%% Loops and one dimensional arrays %% Introduction % % So far, we have covered how to use linspace % to construct a one-dimensional array useful for plotting. In this lab, % we want to explore other ways to create one-dimensional arrays. %

% We will also learn how to retrieve particular entries of an array, as % well as how to use the for loop to control % the flow execution of statements in a script. % %% Using the colon operator to construct arrays % % We have already seen how we can construct one dimensional arrays using % the linspace command. We have also seen % how to use square brackets to enclose a list of numbers: % x = [1 3 5 7 9] %% % % Another method to construct an array that is very similar to the % linspace makes use of the : operator. For example, % v = 1:10 %% % % constructs an array of 10 integers between 1 and 10. A more general use % of the colon (:) operator allows us to % specify the step size we wish to take. % v = 0:2:20 %% % % is a list of even numbers between 0 and 20. The step size can also be % negative. For example, % v = 5:-1:-5 %% % % or % v = 50:-5:0 %% % % Querying the help documentation on the % the colon operator gives us: %
% >> help colon
%  :  Colon.
%     J:K  is the same as [J, J+1, ..., K].
%     J:K  is empty if J > K.
%     J:D:K  is the same as [J, J+D, ..., J+m*D] where m = fix((K-J)/D).
%     J:D:K  is empty if D == 0, if D > 0 and J > K, or if D < 0 and J < K.
% .............
% 
%
%

Like linspace, the colon operator % automatically fills in entries in an array with equally spaced points. % However, we can get some unexpected results if the endpoints we specify % are not evenly divided by the stepsize we specify. Consider the % following example % v = 0:3:10 %% % % We notice that the endpoint 10 does not appear in the array. The reason % for this is that 10 is not evenly divided by 3, so is not reached in % steps of 3 starting from 0. Rather than shorten the last step size, % Matlab only includes those values between the specified endpoints that % are even multiples of the step size and shifted by the first endpoint. %

% Carrying the above example further, we see that it is possible to % construct an array that contains only a single element : % v = 10:15:20 %% % % In this case, one step already takes us beyond the specified endpoint, % and so the vector only contains the starting point 10. %

% What happens if the step size does not have the same sign as our end % value minus our start value? In this case, Matlab will create an % empty array. The following example illustrates this. % v = 0:-1:10 %% % % Here, 10-0 is positive, but our stepsize -1 is negative, and so an empty array is created. Any % time we use a step size of 0, we will also get an empty array. The % simplest empty array in Matlab is an empty pair of square brackets ([]). % v = [] %% % % The length of an empty array is 0. % length(v) %% The colon operator and the linspace command % % We can use the colon operator to construct the same array that linspace creates. To describe how the colon % operator and the linspace are related, we % consider a vector with a start point a, end % point b and n equally % spaced entries. Using linspace, we create % this vector as % a = -1; % start of the linspace vector b = 1; % end of the linspace vector n = 11; % Number of points in the linspace vector x1 = linspace(a,b,n) %% % % To construct the same same vector using the colon operator, we need to % first compute the step size that linspace used. % step = (b - a)/(n-1) % Step size used by linspace %% % % And then we can construct the same vector using the colon operator. % x2 = a:step:b %% % % A word of warning : Whereas the linspace is guaranteed to have exactly a and b as endpoints, and % exactly n entries, this is not guaranteed when % using the colon operator. We have already seen above that the : operator will leave off the final entry if it % cannot be expressed in terms of multiples of the stepsize. This can % also happen, even when the step sizes are only nearly divide the % interval. % step = (1+1e-15)/5; v = 0:step:1 %% % % In this example, we divided an interval slightly larger than $1$ by the % $5$, and then used this step size to construct an array. The final % element in the array is not 1, as might be expected, but rather is a % value very close to 0.8. To see why this is, we print out the value % 5*step and see that it is in fact larger than % 1, if only by a very small amount. % format long e 5*step %% Indexing arrays % % To retrieve a single entry from a one-dimensional array, we can use the % parenthesis operator () with an argument % indicating the entry we want. First, we will create an array of random % numbers using the rand function. % format short e v = rand(1,10) %% % % We can retrieve the second entry in v as % v(2) %% % % We can also retrieve multiple elements of an array by using an array of % index locations. Suppose we wanted to retrieve every other element in an % array of 7 elements. We pass an integer sequence indicating the entries % we want to the () operator. % v([1,3,5,7]) %% % % Alternatively, we could do the following. % v(1:2:7) %% % % In both cases, we used an array of integers to specify the desired % indices. We can also use the colon operator to reorder the entries in % the array : % w = v(10:-1:1) %% % % In fact we can choose entries in order, even duplicating entries : % w = v([5 7 2 9 9 3 2 2]) %% % % In Matlab, the first entry in an array is always 1. If we try to index the zero-th element of an % array, we can expect an error %

