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MATLAB
Matrix Laboratory
Internet Resources:
Dr. H-C Chen’s CVEN 302 Web Pagehttp://ceprofs.tamu.edu/hchen/cven302.html
For MATLAB, select chap01b.ppt, or clickhttp://ceprofs.tamu.edu/hchen/cven302/chap01b.ppt
Programming
The ideal style of programming is structuredstructured
• Break down a larger goal into tasks• Develop a modulemodule for each task• A module has a single entrance and exit• Modules are reusable
Use MATLAB to solve a problem
• State the problem clearly
• Describe the Input/Output (I/O)
• Algorithm - Numerical Method
• Develop a MATLAB program
• Debugging and Testing
• Documentation
If needed, work the problem by hand
Compiler and Program Execution
Computer Program Compiler
Machine language program
Link/Load
Execute
input
output
executionadinglinking/loncompilatio
MATLAB Execution
Interactive environment - does not require formal compilation, linking/loading, and execution
Scripts - develop and execute m-files that contain MATLAB commands
Important!
You must include commentscomments in your programs you turn in - otherwise we will have great difficulty knowing what you are thinking and doing. You will have
difficulty knowing what you were doing if you need to reuse the programs.
Command Window - enter commands and data - print results
Graphics Window - display plots and graphs
Edit Window - create and modify m-files
MATLAB’s 3 Windows
Two simple MATLAB programs (also called scripts)
Filename: lec2a.m
% Find sum of 1 to b% Comments begin with “%”% cd y:\cven302\clear allsum = 0;b = input('what is the value of b? ')for i = 1 : b sum = sum + iend
Need extension .m called m-file
Filename: lec2b.m
% plot 1/x v.s. tanh x
clca = linspace(0,5);f = 1./a;g = tanh(a);plot(a,f,'-', a, g,'--')axis([0 3 0 2])xlabel ('x')ylabel ('y')title ('Plot 1/x and tanh x')text(0.7, 1.6, '1/x')text(2, 1.1, 'tanh x')
Now we have seen Command Window, Graphics Window, and Edit Window
There is another window which is very useful:the Help Window
You can also type help in Commend Window
e.g.,
help input
» help input
INPUT Prompt for user input. R = INPUT('How many apples') gives the user the prompt in the text string and then waits for input from the keyboard. The input can be any MATLAB expression, which is evaluated, using the variables in the current workspace, and the result returned in R. If the user presses the return key without entering anything, INPUT returns an empty matrix. R = INPUT('What is your name','s') gives the prompt in the text string and waits for character string input. The typed input is not evaluated; the characters are simply returned as a MATLAB string. See also KEYBOARD.
MATLAB’s basic component is a matrixi.e., all variables are treated as matrices
All operations are vectorized (optimized for vector use)
Loops run much slower in MATLAB than in Fortran (not a vector operation)
Scalar, vector, and matrix
Scalar: k = 511 matrix
Matrix:
8642
9753
4321
a
m n matrix
m rows (m = 3)n columns (n = 4)
2
3
1
c 4321b
14 matrix, or a row vector
Vector:
31 matrix, or a column vector
Variable
• may consist up to 31 characters
• starting with a letter and followed by any combination of letters, digits, and underscores (Variable names must start with a letter)
• punctuation marks and spaces should not be included
e.g., CVEN_302, TIME, Time, time, velocity, force_x, force_y
• All numbers are double precision
• Text is stored as arrays of characters
• You don’t have to declare the type of data (defined when running)
• Variables are case sensitive
Data Type
e.g., CVEN_302, TIME, Time, time, velocity, force_x, force_y
Arithmetic operation of scalars
Addition: a+b subtraction: a-bmultiplication: a*b division: a/bexponentiation: a^b= ab
abs(x) : absolute value sqrt(x) : square root = x^(1/2)sin(x) : sine asin(x) : inverse sinesinh(x) : hyperbolic sine asinh(x) : inverse hyperbolic sinelog(x) : natural logarithm log10(x) : common logarithmexp(x) : exponental: ex (e = 2.7183….)
