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%% Lesson 3b: Advanced Indexing
%
% Objectives:
% - Review basic vector indexing
% - Introduce multidimensional indexing and linear indexing
% - Introduce logical indexing
clear; clc; close all;
%% Review of basic vector indexing
% Indexed expressions can appear on the right-hand side of an assignment or the
% left-hand side (rvalues and lvalues in C).
x = [2 4 8 16 32 64 128];
x(3);
x(4:6);
x(:);
x(end);
x(2) = -4;
%% Vectors indexing vectors
% You can use a vector to index another vector. This can be useful if there is
% a specific index pattern you want from a vector.
x([1 3 5 7]); % Returns a vector with the 1st, 3rd, 5th, and 7th
% entries of x
x(1:2:7); % Same thing uses the colon operator to create the
% index vector
x([2:4 3:6]);
x([1:4]) = 7; % Changes the values from the 1st and 4th entries
% of x to 7
%% Indexing matrices with 2 subscripts
% You can index a matrix using two indices, separated by a comma.
A = [16 2 3 13; 5 11 10 8; 9 7 6 12; 4 14 15 1];
A(1,1); % Returns entry in first row and first column
A(1:3,1:2); % Returns the entries in the first three rows AND
% first two columns
A(end,:);
% What if I want to index the (2,1), (3,2), (4,4) entries?
A([2 3 4],[1 2 4]); % Will this work?
%% Linear Indexing
% Instead you should index linearly! Linear indexing only uses one
% subscript
A(:); % What does this return and in what form and order?
reshape(A, [], 1); % What about this?
A(14); % What value does this return?
%% Non-rectangular indices
% Now that we know about linear indexing, what values does this return?
A([2 7 16]);
% Luckily enough, MATLAB has a function that would calculate the linear
% index for you!
ind = sub2ind(size(A),[2 3 4],[1 2 4]);
A(ind);
% Note that sub2ind follows a common naming convention in MATLAB for converting
% one type to another. For example, there is also ind2sub, str2int, int2str,
% zp2tf, tf2zp, etc.
%% Logical Indexing
% Logical indexing allows us to use boolean logic to build more complex
% indices. This is based off of boolean logic: we have two boolean values TRUE
% (represented by 1) and FALSE (represented by 0), and the boolean operators
% AND, OR, and NOT.
%
% Logical scalar and matrix literals can be constructed from double and matrix
% literals using the logical() function, or they may be generated from other
% operations. (Note that logical and double values have different data types.)
B = eye(4); % Regular identity matrix
C = logical([1 1 1 1; 1 0 0 0; 1 0 0 0; 1 0 0 0]); % Logical entries
islogical(B);
islogical(C);
%% Some logical operators
B = logical(B); % Converts B into a logical array
B & C; % and(B,C) is equivalent to B&C
B | C; % or(B,C) is equivalent to B|C
~(B & C); % not(and(B,C)) is equivalent to ~(B&C)
not(B & C); % you can use both representations intermittenly
%% Functions that generate logical values
% The following functions are analogous to the ones() and zeros() functions
% for double matrices.
true(2,5); % creates a 2x5 logical matrix of trues(1s)
false(2,5);
%% Generating logical values from predicates
% Relational (comparison) operators generate logical values. Relational
% operators on matrices will perform the operation element-wise and return a
% logical matrix.
%
% This is not anything really special, since we've seen element-wise operations
% on matrices. The only change is that instead of functions that return
% numbers, we have functions that return booleans (predicates).
D = [1 2 3; 2 3 4; 3 4 5];
E = [1 5 6; 9 2 8; 7 3 4];
D == 3; % Returns true every place there is a 3.
D < E; % Elementwise comparison
D < 3; % Elementwise comparison against a scalar.
% (Note: this is broadcasting!)
D < [7; 4; 1]; % Another example of broadcasting: comparison
% of each row against a scalar.
%% Logical indexing
% Logical indexing uses a logical value as the index. Elements in the indexed
% matrix where the logical index is TRUE will be returned, and FALSE values
% will be disregarded. The returned matrix will always be a column vector,
% similar to indexing with (:).
A(A > 12); % returns entries of A which are greater
% than 12 in a column vector
% To return the linear indices of the TRUE values in a logical matrix.
find(A > 12);
A(find(A > 12)); % Gives the same results as above

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%% Lesson 3a: Meshes & Broadcasting
%% Meshgrid
% Meshgrid is quite hard to understand. Think of it as a way to replicate
% arrays, like the following example:
a = 1:3;
b = 1:5;
[A,B] = meshgrid(a,b);
%% Using meshgrid for functions of multiple variables
% You have created two arrays A and B, note that in A, the vector a is copied
% row-wise, while the vector b is transposed and copied column-wise. This is
% useful, because when you lay one above the other, you essentially create a
% CARTESIAN PRODUCT, and it is useful when we need to plot a 3D graph.
%
% Here is a more complicated example to show you when meshgrid is useful
a1 = -2:0.25:2;
b1 = a1;
[A1,B1] = meshgrid(a1);
% Here we plot the surface of f(x) = x*exp^(x.^2+y.^2)
F = A1.*exp(-A1.^2-B1.^2);
surf(A1, B1, F);
%% Broadcasting: an alternative to meshgrid
% Broadcasting, like meshgrid, can be used as a way to write functions of
% multiple variables, without generating the intermediate matrices A and B.
% The way this works is that operations performed on a set of N orthogonal
% vectors will automatically generate an N-dimensional result.
%
% See the following example, which recreates the above example. Note that b1 is
% transposed.
F2 = a1.*exp(-a1.^2-(b1.').^2);
surf(A1, B1, F2);
% Check that this matches the previous result.
error = rms(F - F2, 'all');
% Note: broadcasting doesn't generate the domains A1 and B1, so meshgrid() is
% more useful when we need to supply the domain to a function like mesh() or
% surf(). But when we only need the result, broadcasting is somewhat simpler.
%
% For homework assignments that require functions of multiple variables, use
% whichever method is more intuitive to you.

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