% v(0)
% 
%
% Subscript indices must either be real positive integers or logicals.
% 
%
%

If we use an index that is larger than the number of elements that we % have, we can also expect an error: %

% v(12)
% 
%
% Index exceeds matrix dimensions.
% 
% %% % % What if you want to get the last three entries in an array? Since the % length of an array is often one of the variables in the script you are % writing, you might consider doing the following. % n = length(v); v([n-2 n-1 n]) %% % % Here, we used the Matlab length function % to find out how long v is. %

% The above situation arises so often, that Matlab has built into the % language a way to get the last entries of a vector by allowing the use of % the end keyword inside of arrays. This % statement retrieves the last entry in an array % v(end) %% % % To retrieve the last three entries in an array, you can use the end as if it were the actual variable or constant % storing the length of the array. For example, % v([end-2 end-1 end]) %% % % or simply % v(end-2:end) %% Simple array concatenation % % We can always construct a new array by glueing two arrays together. For example, suppose we have two row vectors u and v. We can construct a % new array w by glueing, or % concatenating u and v together using square brackets. % u = -5:-2 %% v = 21:27 %% w1 = [u v] %% w2 = [v u] %% % % The resulting arrays w1 and w2 each have length equal to the sums of the lengths % of u and v. %

%

Example : Approximating $\pi$


%

Suppose we % want to approximate the value of $\pi$ by % computing the length of an n-sided the polygon whose vertices are on the % quarter circle of radius 2, as shown below. %

%

%
%

To approximate $\pi$, we compute % the perimeter of the polygon described by this quarter circle.

%
%

%

We construct arrays x and y containing the coordinates of each vertex of the % quarter of a circle (including the origin), and then compute the distance % between each neighboring pair of vertices $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$. For % convenience, we use array concatenation to add the first point $(x_1,y_1)$ to the end of our vertex list so % that we have $x_{n+1} = x_1$ and $y_{n+1} = y_1$. We can then compute the % length of each side of the polygon with a single Matlab statement and % then sum the lengths to get the perimeter of the polygon. By subtracting % out the length of the edges that lie on the x and y axes, we can get an % approximation to $\pi$. % % --------------------------------------------------------- % Approximate pi by computing the perimeter of a semicircle % --------------------------------------------------------- n = 8; % Number of vertices on the polygon R = 2; % Radius of the circle th = pi*(0:n)/(2*n); % Angles 'theta' between 0 and pi/2 x = R*cos(th); % x-coordinates along the semicircle y = R*sin(th); % y-coordinates along the semicircle % Construct the vertex list, including the point (0,0). Wrap the first % point around to the end of the list. x = [x 0 x(1)]; y = [y 0 y(1)]; % Compute distances between each neighboring pair of points d = sqrt((x(1:end-1) - x(2:end)).^2 + (y(1:end-1) - y(2:end)).^2); % Approximate pi by adding up the lengths of the edges of the polygon, and % and subtracting out the segments that lie along the x- and y-axis. pi_approx = sum(d)-4; % Print out the results fprintf('Approximation to pi %24.16f\n',pi_approx); fprintf('Matlab value ''pi'' %24.16f\n',pi); fprintf('Error %24.8e\n',abs(pi_approx - pi)); %% % % This approximation uses the sum function % which we will discuss in more detail below. % %% Introduction to the 'for' loop % % Up until now, commands in a script have been executed in the order in % which they appear in a script. We can control this order of execution % using flow control statements. The for loop is a type of flow control statement that % allows us to execute a set of commands multiple times. For example, in % the following code fragment, the statement fprintf('Hello, World\n') gets executed 10 times, % even though it only appears once in the code fragment. % for i = 1:10, fprintf('Hello, World!\n'); end %% % % The for loop has three basic components : % the enclosing keywords for and end which enclose the set of statements to be % carried out multiple times by the loop (the statement % fprintf('Hello, World\n'), % in the above example), a loop index which is % the variable that stores the current value of the loop counter (i in the above example), and an index array % containing the range of values over which the loop index should vary (the % array 1:10 in the above example). % See below for the "Anatomy of a "for" loop" %

