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lessons/lesson02/numerical_calculus.m

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+%% Lesson 2c: Numerical estimation of integrals and derivatives
+% Numerical integration and differentiation are a staple of numerical
+% computing. We will now see how easy these are in MATLAB!
+clc; clear; close all;
+
+%% The diff and cumsum functions
+% Note the lengths of z, zdiff, and zcumsum! (Fencepost problem)
+z = [0 5 -2 3 4];
+zdiff = diff(z);
+zcumsum = cumsum(z);
+
+%% Setup: A simple function
+% Let's start with a simple example: y = x.^2. The domain is N points 
+% linearly sampled from lo to hi.
+lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');
+
+%% Numerical (Approximate) Derivatives
+% We can calculate a difference quotient between each pair of (x,y) points
+% using the diff() function.
+dydx = diff(y)./diff(x);    % difference quotient
+
+figure();
+plot(x(1:end-1), dydx);
+
+%% Numerical (Approximate) Integrals
+% Now suppose we want to approximate the cumulative integral (Riemann sum)
+% of a function.
+xdiff = diff(x);
+dx = xdiff(1);              % spacing between points
+dx = (hi-lo) / (N-1);       % alternative to the above (note: N-1)
+
+Y = cumsum(y) * dx;         % Riemann sum
+
+figure();
+plot(x, Y);
+
+%% Error metrics: Check how close we are!
+% dydx should be derivative of x.^2 = 2*x
+dydx_actual = 2 * x;
+
+% Y should be \int_{-2}^{x}{t.^2 dt}
+Y_actual = (x .^3 - (-2)^3) / 3;
+
+% Slightly more accurate -- can you figure out why?
+% Y_actual = (x .^3 - (-2-dx)^3) / 3;
+
+% Error metric: MSE (mean square error) or RMSE (root mean square error)
+% Try changing N and see how the error changes. Try this with both the
+% integral and derivative.
+
+% estimated = dydx;
+% actual = dydx_actual(1:end-1);
+estimated = Y;
+actual = Y_actual;
+mse = mean((estimated - actual) .^ 2);
+rmse = rms(estimated - actual);
+
+%% Fundamental Theorem of Calculus
+% Now, use the approximate derivative to get the original function, y back 
+% as yhat and plot it.  You may need to use/create another variable for
+% the x axis when plotting.
+figure();
+yhat = diff(Y)./diff(x);    % differentiate Y
+plot(x(1:end-1), yhat);
+
+figure();
+yhat2 = cumsum(dydx) * dx;  % integrate dydx
+plot(x(1:end-1), yhat2);

lessons/lesson02/vectorized_operations.m

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+%% Lesson 2a: More vector and matrix operations
+%
+% Objectives:
+% * Understand how to perform vector operations in MATLAB
+% * Understand arithmetic and basic functions in MATLAB
+
+%% Vector operations
+% In lesson 1, we saw how to create a vector with the colon operator and
+% linspace. Now let's perform some operations on them!
+% 
+% There are two common classes of operations that you can perform on vectors:
+% element-wise operations (which produce another vector) and aggregate
+% operations (which produce a scalar value). There are also many functions that
+% don't fall under these categories, but these cover many of the common
+% functions.
+
+%% Element-wise operations
+% Many operations that work on scalars (which are really degenerate matrices)
+% also work element-wise on vectors (or matrices).
+x = 0:0.01:2*pi;    % Create a linearly-spaced vector
+y = sin(x);         % sin() works element-wise on vectors!
+y = abs(x);         % same with abs()!
+y = x .^ 4;         % element-wise power
+y = power(x, 4);    % same as above
+
+plot(x, y);         % Plot y vs. x (line graph)
+title('y vs. x');
+
+%% Aggregate operations
+% Another common class of operations produce a single output or statistic about
+% a vector (or matrix).
+length(x);              % number of elements in x
+sum(x);                 % sum of the elements of x
+mean(x);                % average of the elements of x
+min(x);                 % minimum element of x
+diff(x);                % difference between adjacent elements of x
+
+%% Exercise 1 : Vector operations
+T = 1e-6;           % Sampling period (s)
+t = 0:T:2e-3;       % Time (domain/x-axis)
+f0 = 50;            % Initial frequency (Hz)
+b = 10e6;           % Chirp rate (Hz/s)
+A = 10;             % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);
+
+%% Exercise: Numerical calculus
+% See numerical_calculus.m.
+
+%% Basic indexing in MATLAB
+% The process of extracting values from a vector (or matrix) is called
+% "indexing." In MATLAB, indices start at 1, rather than 0 in most languages
+% (in which it is more of an "offset" than a cardinal index).
+
+%% Exercise 2 : Basic indexing
+% The syntax for indexing is "x(indices)", where x is the variable to index,
+% and indices is a scalar or a vector of indices. There are many variations on
+% this. Note that indices can be any vector
+x(1);                   % first element of x
+x(1:3);                 % elements 1, 2 and 3 (inclusive!)
+x(1:length(x));         % all elements in x
+x(1:end);               % same as above
+x(:);                   % same as above
+x(end);                 % last element of x
+x(3:end);               % all elements from 3 onwards
+x([1,3,5]);             % elements 1, 3, and 5 from x
+
+x(1:2:end);             % all odd-indexed elements of x
+ind = 1:2:length(x);
+x(ind);                 % same as the previous example
+
+%% Exercises to improve your understanding
+% Take some time to go through these on your own.
+x([1,2,3]);             % Will these produce the same result?
+x([3,2,1]);
+
+x2 = 1:5;
+x2(6);                  % What will this produce?
+x2(0);                  % What will this produce?
+x2(1:1.5:4);            % What will this produce?
+ind = 1:1.5:4;
+x2(ind);                % What will this produce?
+
+z = 4;
+z(1);                   % What will this produce?
+
+%% Matrix operations
+% Matrices is closely related to vectors, and we have also explored some matrix
+% operations last class. This class, we are going to explore functions that are
+% very useful but are hard to grasp for beginners, namely reshape, meshgrid,
+% row-wise and column-wise operations.
+
+%% Reshape
+% Change a matrix from one shape to another. The new shape has to have the same
+% number of elements as the original shape.
+%
+% When you are reshaping an array / matrix, the first dimension is filled
+% first, and then the second dimension, so on and so forth. I.e., elements
+% start filling down columns, then rows, etc.
+M = 1:100;
+N1 = reshape(M,2,2,[]);    % It would create a 2*2*25 matrix
+N2 = reshape(M,[2,2,25]);  % Same as N1
+N2(:,:,25);                % Gives you 97,98,99,100
+N2(:,1,25);                % Gives you 97 and 98
+
+%% Row-wise / Column-wise operations
+% Vector operations can also be performed on matrices. We can perform a vector
+% operation on each row/column of a matrix, or on a particular row/column by
+% indexing.
+H = magic(4);    % create the magical matrix H
+sum(H,1);        % column wise sum, note that this is a row vector(default)
+fliplr(H);       % flip H from left to right
+flipud(H);       % flip H upside down
+H(1,:) = fliplr(H(1,:)); % flip only ONE row left to right
+H(1,:) = [];     % delete the first row
+
+%% Exercise 7 : Matrix Operations
+H2 = randi(20,4,5);   % random 4x5 matrix with integers from 1 to 20
+sum(H2(:,2));
+mean(H2(3,:));
+C = reshape(H2,2,2,5);
+C(2,:,:) = [];