 1]);nj=fscanf(fid1,'%g',[1 1]);
a=fscanf(fid1,'%g',[ni nj]);b=fscanf(fid1,'%g',[ni nj]);
c=a.*b;
d=a*b;
fprintf(fid2,'%4.1f %4.1f %4.1f \n',c);
fprintf(fid2,'%4.1f %4.1f %4.1f \n',d);
pcolor(d);shading interp;han=colorbar('vert');
hold on
[c,h]=contour(c,9:25);clabel(c);
xlabel('row');ylabel('col');title('Matrix Product');
status=fclose('all');
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3-D PLOT
Function
plot3(x,y,z)
Description
Generates a line in 3-D through the points whose coordinates are the
elements ofrm bold x ,rm bold y , andrm bold z and then produces a 2
-D projection of that line on the screen.
 t = 0:pi/50:10*pi;
 plot3(sin(t),cos(t),t)
 title('Plot3')
 set(gcf,'color','w')
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2.3 RANDOM NUMBER GENERATING FUNCTIONS
UNIFORM RANDOM NUMBERS
Random numbers are not defined by an equation; instead they can be characterized by the
distribution of values. For example, random numbers that are equally likely to be any value
between an upper and lower limit are called uniform random numbers.
The rand function in MATLAB generates random numbers uniformly distributed over the
interval [0,1]. A seed value is used to initiate a random sequence of value; this seed value is
initially set to 0, but it can be changed with the seed function.
The rand function generates the same sequence of random values in each work session if the
same seed value is used. The following command generate and print two sets of ten random
Function
Description
rand(n)
Generates an n  n matrix containing random numbers between 0
and 1.
rand(m,n)
Generates an m n matrix containing random numbers between 0
and 1.
rand('seed',n)
Sets the value of the seed n .
rand('seed')
Returns the current value of the random number generator seed.
numbers uniformly distributed between 0 and 1; the difference between the two sets is
caused by the different seeds:
 rand('seed',0)
 set1=rand(10,1);
 rand('seed',123)
 set2=rand(10,1);
 [set1 set2]
ans =
0.2190 0.0878
0.0470 0.6395
0.6789 0.0986
0.6793 0.6906
0.9347 0.3415
0.3835 0.2359
0.5194 0.2641
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0.8310 0.6044
0.0346 0.4181
0.0535 0.1363
Random sequences with values that range between values other than 0 and 1 are often
needed. To illustrate, suppose that we want to generate values between -5 and 5.
We first generate a random number r (which is between 0 and 1), and then multiply it by 10
, which is the difference between the upper and lower bounds(5 -(-5)).
We then add the lower bound (-5), giving a resulting value that is equally likely to be any
value between -5 and 5. Thus, if we want to convert a value r that is uniformly distributed
between 0 and 1 to a value uniformly distributed between a lower bound a and an upper
bound b , we use the following equation:
x  (b  a )  r  a
 data_1 = 2 * rand(1,500)+2;
 plot(data_1)
 hold on
 axis([0 500 0 6]);
 grid on
 title('Random Number-data _1')
 set(gcf,'color','w')
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GAUSSIAN RANDOM NUMBERS
MATLAB will generate Gaussian values with a mean of zero and a variance of 1.0 if we specify
a normal distribution. The functions for generating Gaussian values are as following:
Function
Description
randn(n)
Generates an n  n matrix containing Gaussian (or normal) rando
m numbers with a mean of 0 and a variance of 1.
randn(m,n)
Generates an m n matrix containing Gaussian (or normal) rando
m numbers with a mean of 0 and a variance of 1.
 data_2 = randn(1,500)+3;
 plot(data_2)
 hold on
 axis([0 500 0 6])
 title('Random Number-data _2')
 xlabel('index, k')
 set(gcf,'color','w')
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