%
% Anatomy of a for loop %
%

% The number of times the loop executes the statements enclosed by the % for and end % is determined by the length of the array describing the loop range. % Consider each of the following examples : % % Example 1 for i = 0:5:10, fprintf('Three brown bears\n'); end %% % Example 2 for x = [3.1 5.6], fprintf('A couple of ducks\n'); end %% % Example 3 for j = 10:-2:0, fprintf('Six Simple Simons sitting on a stump\n'); end %% % Example 4 for j = 1:-1:10, fprintf('I am being ignored (:-((\n'); end %% % % In each case, the statement enclosed by for % loop was executed once for each entry in the loop index array. In the % last example, the loop index array is empty, and so the enclosed % statement did not get executed at all. % %% The 'for' loop - using the loop index variable % % In the above examples, we did not make use of the actual values in the % loop index array, only the length of the array. But in most cases, the % statements enclosed by the for loop will % actually involve the value of the loop index variable as well. Each of % the following loop examples illustrates how we can make use of the loop % index variable. % clear all; %% % Example 1 - Print out the value of the loop index for i = 0:2:20, fprintf('i = %d\n',i); end %% % Example 2 - Print out the entries in an array 'x', in reverse order. x = rand(1,10); for k = 10:-1:1, fprintf('x(%2d) = %8.4f\n',k, x(k)); end %% % Example 3 - Fill in entries in an array 'y' and print each value for j = 1:8, y(j) = sin(pi*(j-1)/2); fprintf('y(%d) = %16.8f\n',j,y(j)); end %% % Example 4 - Cumulative sums of the integers 1 through 10. s = 0; for k = 1:10, s = s + k; fprintf('Sum of the numbers 1 to %2d : %5d\n',k, s); end %% % % In all of the above examples, the loop index is an integer. This % is by far the most common loop index array for the for % loop. However, there is nothing in Matlab to prevent us from trying the % following unusual (and not particularly recommended) ways to use % the loop index. % % Example 5 - A confusing example in which 'x' is an array and 'y' takes on % different values in the array 'x'. x = linspace(-1,1,7); for y = x, fprintf('y = %f\n',y); end %% % Example 6 - Let's see what we can do with strings. for c = ['Hello', 'Goodbye','See Ya!'], fprintf('%s',c); end %% % % In the first case, the loop index array x is an % array containing non-integer values, but the loop index y still takes on each value, and the print statement % is executed the expected number of times, e.g. one time for each entry in % x. %

