Lecture 14: M/O/F/ for Engineering Applications - Part 2

1. Lecture 14: M/O/F/ for Engineering Applications - Part 2 BJ Furman 28NOV2011

2. The Plan for Today • Matlab/Octave/FreeMat (M/O/F) for engineering applications – part 2 • Recap M/O/F key concepts • Element-by-element operations (dot operator) • Function examples • 2D graphs • File IO • Resources for more information

3. Learning Objectives • Distinguish between matrix and array arithmetic, and use them appropriately • Explain the differences between script files and functions • Explain the basic elements of creating a 2D graph • Explain how to read from and write to files

4. Last Lecture • Overview of Matlab/Octave • Useful commands • The array as a fundamental element • creating arrays • indexing elements • vectors • colon operator • linspace • logspace • extracting sub-arrays using indexing and the colon operator • special matrices • ones(), zeros(), diag(), eye() • Introduction to plotting in M/O/F via script file • comparison to Ch and Excel

5. Arrays, Vectors, and Matrices • Array : • A collection of data elements arranged in rows and columns • Vector: • A one-dimensional array (single row or single column of data elements) • Matrix : • A two-dimensional array (more than one row and/or column of data elements) >> A=[1:9] row or column? column vector >> B=[1:9]’ >> C=[1:3; 4:6; 7:9] C = 1 2 3 4 5 6 7 8 9

6. Colon Operator vs. linspace() • What will the following do? • D=0 : 3 : 10 • F=linspace(0, 10, 11) • Observations about the two methods: • both methods produce vectors with equally spaced elements • colon operator method allows you to specify the first element and the interval spacing between elements, but notthe number of elements • If just start : end, then spacing is 1 • linspace method allows you to specify the first and last elements of the vector, but notthe spacing between elements base : increment : limit start : end : n

7. Review of length() and size() • What will the following do? • length(ones(1,3)) • length returns the number of elements of the vector • size(zeros(2,3)) • size returns the size of the dimensions of its argument

8. Review of Array Manipulation • Given G=[2 4 6; 8 10 12], what is: • G(2,3) • G(: , 2) • G(4) • G(1,1)=0 • G(1, :)=0 • Observations: • Array indexing begins with 1 (contrast with C) • : means “all of” the elements in that dimension • Extract elements by indexing • Extract sub-arrays using vectors as the indexing arguments

9. More Array Manipulation • Suppose H=1:9 (what will this produce?) • How could we form into a 3x3 matrix? • Reshape function • I=reshape(H,3,3) • How to transpose the rows to be the columns? • Reverse the order of the rows of I • J=I(3:-1:1,:) • Reverse the order of the columns of I (how?) • K=I(:, 3:-1:1) • Reverse the order of all the elements of I (how?) • Pseudocode: • Index I in reverse order • Reshape I = I’ L = reshape(I(9:-1:1),3,3)

10. Matrix and Array Arithmetic • Arithmetic operators: • + - * / \ ^ ’ • addition and subtraction are done element-by-element (same for matrix and array arithmetic) • Unless one is a scalar, the operands must be of the same size • scalar  (matrix or array) --> ? • (matrix or array)  (matrix or array) --> ? • (matrix or arrays) must be of the same size • For the other operators, need to distinguish between matrix and array operation • Matrix arithmetic operations • per rules of linear algebra • rows and columns must conform • For example, A x B: must have column and row agreement • Array arithmetic operations • element-by-element • Denote with dot operator: .* ./ .\ .^ .’ (array transpose) add. sub. mult. right div. left div. expon. algebr. transpose B A

11. Matrix and Array Arithmetic Examples • Scalar and matrix operands • If L=ones(1,5) and M=ones(1,4) • N = 2*L --> ? • N – 1 --> ? • Non-scalar operations • If O = [ 1:5 ] • O + M --> ? • L * O --> ? • L* O’ --> ? ??? Error using ==> + Matrix dimensions must agree. (1x5 * 1x5 does not work!) ??? Error using ==> * Inner matrix dimensions must agree. 15 Same as sum(L .* O) (1x5 * 5x1 works! Inner matrix dimensions agree. Results in a 1x1)

12. Array Operations • Element-by-element array operation • Ex: Given a set of distances and times, calculate average speeds and maximum of averages • How would you do this in C? • Pseudocode: • Calculate avg. speeds: speed[i] = distance[i] / time[i], for i=1 to 4 • Determine maximum speed • M/O/F (vectorize!): • distances=[120, 213, 87, 35] (in miles) • times=[ 2, 3.8, 0.9, 0.6] (in hours) • speeds=distances ./ times %( note: ‘dot /’  divide element-by-element) • max_speed=max(speeds) • To get the maximum speed and its index: • [max_speed, i] = max(speeds)

13. i2 i1 R2 +V i3 R3 R1 Circuit Analysis Equations • Matrix operations • Matrix division • Recall the circuit analysis • R1=10k • R2=R3=5k • V=10V • Matrix solution Think of it like inverting R and multiplying on the left side of V If we had iR = V instead, we’d use ‘right’ division to solve for i: ( i = R / V ) Think of it like inverting R and multiplying on the right side of V: i = VR-1

14. Circuit Analysis Solution • Circuit analysis solution: • Build R, build V, solve for i • Build R • all at once • R=[1 -1 1; 0 0 10e3; 0 10e3 0] or • build by rows and combine • eq1 = [ 1 -1 1] • eq2 = [0 0 10e3] • eq3 = [0 10e3 0] • R = [eq1; eq2; eq3] • Build V • V = [0 10 10]’ (note: transposed) • Solve I = R \ V • I = R \ V • R1=10k • R2=R3=5k • V=10V

