1. MATLAB Software

2. Introduction Matlab (Matrix Laboratory) –
An interactive program that is suitable for run computations, draw graphs and much more (graphical interface).
A programming language for technical computing (scripts).

3. Introduction

4. Command Window Matlab as a calculator: Lets run our first command:
Command line
Matlab as a calculator:
Lets run our first command:
1 + 1

5. Command Window Previous Command line Output (answer)
Current Command line
The output is displayed
Get the previous command by pressing up arrow keyboard
Can clear window using the command clc

6. All workspace variables
Workspace Window
Workspace Window
All workspace variables
Variable value

7. Command History Window

8. Current Directory Window
Files

9. Changing The Current Directory

10. Variable Format
format short g Best of fixed or floating point format with 5 digits.
format long g Best of fixed or floating point format with 15 digits for double and 7 digits for single.
format short e Floating point format with 5 digits.
format long e Floating point format with 15 digits for double and 7 digits for single.

11. Starting MATLAB The various forms of help available are:
help Types a matlab help text
helpwin Opens a matlab help GUI
demo Starts the matlab demonstration

12. Matrices Variables: Array a = 1; speed = 1500; K = [0.1 0.15 0.2];

13. Matlab Special Numberspi …
i and j
Imaginary unit
eps
Relative precision 2-52
Inf
Infinity (divide by zero or larger then realmax)
NaN
Not a Number (0/0 Inf/Inf)

14. Matrices Two Dimensional Matrices: Multi Dimensional Matrices:
1 2 3
4 5 6
Multi Dimensional Matrices:

15. Useful Matrix Generators
zeros a matrix filled with zeros
ones a matrix filled with ones
rand a matrix with uniformly distributed random elements
eye identity matrix

16. Useful Matrix Generators
>> a = zeros(2,3)
a =
0 0 0
>> b = ones(2,2)
b =
1 1

17. Useful Matrix Generators
>> u = rand(1,5)
u =
>> eye(3)
ans =
1 0 0
0 1 0
0 0 1

18. Subscripting >> u(3) ans = 0.1763 >> u(1:3)

19. Subscripting >> a = [1 2 3;4 5 6;7 8 9] a = 1 2 3 4 5 6 7 8 9
ans = 8

20. Subscripting
>> a(2:3,3)
ans = 6 9
>> a(2,:)
4 5 6

21. Deleting Rows or Columns
>> a(:,2) = []
a =
1 3
4 6
7 9

22. Matlab Arithmetic Operators
Plus  +
Minus -
multiply *
power ^
Backslash or left matrix divide \
Slash or right matrix divide /

23. Matlab Arithmetic Operators
multiply .*
Array power                             .^
Left array divide .\
Right array divide   ./

24. Matlab Arithmetic Operators
>> [1 2;3 4] / 2
ans =
>>2 \ [1 2;3 4]

25. Matlab Arithmetic Operators
>> [1 2;3 4] ^ 2
ans =
>> [1 2;3 4] .^ 2

26. Matlab Arithmetic Operators
>> [1 2;3 4] .\ [5 6;7 8]
ans =
>> [1 2;3 4] ./ [5 6;7 8]

27. Simple Operations on Matrices
Finding the maximal number in a vector
>> x = 1 : 50;
>> max(x) 50
Finding the minimal number in a vector
>> min(x) 1
Finding the mean of a vector
>> mean(x)
25.500

28. Simple Operations on Matrices
Finding the maximal numbers in each matrix column
>> x = [1 2 3;7 8 9;4 5 6];
>> max(x)
ans =
Think of one way to get the maximal element in the entire matrix…

29. Simple Operations on Matrices
Finding the mean of each matrix column
>> mean(x)
ans =
How do we get the mean of each matrix row?