% The second example is less familar, but only because we have not worked % with strings much. Here, the loop in index is the array of characters % s = ['Hello', 'Goodbye','See Ya!'] %% % % We can index into this array in exactly the same way as with numeric % data. For example, % s(1:10) %% % % We can also use the length function to get % the length of s. This length tells us the % number of times the statements in the for % will be executed. % length(s) %% % % So in this case as well, the for loop above % is behaving exactly as expected. Notice that we did not put a new line % character into the fprintf statement in the % the loop. If we had, our output for the odd 'string' loop would look like % the following. We have added a loop counter here as well to see that the % in fact, the number of times that the code inside the loop is called is % equal to the length of the loop index array. In this case, the variable % k acts like the loop index variable and is % incremented by 1 each time the code in the loop is executed. % % add a new line character to the fprintf statement. k = 1; for c = ['Hello', 'Goodbye','See Ya!'], fprintf('%2d %s\n',k,c); k = k + 1; end %% % % A final, entertaining example involves the use of a "cell" array. We % will not discuss cell arrays in detail now, but suppose you really wanted to % loop over a list of words. The following syntax will allow % you to do this: % for word = {'Howdy', 'Later', 'Cheerio'}, fprintf('%s\n',word{:}); end fprintf('The last word is ''%s''\n',word{:}); %% % % As this example illustrates, the value of the loop index after we % have completed the loop is the last value in the loop index array. % %% Rules for using the 'for' loop % % Here are some basic principles to follow when using the % for loop. %
%
Rules and guidelines for the for loop.
%
    %
  1. The for loop may enclose % multiple Matlab statements,
  2. %
  3. The number of times that enclosed statements get % executed is equal to the length of the loop index array,
  4. %
  5. The loop index variable may be used by the enclosed % statements,
  6. %
  7. One for loop may be % nested inside of one another loop. Statements in the inner for loops may use the loop index variable from % the outer loop. %
  8. Loop index arrays most commonly contain % integer values. Loop indices are often named i, % j, k, m or n. Avoid the use of % variable names x, y, % z or w for loop % indices.
  9. %
%
%

%

Example : Finding the maximum distance between pairs of % points


%

For a more complete example that requires the use of a for loop, we consider the problem of finding the % maximum distance between all possible pairs of points with coordinates % described by the arrays x and y. We will also include a loop in which we plot the % line between a point in the array and each of the other points. % % ------------------------------------------------------------------- % Compute the maximum distance between all possible pairs of points % in a set of points (xi,yi), i = 1,...n. % ------------------------------------------------------------------- n = 15; x = rand(1,n); % x-coordinates, chosen at random from [0,1] y = rand(1,n); % y-coordinates, chosen at random from [0,1] dmax = 0; for i = 1:n, % Compute the distance between the point (x(i),y(i)) and the % array of points (x,y). d = sqrt((x(i) - x).^2 + (y(i) - y).^2); % Find the maximum value in the array [d dmax] (see 'max' function % below). dmax = max([d dmax]); % Plot the line connecting the points (x(i),y(i)) each point in (x,y) for j = 1:n, plot([x(i) x(j)],[y(i),y(j)],'r-'); hold on; end end fprintf('The maximum distance is %12.4e\n',dmax); % Plot a symbol at each point and add a title and axes labels. plot(x,y,'bp','markersize',15,'linewidth',2); set(gca,'fontsize',16,'fontweight','bold') title('Point star'); xlabel('x-coordinate','fontsize',16); ylabel('y-coordinate','fontsize',16); axis([0 1 0 1]); % Fix aspect ratio so the plot is square. axis square; %% Simple array functions for one-dimensional arrays % % Matlab has several functions which operate on one-dimensional arrays. We % have already see one such function, the length function, which returns the number of % elements in the array. % x = 0:25:1000; length(x) %% % % Other such Matlab functions are convenient in that they allow us to avoid % the use of for loops in many cases. A % partial list of such functions include min, % max, sum, prod, cumsum, cumprod, mean, median, std and sort. Below, we use an example to illustrate how % each function works. % x = rand(1,1000); v = 1:10; %% % %

Example 1 : The minimum and maximum values of an array


% fprintf('The minimum value in array x is %16.8f\n',min(x)); fprintf('The maximum value in array x is %16.8f\n',max(x)); %% % %