Elementary math functions
» a = 5;» b = 3;» a*bans = 15» d = ans * 3d = 45» e = ans * 3;» ee = 45» a^2ans = 25
» a/bans = 1.6667» format long» a/bans = 1.66666666666667» format short» a/bans = 1.6667
» exp(2)ans = 7.3891» 10^2ans = 100» log(100)ans = 4.6052» log10(100)ans = 2» pians = 3.1416
; - result not displayedformat long - 15 digitsformat short - 5 digitsans - result from previous calculationpi -
Ctrl-c : terminate a running program
2e
100ln
100log
Array Operations
• An array operation is performed element-by-An array operation is performed element-by-element - Need “element - Need “..” in front of the operator” in front of the operator
B(5);A(5) C(5)
B(4);A(4) C(4)
B(3);A(3) C(3)
B(2);A(2) C(2)
B(1);A(1) C(1)
MATLAB: C = AMATLAB: C = A..B;B;
Arithmetic operation of Arrays
Addition: a+b subtraction: a-bmultiplication: a.*b division: a./bexponentiation: a.^b = ab
MATLAB variables and matrix operation
2
3
1
c 654b
» b = [4 5 6]b = 4 5 6» c = [1 3 2]c = 1 3 2
» c = [1; 3; 2]c = 1 3 2» d = c'd = 1 3 2
- transpose
; - a new line
» b * cans = 31» b.*dans = 4 15 12» b./dans = 4.0000 1.6667 3.0000» f = [b; d; [3 6 8]]f = 4 5 6 1 3 2 3 6 8
2
3
1
c 231d
654b
b.*d = [4*1 5*3 6*2]b./d = [4/1 5/3 6/2]
bd[3 6 8]
» f(2,3)ans = 2» size(f)
size - size of a matrix
Matrix Operation (conti.)
497 ; 321 yx
497321 yxz
497
321 ; yxu
s = [1 2 3 4; 5 6 7 8]t = [1 2 3 4.... 5 6 7 8]
continue
Colon Operator
Creating new matrices from an existing matrix
C = [1,2,5; -1,0,1; 3,2,-1; 0,1,4]
F = C(:, 2:3) = [2,5; 0,1; 2,-1; 1,4]
E = C(2:3,:) = [-1 0 1; 3 2 -1]
G = C(3:4,1:2) = [3,2; 0,1]
410
123
101
521
C
123
101E
41
12
10
52
F
10
23G
all columns
rows 2 to 3
» x = 1:10x = 1 2 3 4 5 6 7 8 9 10» t = 1 : 2 : 10t = 1 3 5 7 9» k = 5:-1:-3k = 5 4 3 2 1 0 -1 -2 -3
a : step : b gives number from a to b with the specified step between elementsstep = 1 if not specified
What if the step is not easy to calculate, or is an odd number?
y = linspace (a, b, n_pts)
» y = linspace (0, 1, 4)y = 0 0.3333 0.6667 1.0000
2 end points are included in n_pts.
y = linspace (0,1)
a number of 100 for n_pts is assigned.