30. Simple Operations on Matrices
Finding the size of a matrix
>> x = 1 : 50;
>> size(x)
ans =
Finding the length of a vector
>> length(x) 50

31. Simple Operations on Matrices
x =[ 2 4 6;3 6 9];
size(x)
2 3
length(x) 3

32. Sub-array searching The “find” operation >> x = [2 8 7 6 4 2 3];
>> find(x == 2)
1 6
>> find(x > 3)

33. Sub-array searching >> x = [1 2 3 7 8 9 4 5 6];
>> [h, w] = find(x > 5)
h =
w =

34. 3D arrays - example

35. 3D arrays - example

36. Interpolation Function
One-dimensional data interpolation
yi = interp1(x,y,xi, 'linear')
yi = interp1(x,y,xi, 'spline')
yi = interp1(x,y,xi, 'cubic')

37. Interpolation Function
>> x=[ ];
>> y=[ ];
>> yi = interp1(x,y,2.5, 'linear')
yi = 5

38. Interpolation Function
Two-dimensional data interpolation
ZI = interp2(X,Y,Z,XI,YI)

39. Interpolation Function
Cubic spline data interpolation
yi = spline(x,y,xi)
>> x=[ ];
>> y=[ ];
>> yi = spline (x,y,2.5)
yi = 5

40. Learning Polynomials
MATLAB represents polynomials as row vectors containing coefficients ordered by descending powers. For example, consider the equation
p(x) = x x + 5
To enter this polynomial into MATLAB, use
» p=[ ];

41. Polynomial Roots » p = [1 0 -2 -5];
The roots function calculates the roots of a polynomial:
r = roots(p)
r = i i

42. Polynomial Roots
Function poly returns roots to the polynomial coefficients:
p2 = poly(r)
p2 = e
poly and roots are inverse functions

43. Polynomial Evaluation
The polyval function evaluates a polynomial at a specified value. To evaluate p(5), use
polyval(p,5)
ans =
110

44. Polynomial Evaluation
It is also possible to evaluate a polynomial in a matrix sense.
X = [2 4 5; ; 7 1 5];
p(X) = X X + 5
Y = polyvalm(p,X)
Y = p(1)*X^3 + p(2)*X^(2) + p(3)*X + P(4)*I
Y =

45. Polynomials Convolution
a(x) = x x && b(x) = 4x x + 6
a = [1 2 3];
b = [4 5 6];
c = conv(a,b)
c =

46. Polynomials Deconvolution
The syntax for deconv is
[q,r] = deconv(c,a)
q =
r =

47. Polynomial Derivatives
To obtain the derivative of the polynomial
p = [ ];
q = polyder(p)
q =

48. Polynomial Derivatives
b = [2 4 6];
c = polyder(a,b)
c =

49. Polynomial Derivatives
[q,d] = polyder(a,b)
q =
d =
q/d is the result of the operation.

50. Polynomial Integrates
polyint Integrate polynomial analytically.
>> p=[ ];
>> polyint(p,-5)
ans =

51. Residue

52. Residue
[r,p,k] = residue(b,a)

53. Residue
b = [ ]
a = [ ]
[r, p, k] = residue(b,a)

54. Residue
r =
1.3320
p =
1.5737
k =

55. Polynomial Curve Fitting
polyfit Fit polynomial to data.
P = polyfit(X,Y,N)
P(1)*X^N + P(2)*X^(N-1) P(N)*X + P(N+1).

56. Polynomial Curve Fitting
>> x = (0: 0.2:1)
x =
>> y=x.^2+2.*x+1
y =

57. Polynomial Curve Fitting
>> p = polyfit(x,y,2)
p =

58. Polynomial Curve Fitting
pp = spline(x,y)
>> x = (0: 0.2:1)
x =
>> y=x.^2+2.*x+1
y =

59. Polynomial Curve Fitting
>> pp = spline(x,y)
pp =
form: 'pp'
breaks: [ ]
coefs: [5x4 double]
pieces: 5
order: 4
dim: 1

60. Polynomial Curve Fitting
>> yy = ppval(pp, 0.6)
yy =
2.5600

61. Integrate int(S,v) is the indefinite integral of S with respect to v.
>> int('1/(1+x^2)')
ans =
atan(x)

62. Integrate
int(S,v,a,b) is the definite integral of S with respect to v from a to b.
>> int('y/(1+x^2)','x',0,1)
ans =
1/4*pi*y

