% NOTES
%
% MATLAB Preliminaries
%
% MATLAB File types
% Scripts are used for storing several executable statements.
% Functions are used for functions and subroutines.
% MAT-files are used for storing data.
%
% Editing
%
% The command window.
%
% 1. The command window is used for immediate mode entering and display
% of variables and execution of a MATLAB command.
%
% 2. The command window is used for execution of a functions, subroutines
% and programs that are stored in an M-File .
%
% The edit window.
%
% 1. The edit window is write and edit functions, subroutines and programs
% which are to be stored in an M-File.
%
% The graph window
%
% 1. The graph window is used to view plotted data.
%
% Diary
% Diary <filename> directs output to the file
% Diary off suspends the diary
% Diary on turns diary on again
%
% Conservation of resources
%
% Most numerical experiments involve columns of numbers which might extend
% several pages. It is not necessary to see all of the computations in a project.
% Judicious use of diary and files commands such as X(1:5:N) will reduce the
% amount of paperwork. Before you hand in a project, determine how little is required.
%
%
% Obtaining Help
%
% HELP topic gives help on the topic.
% HELP word where 'word' is a filename,
% displays the first comment lines in the M-file 'word.m'.
%
% Housekeeping
%
% ; Place at the end of line to suppress the computer echo.
% , Used at the end of a line if the computer echo is desired.
% % This is a comment.
%
% format short output displays four decimal places
% format long output displays fourteen decimal places
% who list of variables
% what list of files
% clear clear workspace
% clear variables clear variables
% clear functions clear functions
% clc clear display
% clg clear graph
%
% Assignment Statements
%
% x = 2;
% y = x^2 -3*x + 2;
%
% Form of a MATLAB Program M-File of executable statements
%
% % <program-name>
% {<specification-statements>}
% {<executable-statements>}
%
% Form of a MATLAB Subroutine M-File
%
% function (output-list) = <subroutine-name> (argument-list)
% {<specification-statements>}
% {<executable-statements>}
%
% Example, store the following subroutine in the M-file; sroot.m
%
% function r = sroot(A)
% % Newton's method to find A^(1/2)
% p0 = 1; % Starting value.
% for k=1:50,
% p1 = (p0+A/p0)/2;
% disp(p1);
% if abs(p1-p0)/p1 < eps, break, end;
% p0 = p1;
% end
%
% Form of a function (or subroutine) call.
%
% sroot(5)
%
% 3
% 2.33333333333333
% 2.23809523809524
% 2.23606889564336
% 2.23606797749998
% 2.23606797749979
% 2.23606797749979
%
%
% Form of a MATLAB function M-File
%
% function <output-list> = <function-name> (argument-list)
% {<specification-statements>}
% {<executable-statements>}
% <output-name> = <returned computation>
%
% Store the following functions in the M-files: f.m and G.m
%
% function y = f(x)
% y = exp(-x./10) + sin(x);
%
% function W = G(Z) % Z is a 1 by 2 vector
% x = Z(1);
% y = Z(2);
% W = [x.^2-y.^2 2*x.*y];
%
% Form of a Function call.
% f(pi/2)
%
% ans =
% 1.85463599915323
%
% G([2 1])
%
% ans =
% 3 4
%
% Arrays
%
% All variable in MATLAB are treated as arrays. A scalar is a one by one array.
% Initialization can fill an array with either zeros or ones.
%
% Z = zeros(1,5) % Initialize a row vector with zeros.
%
% Z =
% 0 0 0 0 0
%
% W = zeros(3,1) % Initialize a column vector with zeros.
%
% W =
% 0
% 0
% 0
%
% M = ones(2,4) % Initialize a matrix with ones.
%
% M =
% 1 1 1 1
% 1 1 1 1
%
% size(M)
%
% ans =
% 2 4
%
% Vectors
%
% Vectors can be entered and stored as a matrix with one row or one column
%
% V = [1,2,3,4]
%
% V =
% 1 2 3 4
%
%
% length(V)
%
% ans =
% 4
%
% sum(V) % The sum of the elements of V.
%
% ans =
% 10
%
% mean(V) % The mean of the elements of V.
%
% ans =
% 2.5000
%
%
% Transpose
%
% W = [1,
% 2,
% 3,
% 4]
%
% W =
% 1
% 2
% 3
% 4
% W'
%
% ans =
% 1 2 3 4
%
% Colon
%
% Used for creating a list and it is used for selecting part(s) of a list.
%
% X = 1:20; % A list of integers from 1 to 20.
% Y = X.^2; % A list of their squares.
% Y(10:20) % Display only the last 11 entries.
