A 30 Minute Introduction to Octave ENGR Engineering Mathematics Tony Richardson

Transcription

1 A 30 Minute Introduction to Octave ENGR Engineering Mathematics Tony Richardson Introduction This is a brief introduction to Octave. It covers several topics related to both the statistics and linear algebra portions of the course. It is recommended that you try the examples as you read through this introduction. Basic Operations Define a scalar (a single number) and a matrix: > a = 2; > [ 1 2; 3 4] # Spaces between cols, semicolons between rows a is a scalar while A is a 2x2 matrix whose first row contains the elements 1 and 2. Matrices are entered in row order with spaces (or commas) used to separate columns and semicolons (or a carriage return) between rows. By default the result of the definition is displayed unless the line of input ends with a semicolon (as when a is defined above). Comments begin with a hash symbol (#). We can access a particular element by specifying indices inside parenthesis (row #, column #): > A(1,1) = 6 # Change a single element in the matrix > B(3,3) = 5 # Define a new matrix by defining a single element B = The functions zeros(m,n), ones(m,n), eye(m,n) and rand(m,n) can be used to define m x n matrices of all zeros, all ones, the identity matrix, and a matrix of random numbers. The special variables i, pi, and e represent 1, , and respectively: > [2+2i cos(2*pi); log(e) e^(i*pi)] i i i i The ':' (colon) operator can be used to form a vector whose elements increase (or decrease) incrementally: > x = [1:6] x = Anthony M. Richardson 1 of 10 8/20/2009

2 > y = [9:-2:1] y = You can also use vectors as indices to select a submatrix: > rand(4,4) # Define a new random matrix > A ([1 2],1) # Select first two elements from the first column > A([3:rows(A)],[2:columns(A)]) # Cols 2-4 of rows A(:) would be a column vector with all the elements of A stacked in a single column. The reshape function can be used to reshape a vector or matrix: > reshape([1:10],2,5) > reshape([1:10],5,2)' # Note the transpose (') operation! Matrix Operations Octave's native data type is the matrix, so the +, -, and * operators perform matrix addition, subtraction, and multiplication. > [ # Define a matrix separating rows by newlines > > > ] > B = A' # B is the transpose of A (rows and columns swapped) B = Anthony M. Richardson 2 of 10 8/20/2009

3 > A*B # Multiply matrix A times B > A + A # Add A to itself > A.* A # Element-by-element multiplication Notice that element-by-element multiplication uses.* instead of *. The slash and backslash operators correspond to right and left multiplication by a matrix inverse. For example A\b is equivalent to inv(a)*b and B/A is equivalent to B*inv(A). The slash and backslash operators require fewer calculations and are less subject to numerical error than calculation and multiplication by the inverse. Getting Help Octave has an extensive on line help system. Typing doc brings up the interactive help system. You can get help on any command by typing either doc command or help command. Note: The doc command does not work within QtOctave, the same interactive help is available from the Help menu or by just pressing the F1 function key on your keyboard. Programming Octave provides a full programming environment with loops, selection, and functions. Listing 1 shows an example Octave function that calculates e x from its power series expansion. (Note that the exp(x) function is built into Octave, this is just an example to illustrate Octave programming.) The code can be typed in at the Octave prompt, but typically it would be saved in a file using a text editor (Notepad for example). Save the code in Listing 1 in a file named myexp.m. The function can be loaded and run using the following command: > myexp(1) # Compute exponential using power series definition # e(x) = 1 + x/1! + (x*x)/2! + (x*x*x)/3! + (x*x*x*x)/4! +... function [y] = myexp(x) y = 0; n = 1; term = 1; # Loop until term is numerically equal to 0 while term!= 0 y = y + term; term = term*(x/n); n = n + 1; end endfunction Listing 1 Anthony M. Richardson 3 of 10 8/20/2009

4 The Octave m-file should be in the current directory or in directory contained in the LOADPATH variable. A while loop is shown in Listing 1. The!= is one of the Octave comparison operators, it represents not equal to. Other comparison operators are: ==, <, >, <=, >= and represent equal to, less than, greater than, less than or equal to and greater than or equal to respectively. Octave functions should normally be written to accept matrix arguments. The function in Listing 1 does not. Listing 2 shows a matrix version of the exponential function. The matrix version of this function is very similar to the scalar version in Listing 1. y and term are matrices instead of scalars now. Instead of a while loop a do-until loop is now used instead. The body of the loop will be executed until the condition (term == 0) becomes true. A matrix is not considered true until all matrix elements are true. Notice that elementby-element multiplication is used to update the elements in the term matrix. We can compare the built-in exp() function and our own mymatexp() expression as follows: > exp([0 1; 2 3]) > mymatexp([0 1; 2 3]) # Compute exponential using power series definition (matrix version) function [y] = mymatexp(x) # allocate space for y y = zeros(size(x)); n = 1; term = ones(size(x)); # loop until all elements in the term matrix are 0 do y = y + term; # use element-by-element multiplication to update the terms term = term.*(x/n); n = n + 1; until (term == 0) endfunction Listing 2 Listing 3 shows Octave code that can be used to compute the sum of two matrices. (Again this operation is built into Octave and doesn't need to be defined.) This code listing demonstrates the if selection statement and for loops. Plotting It is quite easy to create plots in Octave with the plot function. Arguments to plot must be vectors (or matrices). The following commands plot 10 cycles of the sine function, by first creating an angle vector (theta) that contains 1024 values between 0 and 20π. > theta = [0:1023]/1024*10*2*pi; > plot(sin(theta)); This produces the plot shown in Figure 1. Note that the x-axis is labeled with the index of the angle vector. We can plot the sine function versus the angle and add axis labels and a title with the following commands: Anthony M. Richardson 4 of 10 8/20/2009