Some useful special matrices
» eye (3)ans = 1 0 0 0 1 0 0 0 1» ones (3)ans = 1 1 1 1 1 1 1 1 1» ones (1,4)ans = 1 1 1 1
» zeros (2,3)ans = 0 0 0 0 0 0
eye (m,n) eye (n)ones (m,n) ones (n) zeros (m,n) zeros (n)
Plotting
plot (x, y) plot(x1, y1, x2, y2)
plot (x, y, ‘color symbol line style’)
» x = linspace(0, 2*pi);» y = sin (x);» z = cos (x);» plot (x, y)» plot (x, y, x, z)» figure (2)» plot (x, y, 'r o -'); grid on» hold on» plot (x, z, 'b x :')
red, circle, solid
blue, x-mark, dotted
figure or figure (#) : open a figure
try help plotadd grids
try grid off
» xlabel ('x')» ylabel ('y')» title ('sine and cosine')» text (2, 1, 'This is a sine fuction')» axis ([0 2*pi -2 2])» hold off
xlabel (‘ label ‘) ylabel (‘ label ‘)title (‘ title of the plot ‘)text ( x_location, y_location, ‘ text ‘)axis ([ x_min x_max y_min y_max ])
Plotting (continue)
- text string
hold on: hold the plots to avoid overwriting
subplot ( m, n, figure number ) - break the figure window into m by n small figures, and plot the specified figure
Plotting (continue)
» figure» subplot (3, 2, 1)» plot (x, y)» subplot (3, 2, 2)» plot (x, z)» subplot (3, 2, 4)» plot (x, y-z)
1 2
3 4
5 6
semilogx (x, y) logarithmic scales for the x axis.
semilogy (x, y) logarithmic scales for the y axis.
loglog (x, y) logarithmic scales for the x and y axes
x = 10.^[-1:0.1:2];y = exp(x);figure(1)semilogy(x,y,':^')
figure(2)loglog(x,y,'-s')grid on
More control on plotting
get(H) get object properties
set(H,'PropertyName',PropertyValue,...)
set object properties
figure(2)h = loglog(x,y,'-sr')grid on
get(h)set(h, 'LineWidth',2)set(h, 'MarkerFaceColor', [0 0 1])
who list of current variablesclc clear the command windowclf clear the graphical windows clear x clear the variable xclear all clear all variablesclose close the current figureclose all close all figurescd y:\cven302\ change directorydir list all fileswhat list all m-filesCTRL-C Abort
semicolons (;) at the end of line: Calculation results will not be displaced on the command window
Some useful commands
Initializing Variables
• Explicitly list the values• reads from a data file• uses the colon “:” operator• reads from the keyboard
Input and output (I/O)
a = input (‘ enter a value or string ‘) - wait for input from the keyboard
load file_name- input an existing data file (ASCII format)
diary file_name- save anything in the Command Window
diary off
save file_name variable(s) -ascii- save the data of the variable(s) to a file in ASCII format
You may need to go to certain directory before loading and/or saving files
» diary tmp.dat» a = [ 1 2 3; 4 5 6]a = 1 2 3 4 5 6» save a_file.dat a -ascii» load a_file.dat» b = a_fileb = 1 2 3 4 5 6» d = input ('give a value to d: ')give a value to d: 5d = 5» diary off
Save in the tmp.dat file
M-Files: Scripts and Functions
• You can create and save code in text files using MATLAB Editor/Debugger or other text editors (called m-files since the ending must be .m)
• M-file is an ASCII text file similar to FORTRAN or C source codes ( computer programs)
• A script can be executed by typing the file name, or using the “run” command
Functions
distinguished from the script that the first line is of the form
function x = function_name (input arguments)
function [x, y] = function_name (input arguments)
A function has to be stored as a stand-along file ended with “.m”. The name of the function is usually (but not necessary) the same as the name of the file.
Difference between scripts and functions
Scripts share variables with the main workspaceFunctions do not
function t = trap_ex(a, b)t = (b - a) * (a^(-3) + b^(-3)) / 2;
Approximate integral f(x) = 1/x3 using basic trapezoid rule
File name: trap_ex.m
» y = trap_ex (1, 3)y = 1.0370»
dxx
yb
a3
1
bfafab
I
2
Trapezoidal rule
a b
f(a) f(b)
height average width I
function t = trap_ex2(a, b, step)dx = (b - a)/step;x1 = a;t = 0;for i = 1 : step x1 = a + (i-1)*dx; x2 = a + i*dx; t = t + (x2 - x1) * (x1^(-3) + x2^(-3)) / 2;end
File name: trap_ex2.m
Approximate integral f(x) = 1/x3 using basic trapezoid rule with n steps used
» y = trap_ex2 (1,3,1)y = 1.0370» y = trap_ex2 (1,3,2)y = 0.6435» y = trap_ex2 (1,3,100)y = 0.4445» y = trap_ex2 (1,3,1000)y = 0.4444» format long» y = trap_ex2 (1,3,1000)y = 0.44444543209743 Exact solution = 0.444444444444 ….