%
% ans =
% 100 121 144 169 196 225 256 289 324 361 400
%
% A = [1 2 3 4,
% 5 6 7 8,
% 9 10 11 12];
%
% A(1,3) % Select one element in a matrix.
%
% ans =
% 3
%
% A(2:3, 1:2) % Select a sub-matrix.
%
% ans =
% 5 6
% 9 10
%
%
% Arithmetic Operations
%
% + Addition
% - Subtraction
% * Multiplication
% / Division
% ^ Power
%
%
% Arithmetic Operations for arrays (also useful for functions to be plotted)
%
% .+ Element-wise addition
% .- Element-wise subtraction
% .* Element-wise multiplication
% ./ Element-wise division or use ( ).^(-1)
% .^ Element-wise power
%
% A = [1 2,
% 3 4];
% A^2 % Square a matrix.
%
% ans =
% 7 10
% 15 22
%
% A.^2 % Square each element in a matrix.
% ans =
% 1 4
% 9 16
%
% C = inv(A) % Find the inverse of a matrix.
% C =
% -2.0000 1.0000
% 1.5000 -0.5000
%
% A*C % Verify that C is the inverse.
% ans =
% 1.0000 0
% 0.0000 1.0000
%
% Relational Operators
%
% == Equal to
% ~= Not equal to
% < Less than
% > Greater than
% <= Less than or equal to
% >= Greater than or equal to
%
% Logical Operators
%
% ~ Not Complement
% & And True if both operands are true
% | Or True if either (or both) operands are true
%
% Boolean Values
%
% 1 True
% 0 False
%
% IF (Block) Control Statement
%
% Performs the series of {<executable-statement>} following it
% or transfers control to an ELSEIF, ELSE, or ENDIF statement,
% depending on (<logical-expression>).
%
% if (<logical-expression#1>),
% {<executable-statements>}
% elseif (<logical-expression#2>),
% {<executable-statements>}
% end
%
%
% if (<logical-expression#1>),
% {<executable-statements>}
% elseif (<logical-expression#2>),
% {<executable-statements>}
%
% else
% {<executable-statements>}
% end
%
%
% Break
%
% if n==3, break, end
%
% for k=1:100,
% x=sqrt(k);
% if x>5, break, end
% end
%
%
% For loop
%
% sum1 = 0;
% for k = 1:1:10000,
% sum1 = sum1 + 1/k;
% end
% sum1
%
% sum1 =
% 9.78760603604434
%
% sum2 = 0;
% for k = 10000:-1:1,
% sum2 = sum2 + 1/k;
% end
% sum2
%
% sum2 =
% 9.78760603604439
%
% for j = 1:5,
% for k = 1:5,
% A(j,k) = 1/(j+k-1);
% end
% end
% A % The 5 by 5 Hilbert matrix A will be displayed.
%
%
% While (Block) Control Statement
%
% m = 10;
% k = 0;
% while k<=m
% x = k/10;
% disp([x, x^2, x^3]); % A table of values will be printed.
% k = k+1;
% end
%
%
% Pause statement
%
% pause
%
% Mathematical functions
%
% cos(x) cosine (radians)
% sin(x) sine (radians)
% tan(x) tangent (radians)
% exp(x) exponential exp(x)
% acos(x) inverse cosine (radians)
% asin(x) inverse sine (radians)
% atan(x) inverse tangent (radians)
% log(x) natural logarithm base e
% log10(x) common logarithm base 10
% sqrt(x) square root
% abs(x) absolute value
% round(x) round to nearest integer
% fix(x) round towards zero
% floor(x) round towards -
% ceil(x) round towards +
% sign(x) signum function
% cosh(x) hyperbolic cosine
% sinh(x) hyperbolic sine
% tanh(x) hyperbolic tangent
% acosh(x) inverse hyperbolic cosine
% asinh(x) inverse hyperbolic sine
% atanh(x) inverse hyperbolic tangent
% real(z) real part of complex number z
% imag(z) imaginary part of complex number z
% conj(z) complex conjugate of the complex number z
% angle(z) argument of complex number z
% rem(p,q) remainder when p is divided by q
%
%
% Data Analysis
%
% max maximum value
% min minimum value
% mean mean value
% median median value
% std standard deviation
% sort sorting
% sum sum the elements
% prod form product of the elements
% cumsum cumulative sum of elements
% cumprod cumulative product of elements
% diff approximate derivatives (differences)
% hist histogram
% corrcoef correlation coefficients
% cov covariance matrix
%
%
% Graphics
%
% Store the following script in the M-file; sketch.m
%
% X = 0:pi/8:pi;
% Y = sin(X);
% axis([-0.2 3.2 -0.1 1.1]);
% plot(X,Y);
% hold on;
% plot([-0.2 3.2],[0,0],[0,0],[-0.1 1.1]);
% xlabel('x');
% ylabel('y');
% title('Graph of y = sin(x)');
% grid;
% axis;
% hold off;
%
% Issue the command sketch and obtain the graph
%
%
%
% Remark. Data used by the plot command must be two vectors of equal length.