63. Derivative
diff(X,N,DIM) is the Nth difference function along dimension DIM.
>> diff('y^3*sin(x)',3,'x')
ans =
-y^3*cos(x)
>> diff('y^3*sin(x)',2,'y')
6*y*sin(x)

64. 2–D Plot Function >> x = -pi:0.01:pi; >> y=sin(x);
plot(X,Y) plots vector Y versus vector X.
>> x = -pi:0.01:pi;
>> y=sin(x);
>> plot(x,y)

65. 2–D Plot Function

66. 2–D Plot Function
>> plot(x)

67. 2–D Plot Function
If z is complex, plot(z) is equivalent to plot(real(z),imag(z))
>> z=[ i 1+i 2+4i 3+9i i i];
>> plot(z)

68. 2–D Plot Function

69. 2–D Plot Function >> x = -pi:0.01:pi; >> y=sin(x);
>> z=cos(x);
>> plot(x,y,x,z)
>> plot(x,y)
>> hold on
>> plot(x,z)

70. 2–D Plot Function

71. 2–D Plot Function

72. 2–D Plot Function

73. 2–D Plot Function >> x = -pi:0.5:pi; >> y=sin(x);
>> z=cos(x);
>> plot(x,y,'rp')
>> hold on
>> plot(x,z,'kd--')

74. 2–D Plot Function

75. 2–D Plot Function
Axis
>>axes1=axes('FontName','Arial','FontSize',10,'FontWeight','bold','LineWidth',2);
>> box(axes1,'on');
>> hold(axes1,'all');

76. 2–D Plot Function

77. 2–D Plot Function
set(gcf,'Color',[1,1,1])

78. 2–D Plot Function >> x = -pi:0.5:pi; >> y=sin(x);
>> plot(x,y,'k-','LineWidth',2)

79. 2–D Plot Function

80. 2–D Plot Function >>xlabel('\beta')
>>ylabel('y_\beta','Rotation',0)
>>title('plot')
>>text(0.25,0.15,'HELLOW','FontName','Arial','FontSize',10,'FontWeight','bold')

81. 2–D Plot Function

82. 2–D Plot Function >> x = -pi:0.5:pi; >>y=sin(x);
>>z=cos(x);
>>plot(x,y,'rp')
>>hold on
>>plot(x,z,'kd--')
>>legend('sin(x)','cos(x)','Location','NorthWest')

83. 2–D Plot Function

84. 2–D Plot Function

85. 2–D Plot Function Axis scaling and appearance
axis([xmin xmax ymin ymax])
Grid lines for two- and three-dimensional plots
grid on
grid off

86. 2–D Plot Function
subplot divides the current figure into rectangular panes that are numbered row wise.
subplot(m,n,p)

87. 2–D Plot Function >> x = -pi:0.5:pi; >>y=sin(x);
>>z=cos(x);
>> t=x.*sin(x)
>> c=x.*cos(x)

88. 2–D Plot Function >> subplot(2,2,1) >>plot(x,y)
>>plot(x,z)
>>subplot(2,2,3)
>>plot(x,t)
>>subplot(2,2,4)
>>plot(x,c)

89. 2–D Plot Function

90. 2–D Plot Function >> x = -pi:0.5:pi; >>y=sin(x);
>>z=cos(x);
>> t=x.*sin(x)
>> c=x.*cos(x)

91. 2–D Plot Function >> figure(1); >> plot(x,y)
>> plot(x,z)
>> figure(3);
>>plot(x,t)
>> figure(4);
>>plot(x,c)

92. 2–D Plot Function

93. 2–D Plot Function

94. 2–D Plot Function

95. 2–D Plot Function

96. fPlot Function
fplot(FUN,LIMS) plots the function FUN between the x-axis limits specified by LIMS = [XMIN XMAX].
sin(1./x), [ ])

97. fPlot Function

98. Logarithmic scale plot
>>semilogx(x,y)
Create semilogarithmic plot with logarithmic x-axis
>>semilogy(x,y)
Create semilogarithmic plot with logarithmic y-axis

