1. Introduction to MATLAB for Biomedical Engineering BME 1008 Introduction to Biomedical Engineering FIU, Spring 2015 Lesson 2: Element-wise vs. matrix operations save(), load(), matlab script plot(x,y)

2. Creating Vectors (1D arrays) Explicit list of elements >> x1 = [-2 -1 0 1 2 3 4 5] x = -2 -1 0 1 2 3 4 5 Using colon (:) notation >> x2 = -2 : 5 x2 = -2 -1 0 1 2 3 4 5

3. General colon (:) syntax: initial value : increment : final value >> x3 = 1:1.5:6 x3 = 1.0 2.5 4.0 5.5 >> x4 = 1:6 %default increment 1 x4 = 1.0 2.0 3.0 4.0 5.0 6.0

4. In matrix algebra In MATLAB A*B means matrix product A.* B means element-by-element multiplication '. ' (dot) - indicates element-by-element operation Matrix product C = AB

5. Element-wise (array) operators: +, -,.*,./,.^ '+ ' - addition always element wise >> x1 = [1 2 3]; x2 = [4 5 6]; >> x1 + x2 ans = 5 7 9 >> x1 + 10 ans = 11 12 13

6. >> x1 + [10 20] ??? Error using ==> plus Matrix dimensions must agree. '- ' - subtraction always element wise >> x1 = [1 2 3]; x2 = [4 5 6]; >> x1 - x2 ans = -3 -3 -3 >> 3 - x1 ans = 2 1 0

7. '.* ' - element-wise multiplication >> x1 = [1 2 3]; x2 = [4 5 6]; >> x1.* x2 ans = 4 10 18 >> x1 * 10 ans = 11 12 13 >> x1*x2 ??? Error using ==> mtimes Inner matrix dimensions must agree.

8. './ ' - element-wise multiplication >> x1 = [1 2 3]; x2 = [4 5 6]; >> x1./ x2 ans = 0.25 0.4 0.5 >> 1./x1 ans = 1.0 0.5 0.33 >> 1/x1 ??? Error using ==> mrdivide Matrix dimensions must agree.

9. '.^ ' - element-wise power >> A = [5 1 -2]; >> A.^ 3 ans = 125 1 -8 >> A ^ 3 ??? Error using ==> mpower Matrix must be square.

10. Matrix operators *, ^ >> x1 = [1 2 3]; %row vector x1 = 123 >> x2 = [4 5 6]'; %column vector x2 = 4 5 6 >> x1*x2 %matrix multiplication ans = 32

11. >> x1 = [1 2; 3 4] % 2-by-2 matrix x1 = 12 34 >> x1^2 % or x1*x1 ans = 7 10 15 22 %matrix product

12. Quiz 1 1) What is the average of: 1 3, 1.5 3, 2 3, 2.5 3..., 11.5 3, 12 3 ? Hint: First create vector, then cube. 2) What is the maximum element of C, where Hint: Use max() two times.

13. Answers to Quiz 1 1) >> x = [1:0.5:12]; y = x.^3; >> average = mean(y) average = 489.1250

14. 2) >> A=[30 5.5; 2 -7]; B=[-3 6; 8 12]; >> C = A*B C = -46 246 -62 -72 >> max(C) %returns max. element from each column ans = -46 246 >> max(max(C)) %returns max. element from C ans = 246

15. Matlab Scalar functions sin(), cos(), exp(), log(), log10() operate element-wise >> t = [0 : pi/2: 2*pi ] % pi = 3.14 Matlab constant t = 0 1.573.144.716.28 >> y = sin(t) y = 0 1 0 -1 0 >> e = exp([0 1 -1]) %exponential function e = 1.00 2.71 0.36

16. SAVE & LOAD Commands save('filename') saves all workspace variables to disk in current directory in the file 'filename.mat' '.mat' - default extension for Matlab binary file (MAT-file) save('filename','var1','var2') saves only variables 'var1' and 'var2' load('filename') loads workspace variables from disk from the file 'filename.mat'

17. >> clear %removes variables from workspace >> x = [0: 0.1 : 1000]; >> y = exp(-x/500); % x, y vectors with 10000 elements >> save('myResults') >> save('myResults_y','y') >> clear >> whos >> load('myResults') >> whos

18. Manual Loading & Saving

19. Importing data from other file formats (.txt,.xls,.jpg,...)

20. Matlab script text file with a series of Matlab commands Matlab program with no input and output arguments has extension '.m' (M-file) (1) New or Open (2) Edit in Matlab editor (4) Run(3) Save

21. plot(x,y) command % Moore's Law -transistor count double every two years transistors= [50 100 200 400 800 1600... 3200 6400 12800]; years=[2000:1.5:2012]; plot(years, transistors)

22. plot(years, transistors,'r*--') %'r*--' red, star, dashed line ylabel('transistor count (million)') % y-axis label xlabel('year') % x-axis label title('Moore''s law')

23. Figure editing (1) Edit plot (3) Change properties (2) Select object (line, axis, label)

24. Saving figures '.fig' file, Matlab format, figure can be edited as image file, e.g. '.bmp'

25. Copy & Paste figure to Word or PPT

26. Quiz 2 Given y = sin(x) × e -x/15 and x = [0:0.2:10*pi] hint: exp() max() plot y vs. x find maximum y

27. Answer to Quiz 2 >> x = [0:0.2:10*pi]; >> y = sin(x).*exp(-x/15); >> figure %opens new figure window >> plot(x,y) >> y_max = max(y) y_max = 0.8984