5 > plot(theta, sin(theta)); > xlabel('angle (radians)'); > ylabel('sine(x)'); > title('example Plot'); The result is shown in Figure 2. The plot command has many options, refer to the on-line documentation for details. # Matrix addition function [c] = mymatadd(a,b) [nr,nc] = size(a); if sum(!([nr nc] == size(b)))!= 0 error("matrices are not compatible for addition"); end c = zeros(nr,nc); for i=1:nr for j=1:nc c(i,j) = a(i,j) + b(i,j); end end endfunction Listing 3 Probability and Statistics Figure 1 There is a freely-available statistics package (toolkit) available for Octave. This sections uses some commands from that package. There are commands to generate random numbers from all distributions that we study in this course. Anthony M. Richardson 5 of 10 8/20/2009

6 Figure 2 See the Statistics chapter from the on-line help for details. Here are some examples: > unidrnd(5, 1, 10) # 1x10 matrix of discrete values between 1 and > unifrnd(0, 1, 1, 5) # 1x5 matrix of continuous values between 0 and > binornd(8,.35,1,10) # 1x10 matrix of binomial values with n=8 and p= The mean, std, and var functions compute the mean (average), standard deviation and variance of a data set: > x = normrnd(10, 5, 1, 100); # 100 normally dist variable with mean=10,std=5 > mean(x) # Theoretical value is 10 > std(x) # Theoretical value is 5 > var(x) # Theoretical value is 25 There are also some special plotting functions related to statistics available. Here hist is used to plot a relative frequency histogram of the normally distributed vector x created above: > hist(x,-5:5:25,1); Anthony M. Richardson 6 of 10 8/20/2009

7 This command produces the graph shown in Figure 3. For comparison a histogram of a vector with elements uniformly distributed between -5 and 25 is shown in Figure 4. Figure 4 was generated with the following commands: > y = unifrnd(-5,25,1,100); > hist(y,-5:5:25,1) Figure 3 The boxplot function can be used to create boxplots. Here are the commands to reproduce the boxplot shown in Figure 1.20 of your text: > d = readdataset('ch01/exp01-18.txt'); > boxplot(d,0,['+' 'o'],0); > axis([ ]); The results are shown in Figure 5. Probability density functions (PDF) and cumulative distribution functions (CDF) are available for all of the probability distribution functions that we study are available. This means that all of the data contained in the tables in Appendix A of the text is available. For example, assuming X has a binomial distribution with parameters n = 10 and p = 0.2, the probability X 3 is: > binocdf(3, 10, 0.2) This value is denoted B(3;10,0.2) in the text and can be found in Table A.1 in Appendix A. Anthony M. Richardson 7 of 10 8/20/2009

8 Figure 4 Figure 5 Anthony M. Richardson 8 of 10 8/20/2009

9 Linear Algebra Octave provides a number of different methods for solving systems of linear equations. We can represent the following system: x 1 2x 2 x 3 =1 2 x 1 x 2 x 3 =3 x 1 2x 2 3 x 3 =7 as the matrix equation A x = b. We define matrices A and b in Octave as follows: > [1 2-1; 2-1 1; ] > b = [1 3 7]' b = Here the transpose operator (') was used to conveniently transpose the [1 3 7] row vector into a column vector. (I find this to be more convenient than typing semicolons at the end of each row of a column vector.) Octave provides a variety of methods for solving the matrix equation. We can use the rref() function to transform the augmented [A b] matrix to reduced row echelon form: > rref([a b]) The solution vector x is the last column of the result. > x = ans(:,columns(ans)) x = We can verify that this is indeed the solution by comparing Ax and b. > A*x We can compute x directly using A\b (best method) or inv(a)*b: Anthony M. Richardson 9 of 10 8/20/2009

10 > A\b > inv(a)*b The inverse of A can be computed by transforming [A I] to reduced row echelon form, by evaluating A\I, or by using the inv() function: > rref([a eye(size(a))]) > A\eye(size(A)) eye(size(a)) is an identity matrix with the same dimensions as A. eye(3,3) can also be used to directly obtain a 3x3 identity matrix. > inv(a) Octave provides det() and trace() functions for finding the determinant and trace of a matrix. Check the Linear Algebra section of the on line help for information and for other related routines. Saving a Transcript of Your Octave Session The command diary filename will save a copy of your current Octave session in file filename. This is convenient for creating an electronic copy of your homework. To turn off the diary use diary off. Differences Between MATLAB and Octave The exercises in the Linear Algebra text are for MATLAB. They can be performed under Octave with no change. Octave is designed to be compatible with MATLAB. Anthony M. Richardson 10 of 10 8/20/2009