99. Logarithmic scale plot
>> x = 0:0.1:10;
>> y=10.^x;
>> semilogy(x,y)
>> grid on

100. Logarithmic scale plot

101. Logarithmic scale plot
>>loglog(x,y)
Create a loglog plot
>> x = logspace(-1,2);
>> y=exp(x);
>> loglog(x,y,'-s')
>> grid on

102. Logarithmic scale plot

103. Polar Function >>polar(theta,rho) >>t = 0:.01:2*pi;
>>polar(t,sin(2*t).*cos(2*t),'--r')

104. Polar Function

105. Bar Function Plot bar graph (vertical and horizontal) >>bar(x,y)
>>barh(x,y)

106. Bar Function >> x = -2.9:0.2:2.9; >> y=exp(-x.*x);
>> bar(x,y,'r')

107. Bar Function >> x = -2.9:0.2:2.9; >> y=exp(-x.*x);
>> barh(x,y,'r')

108. 3–D Plot Function mesh(X,Y,Z) draws a wire frame mesh 1.5m 1m y x ];

109. 3–D Plot Function >> x=0:0.5:1.5; >> y=0:0.5:1;
>> mesh(x,y,T)
>> axis([ ])

110. 3–D Plot Function >> x=0:0.5:1.5; >> y=0:0.5:1;
>> surf(x,y,T)
>> axis([ ])

111. 3–D Plot Function >>meshc(x,y,T) >>meshz(x,y,T)
>> waterfall(x,y,T)

112. 3–D Plot Function
The shading function controls the color shading of surface and patch graphics objects.
>>shading flat
>>shading interp

113. 3–D Plot Function >>sphere(16) >>axis square
>>shading flat
>>title('Flat Shading')

114. 3–D Plot Function >>sphere(16) >>axis square
>>shading interp
>>title('Flat Shading')

115. 3–D Plot Function
>> view(az,el)

116. 3–D Plot Function
sets the default two-dimensional view, az = 0, el = 90.
sets the default three-dimensional view, az = -37.5, el = 30.
>> cylinder

117. 3–D Plot Function
>> peaks

118. 3–D Plot Function
>> contour(peaks)

119. 3–D Plot Function >>ezmesh(fun,domain)
x.*exp(-x.^2-y.^2),[ ])

120. M-file
Each M-file function has an area of memory, separate from the MATLAB base workspace, in which it operates. This area, called the function workspace, gives each function its own workspace context.

121. M-file
user_entry = input(' Expression ')

122. Conditionally execute statements
if expression1
statements1
elseif expression2
statements2
else
statements3
end

123. Conditionally execute statements

124. Relational Operators == ~= < > <= >= Equal Not equal
Less than
<
Greater than
>
Less than or equal
<=
Greater than or equal
>=

125. for
for variable = expression
statement
...
end

126. for

127. while
while expression
statement
...
end

128. while

129. Switch-Case switch switch_expr case case_expr 1
statement, ..., statement
case case_expr 2
otherwise
end

130. Switch-Case

131. Continue
Pass control to next iteration of for or while loop

132. Break
Terminate execution of for or while loop

133. Save & Load >>save('filename', 'var1', 'var2', ...)
>>load('filename', 'var1', 'var2', ...)

134. Clear
>>clear('name1','name2','name3',...)

135. Solve solve('eqn1','eqn2',...,'eqnN')
>>[x,y] = solve('x^2 + x*y + y = 3','x^2 - 4*x + 3 = 0')
>>x= solve('p*sin(x) = r')

136. Function
function [out1, out2, ...] = funname(in1, in2, ...)

137. Function

138. Function

139. Function

140. Function

141. Function

142. Nonlinear Equations >>x = fsolve(fun,x0)
fsolve finds a root (zero) of a system of nonlinear equations.
>>x = fsolve(fun,x0)
starts at x0 and tries to solve the equations described in fun.

143. Nonlinear Equations You want to solve the following system for x
starting at x0 = [-5 -5].

144. Nonlinear Equations

145. Nonlinear Equations

146. Nonlinear Equations >>x = fsolve(fun,x0,options)
>> options= optimset ('param1',val.1,'param2',val.2,...)
>>x = fsolve(fun,x0,options)
This structure solves the equations with the optimization options specified in the structure options.