15. Dot Product Example • Another example of element-by-element operations • dot product of two vectors

16. Dot Product Function Development • Define the problem • Create a function that will take two vectors as arguments and will return their vector dot product • Inputs • v1, v2 (three-element row vectors) • Outputs • z (the dot product) • Algorithm • Multiply v1 and v2 element-by-element • Sum the element-by-element products • Return the sum

17. Dot Product Function in M/O/F function [z] = dot_prod(v1, v2) % dot_prod(v1,v2) computes the vector dot product between vectors v1 and v2 % Function dot_prod(v1,v2) computes and returns the vector dot product between vectors v1 and v2 z = sum(v1.*v2); A = [ 1 2 3 ]; B = [ 4 5 6 ]; % what should A dot B result in? A_dot_B = dot_prod(A,B) Write the function Test it out

18. Review of Functions • Functions • Like script M-files, but several differences: • first line (function declaration) must be of the form:function [output args] = function_name(input args) • variables generated in the function are local to the function, whereas for script files, variables are global • must be named, ‘function_name.m’ (same as file name) • Make sure you add comments at the start that describe what the function does (see example code) • Example: root-mean-square function,rms.m

19. Root Mean-Square Function Development • Functions, cont. • Example: root-mean-square function, cont. • Pseudocode: • square each element of x • sum the squares • divide by N • take the square root • Square each element • xs = x .^2 • Sum the squares • sums = sum(xs) • Divide by N • N = length(x) • ms = sums/N • Take the square root • rms = sqrt(ms) • Before you write the function, make sure the name you propose is not already used! • Use: whichnameto check

20. Root Mean-Square Function Implementation • Functions, cont. • Example: root-mean-square function, cont. function [y] = rms(v) % RMS(v) root mean square of the elements of the column vector v % Function rms(v) returns the root mean square of the elements % of the column vector, v vs = v.^2; % what does this line do? Also note semicolon. s = length(v); y = sqrt(sum(vs)/s); H1 comment line (used in lookfor) Comments that will be displayed by help command Let v=sin([0: 0.01*pi: 2*pi]’), one period of a sine wave. The RMS valueof a sine wave is its amplitude*1/sqrt(2) Does rms() work with a row vector? How about a matrix?

21. More Robust Root Mean-Square Function • Functions, cont. • Make rms function more robust • to work with row or column vector or matrix with column vectors of data function [rmsout] = rms2(v) %RMS2(v) Root mean square of v % Function rms2(v) returns a row vector, where % each element is therms value of values in each% column of v vs = v.^2; s = size(v); rmsout = sqrt(sum(vs,1)/s(1));

22. File I/O with M/O/F • Data Input - simplest method • load command • Ex: load(‘data_file.txt’) • reads on a row-by-row basis • data values separated by spaces or commas and rows terminated by new line • columns must have the same number of elements • data is stored in workspace in an array with same name as the argument used in the load function • Ex. Portland International Airport monthly rainfall • load (‘PDXprecip.dat’) % must be in search path!

23. File I/O with M/O/F, cont. • Data Output - simplest method • Save command • Ex: save(‘data_file_name’) • Saves all the variables into a .mat file named ‘data_file_name’ • Many other commands are available for special purpose file I/O

24. File I/O and Plotting Example % read data into PDXprecip matrix load('PDXprecip.dat'); % copy first column of PDXprecip into month month = PDXprecip(:,1); % and second column into precip precip = PDXprecip(:,2); % plot precip vs. month with circles plot(month,precip,'o'); % add axis labels and plot title xlabel('month of the year'); ylabel('mean precipitation (inches)'); title('Mean monthly precipitation at Portland International Airport'); file_io_example.m Adapted from: http://web.cecs.pdx.edu/~gerry/MATLAB/plotting/loadingPlotData.htmlvisited 15NOV2009

25. More on Plotting General Format: plot (x, y, fmt, ...) • Add a red line through the data • Plot multiple sets of data on a single graph and add a legend • grid on • Sub-plots • Format: • subplot (m,n,p) • Figure window divided into m x n matrix of plotting areas • Procedure: • Pick the sub-plot window • Execute plot commands for that sub-plot % plot precip vs. month with circles plot(month,precip,'o',month,precip,'-r'); % copy first column of PDXtemperature into month month = PDXtemperature(:,1); % and second column into high_temp high_temp = PDXtemperature(:,2); % and third column into low temp low_temp = PDXtemperature(:,3); % and fourth column into avg temp avg = PDXtemperature(:,4); % generate the plot plot(month,high_temp,'ko',month,low_temp,'k+',month,avg,‘r-'); % add axis labels and plot title xlabel('Month'); ylabel('temperature (degrees F)'); title('Monthly average temperature for PDX'); % add a plot legend using labels read from the file legend('High','Low','Avg'); multi_plot.m

26. Y X r r ˆ = · v v i x Vector Dot Product Example Find the X and Y components of the vector, V Back

27. Review

28. References Matlab. (2009, November 6). In Wikipedia, the free encyclopedia. Retrieved November 6, 2009, from http://en.wikipedia.org/wiki/Matlab Matlab tutorials: http://www.mathworks.com/academia/student_center/tutorials/launchpad.html GNU Octave. (2009, October 31). In Wikipedia, the free encyclopedia. Retrieved November 6, 2009, from http://en.wikipedia.org/wiki/GNU_Octave Octave main page:http://www.gnu.org/software/octave/ (http://octave.sourceforge.net/ access to pre-built installers) Octave tutorials: http://homepages.nyu.edu/~kpl2/dsts6/octaveTutorial.html, http://smilodon.berkeley.edu/octavetut.pdf FreeMat. http://freemat.sourceforge.net/index.html ftp://www.chabotcollege.edu/faculty/bmayer/ChabotEngineeringCourses/ENGR-25.htm