» x1??? Undefined function or variable 'x1'.
Why?
feval - evaluate function specified by string
function y = my_func(x)% function 1/x^3y = x.^(-3);
function q = basic_trap(f, a, b)% trapezoid rule with input function fya = feval(f, a);yb = feval(f, b);q = (b - a)* (ya + yb)/2;
my_func.m
basic_trap.m
» y = basic_trap ('my_func', 1,3)y = 1.0370
» feval('my_func', 1)ans = 1
Decision Making: Control Flow
(a) For Loops
(b) While Loops
(c) If-Else Structures
z = 0;for i = 1:10 y(11-i) = i; z = z+i;end
For Loops
for x = arraycommands
end
» yy = 10 9 8 7 6 5 4 3 2 1» disp(z) 55
Without this line you will get:Warning: Reference to uninitialized variable z.
disp (x) - display the value of x
Filename: lec2a.m
% Find sum of 1 to b% Comments begin with “%”% cd y:\cven302\clear allsum = 0;b = input('what is the value of b? ')for i = 1 : b sum = sum + iend
While Loops
while expressioncommands
end
eps = 1;count = 0;while (1+eps) > 1 eps = eps/2; count = count + 1;endeps = eps*2display(count-1)
Floating point relative accuracy
(is true)
Determining machine epsilon
52 digits in double precision (64 bit)
eps = 2.2204e-016ans = 52
Floating point
epBdddddN 4321.
a number digit base
exponent
d1 0: normalized floating-point system
Single precision: 32 bits (23 for digit, 8 for exponent, 1for sign
Double precision: 64 bits (52, 11, 1)
What if we omit this operation?
Use “break” to terminate while and for loops prematurely (or set a limit in while loops)
The loop becomes an “infinite loop” because the statement is always true.
eps = 1;count = 0;while (1+eps) > 1 eps = eps/2; count = count + 1;endeps = eps*2display(count-1)
eps = 1;count = 0;while (1+eps) > 1 & count < 100 eps = eps/2; count = count + 1;endeps = eps*2display(count)
If-Else Structures
if expression (is true)commands
end
if expression 1commands 1
elseif expression 2commands 2
elseif expression 3commands 3::
else commands n
end
if expressioncommands 1
elsecommands 2
end
Relational Operators
< less than<= less than or equal to> greater than>= greater than or equal to== equal to~= not equal to
Logical Operators
& and| or~ not
eps = 1;count = 0;while (1+eps) > 1 eps = eps/2; count = count + 1; if count > 100 break endendeps = eps*2display(count)
eps = 1;count = 0;while (1+eps) > 1 & count < 100 eps = eps/2; count = count + 1;endeps = eps*2display(count)
a = input('what is the value of input data? ')if a > 0 sign = 1;elseif a < 0 sign = -1;else sign = 0;end
disp(sign)
print - print out current figure
print -djpeg filename - save current figure in jpeg format
print -djpeg# filename - save current figure in jpeg format with # (resolution level) between 0 and 100 (default 75)
print -dtiff filename - save current figure in tiff format
print -dpsc filename - save current figure in color PostScript format
Print figures to image files
See help print for more
Filename: lec2b.m% plot 1/x v.s. tanh x
clca = linspace(0,5);f = 1./a;g = tanh(a);plot(a,f,'-', a, g,'--')axis([0 3 0 2])xlabel ('x')ylabel ('y')title ('Plot 1/x and tanh x')text(0.7, 1.6, '1/x')text(2, 1.1, 'tanh x')
» cd y:\cven302» print -djpeg myfigure1» print -djpeg100 myfigure2