%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIRST DAY MATLAB worksheet
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for MTH 451-551
%%% this file helps to get familiar with basic elements of MATLAB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%% M. Peszynska,  2017/09/25
%% Copyright Department of Mathematics, Oregon State University
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%% INSTRUCTIONS: please type every line yourself
%%%               ignore all lines and text beginning with %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
diary yourname
doc doc
%% allocate (reserve, initialize) space for a vector
n = 100;  x = zeros(n,1);
%% how large can a vector be ? 
n = 10^5; x = zeros(n,1);
n = 1e9; x = zeros(n,1);
%%%%%%%%% EXERCISE: push it further. How far can you go ?
%% how to input a matrix
A = [2 3 5; 7 11 13; 17 19, 23]
%% how to access its elements
A(1), A(3)
%% the transpose of a matrix
A'
%% how to access the first column and rows from 1:2
A (1:2,1)
%% how to access the first column and all rows
A (1:3,1)
A (:,1)
%% how to get a matrix with random entries
B = rand (3,5)
%%%%%%%%%% EXERCISE: swap the first and last columns of B
%% input a column vector
x = [1; 2; 3]
%% input a row vector 
y = [1 2 3]
%% change from a row vector to column vector
x = y'
%% operations on vectors: multiply a vector by a scalar
pi * x
y = y'
%% (now y is a column vector)
%% dot (inner) product of two column vectors
x, y, x'*y
%% compute outer product 
x * y'
%% compute a vector as a component-by-component product of x and y
x.*y
%%%%%%%%%% EXERCISE: do SAXPY z=x+alpha*y  where alpha is a scalar  
%% how to set-up a simple loop to compute the sum
n = 3;
s = 0;
for j=1:n s = s + x(j); end
s
%% compare to MATLAB's version
sum(x)
%%%%%%%%% EXERCISE: write loops to calculate inner product, SAXPY,  
%%%%%%%%%% EXERCISE: compute Frobenius norm of matrix A directly
%% how to get the size of a vector/matrix if forgotten
size(x)
%% identity matrix
eye(4)
%% eigenevalues of a matrix
eig (A)
%% various norms of matrices: (check 'fro')
norm(A), norm(A,1), norm(A,2), norm(A,infty) 
%%%%%%%%%% EXERCISE: compare these norms with eigenvalues 
%%%%%%%%%% EXERCISE: verify the inequality between 
%%%%%%%%%%%% norm(A,2) and product of norm(A,1),norm(A,inf)
%%%%%%%%%%%% (make some observations, then a conjecture). 
%%%%%%%%% WHAT is "i" and play with complex matrices
i
A = [ 1, i; -2+3*i; 0]
norm(A)
norm(A,1)
eig(A)
%%%%%%%%% EXERCISE: experiment with unitary matrices
%%%%%%%%% EXERCISe: what about A=[1 1; 0 1] ?
%% how to plot
x = 0:.01:6*pi;
plot(x,sin(x));
%%%%%%%%%% EXERCISE: plot exp(x) and use log log plots 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% EXERCISE: what does A do to a vector ? 
%% set up some 2x2 matrix of your choice
theta = linspace(0,2*pi);
u=[cos(theta);sin(theta)];
size(u);
plot(u(1,:),u(2,:)); axis equal;
u=A*u;
%% now change u to be an ellipse. How ? 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% understand floating point precision 
%% find x for which ((1-x)-1)/x is not what it "should" be
%% find a and b for which a+b
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HOMEWORK
%%%%%%%% Go to mathworks.com/moler/chapters.html
%%%%%%%% Read Chapter 1: Introduction to Matlab. Solve 1.2 and 1.38
%%%%%%%% Read Chapter 2: Linear Equations. Solve 2.1 and 2.5a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%