Example 2 : The total sum and product of the entries


% fprintf('The sum of the values in v is %16.8e\n',sum(v)); fprintf('The product of the values in v is %16.8e\n',prod(v)); %% % %

Example 3 : The cumulative sums and products of the entries


% fprintf('The array of cumulative sums of the entries of v are\n'); cumsum(v) %% fprintf('The array of cumulative products of v are\n'); format short e; cumprod(v) %% % %

Example 4 : The mean, median and standard deviation


% fprintf('The mean of the values of x : %16.8f\n',mean(x)); fprintf('The median of the values of x is : %16.8f\n',median(x)); fprintf('The standard deviation of the values of x is : %16.8f\n',std(x)); %% % %

Example 5 : A sorted list of the numbers.


% y = sort(x); %% % % To see that we really have a sorted list, we will compare the first entry % of y to the minimum value of x, and the last entry of y to % the maximum value of x. % fprintf('The minimum value of x : %24.16f\n',min(x)); fprintf('The first entry of y : %24.16f\n',y(1)); %% fprintf('The maximum value of x : %24.16f\n',max(x)); fprintf('The last entry of y : %24.16f\n',y(end)); %% % % Most of the functions illustrated above can be used in many ways, not % just in the manner shown above. For example, the functions min and max can each % take two array arguments, allowing us to easily compare the size of % corresponding entries in two arrays. %

% In the following example, we compare the entries in an array x of random values to the value 0.5 and create a new % array in which the values in x larger than 0.5 % have been replaced by 0.5. % format short; x = rand(1,10) %% y = min(0.5,x) %% Lab exercises % %
% Associate each of the Matlab functions % min, % max, % sum, % prod, % cumsum, % cumprod, % mean, % median, % std % with exactly one mathematical expression below. Then, compute the value % of each mathematical expression by two different ways: %
    %
  1. Using a single Matlab function
  2. %
  3. Using a for loop %
% Compare your results to make sure you get the same result for each case. %

% Put all of your commands into a script containing your solution to each % problem. To start, first create an array $x$ of 100 random values using the rand function. For example, %
%
% x = rand(100,1);
% 
% You will use this array for each of the exercises. %

% Note: In most cases, it will be preferable to use the single % Matlab command than the for loop!. %
    %
  1. % $\displaystyle{\sum_{k=1}^{n} x_k}$
  2. %
  3. % $\displaystyle{\min_{1 \le k \le n} x_k}$
  4. %
  5. % $\displaystyle{\frac{1}{n}\sum_{k=1}^n {x_k}}$
  6. %
  7. % $\displaystyle{y_j = \prod_{k=1}^j {x_k}}$, % for $j = 1,2,...,n$
  8. %
  9. % $\displaystyle{\sqrt{\frac{1}{n-1}\sum_{k=1}^n (x_k - \mu)^2}}$, % where $\displaystyle{\mu = \frac{1}{n}\sum_{k=1}^n % x_k}$
  10. %
  11. $\displaystyle{n}$ %
  12. Find the value $\sigma$ % in $x$ such % that half of the values $x_k$ are % less than $\sigma$ and half are % greater than $\sigma$.
  13. %
  14. % $\displaystyle{\max_{1 \le k \le n} x_k}$
  15. %
  16. % $\displaystyle{y_j = \sum_{k=1}^j {x_k}}$ % for $j = 1,2,...,n$
  17. %
  18. % $\displaystyle{\prod_{k=1}^{n} x_k}$
  19. % %
% % Example : Problem 1 below can be computing using the 'sum' function % and a 'for' loop. n = 100; x = rand(1,n); s = 0; for i = 1:n, s = s + x(i); end fprintf('Result from ''sum'' function : %24.16f\n',sum(x)); fprintf('Result from the ''for'' loop : %24.16f\n',s); %% % %

Compare your answers with the solutions.

%