147. Nonlinear Equations Display 'off' | 'iter' | 'final' MaxIter
Positive integer
TolFun

148. Nonlinear Equations

149. Nonlinear Equations

150. dsolve dsolve('eqn1','eqn2', ...)
>> S = dsolve('Dx = -a*x','x(0) = 1')
>> [x y] = dsolve('Dx = y', 'Dy = -x', 'x(0)=0', 'y(0)=1')

151. Numerical Solution of ODE
Initial Value Problems (IVP)

152. Numerical Solution of ODE
[T,Y] = solver(odefun, [t0 tf],y0,options)

153. Numerical Solution of ODE
options = odeset('name1',value1,'name2',value2,...)

154. Numerical Solution of ODE

155. Numerical Solution of ODE

156. Numerical Solution of ODE

157. Numerical Solution of ODE
Boundary Value Problems (BVP)

158. Numerical Solution of ODE
(BVP)
(IVP)

159. Numerical Solution of ODE
sol = bvp4c(odefun,bcfun,initguess,options)

160. Numerical Solution of ODE
initguess = bvpinit([t0 tf],[y10 y20]);
options = bvpset('name1',value1,'name2',value2,...)

161. Numerical Solution of ODE

162. Numerical Solution of ODE

163. Numerical Solution of ODE

164. Optimization Unconstrained nonlinear optimization
[x,fval] = fminsearch(fun,x0,options)
fminsearch attempts to find a minimum of a scalar function of several variables, starting at an initial estimate.

165. Optimization
A classic test example for multidimensional minimization:

166. Optimization

167. Optimization

168. Optimization Constrained nonlinear optimization
[x,fval] = fminbnd(fun,x1,x2,options)
fminbnd attempts to find a minimum of a function of one variable within a fixed interval.

169. Optimization

170. Optimization

171. Optimization Other optimization functions: >> fmincon
>> fminunc
>> linprog

172. EXCEL Software

173. Introduction to Excel
Sheet, Row, Column, Cell

174. Introduction to Excel Name Box (Cell Address)
Formula Bar (Contents of Cell)

175. Introduction to Formulas
All Formulas begin with =
Basic Formulas (numbers, cells, or both)
Add +
Subtract -
Multiply *
Divide /
=10+5
=10-5
=10*5
=10/5
=B3+C3
=B3-C3
=B3*C3
=B3/C3
=B3+5
=B3-5
=B3*5
=B3/5

176. Introduction to Formulas
Formulas can reference different “Sheets”
=A10*Sheet2!B5 is the value from Cell A10 of our current worksheet multiplied by the value of Cell B5 from Sheet 2

177. Introduction to Formulas
All Formulas begin with =
Automatic Functions (Toolbar button Σ)
Menu Insert/Function for more options

178. Figures in Excel

179. Figures in Excel

180. Figures in Excel

181. Figures in Excel

182. Figures in Excel

183. Matrix Operations in Excel
Select the cells in which the answer will appear

184. Matrix Multiplication in Excel
Enter “=mmult(“
Select the cells of the first matrix
Enter comma “,”
Select the cells of the second matrix
Enter “)”

185. Matrix Multiplication in Excel
Enter these three key strokes at the same time:
Control
Shift
Enter

186. Matrix Commands in Excel
Excel can perform some useful, albeit basic, matrix operations:
Addition & subtraction;
Scalar multiplication & division;
Transpose (TRANSPOSE);
Matrix multiplication (MMULT);
Matrix inverse (MINVERSE);
Determinant of matrix (MDETERM);

187. Excel link for use with MATLAB

188. Excel link for use with MATLAB

189. Excel link for use with MATLAB

190. Excel link for use with MATLAB

191. Excel link for use with MATLAB

192. Excel link for use with MATLAB

193. Excel link for use with MATLAB

194. Excel link for use with MATLAB

195. Optimization

196. Optimization

197. Optimization

198. Optimization

199. Optimization

200. Optimization

201. Optimization

202. Optimization

203. Optimization

204. Optimization

205. Optimization

206. Optimization
Optimum distribution