%
% X =
% 0 0.3927 0.7854 1.1781 1.5708 1.9635 2.3562 2.7489 3.1416
% Y =
% 0 0.3827 0.7071 0.9239 1.0000 0.9239 0.7071 0.3827 0.0000
%
%
% Polynomial functions
%
% The coefficients of a polynomial are stored as a coefficient list.
%
% C = [1 -3 2]
% C =
% 1 -3 2
%
% If C is a vector whose elements are the coefficients of a polynomial,
% then polyval(C,x) is the value of the polynomial evaluated at x.
%
% for x=0:0.5:3,
% disp([x,polyval(C,x)]),
% end
%
% 0 2
% 0.5000 0.7500
% 1 0
% 1.5000 -0.2500
% 2 0
% 2.5000 0.7500
% 3 2
%
%
% Finding the roots of a polynomial. Let p(x) = x5 - 10x4 + 35x3 - 50x2 + 24
%
% C = [1 -10 35 -50 24]
% C =
% 1 -10 35 -50 24
%
% roots(C)
% ans =
% 4.0000
% 3.0000
% 2.0000
% 1.0000
%
% Other operations with polynomials are:
%
% conv polynomial multiplication
% polyfit polynomial curve fitting
%
%
% Graph the polynomial p(x) = x5 - 10x4 + 35x3 - 50x2 + 24
%
% Store the following script in the M-file; sketch.m
%
% C = [1 -10 35 -50 24];
% X=-0.2:0.1:4.2;
% Y=polyval(C,X);
% axis([-0.2 4.2 -2.3 4.3]);
% plot(X,Y);
% hold on;
% plot([-0.2 4.2],[0,0],[0,0],[-2.3 4.3]);
% xlabel('x');
% ylabel('y');
% title('Graph of a polynomial.');
% grid;
% axis;
% hold off;
%
% Issue the command sketch and obtain the graph.
%
%
%
% For function of the independent variable x one can define f
% as a text string:
%
% f = 'sin(x)'
%
% Then use the fplot command
%
% fplot(f,[0 10]);
%
%
%
% 3-dimensional graphs
%
% Store the following script in the M-file; sketch.m
%
% [X Y] = meshdom(-1:0.1:1, -1:0.1:1);
% Z = X.^2 - Y.^2;
% mesh(Z);
% title('Graph of z = x^2 - y^2');
%
% Issue the command sketch and obtain the graph.
%
%
% More commands for matrices
%
% zeros zero matrix
% ones matrix of ones
% rand random elements
% eye identity matrix
% meshdom domain for mesh plots
% rot90 rotation of matrix elements
% fliplr flip matrix left-to-right
% flipud flip matrix up-and-down
% diag extract or create diagonal
% tril lower triangular part
% triu upper triangular part
% ' transpose
%
%
% Solution of a linear system AX = B
%
% A = [1 2 3 4, % Enter the matrix A.
% 2 5 1 1,
% 3 1 2 1,
% 4 1 1 3];
%
% A = [1 2 3 4,
% 2 5 1 1,
% 3 1 2 1,
% 4 1 1 3];
%
% B = [2 1 3 4]' % Enter the vector B.
%
% B =
% 2
% 1
% 3
% 4
%
% X = A\B % Solve the linear system AX = B.
%
% X =
% 0.8333
% -0.2273
% 0.2576
% 0.2121
%
% A*X % Verify that X is the solution.
%
% ans =
% 2
% 1
% 3
% 4
%
%
% Eigenvectors and eigenvalues
%
% A = [1 2 3 4, % Enter the matrix A.
% 2 5 1 1,
% 3 1 2 1,
% 4 1 1 3];
%
% [V E] = eig(A) % V is a matrix of eigenvectors.
% % E is a diagonal matrix of eigenvalues.
%
% V =
% -0.2397 -0.0481 -0.8000 0.5479
% 0.8438 0.0977 0.0966 0.5187
% -0.1880 -0.8201 0.3733 0.3908
% -0.4418 0.5618 0.4597 0.5272
%
% E =
% 3.6856 0 0 0
% 0 1.3717 0 0
% 0 0 -2.9395 0
% 0 0 0 8.8822
%
