1. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/Math/Physics 25 Chp3 MATLAB Functions: Part2

2. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Learning Goals  Understand the difference Built-In and User-Defined MATLAB Functions  Write User Defined Functions  Describe Global and Local Variables  When to use SUBfunctions as opposed to NESTED-Functions  Import Data from an External Data-File As generated, for example, by an Electronic Data-Acquisition System

3. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Functions (ReDeaux)  MATLAB Has Two Types of Functions 1.Built-In Functions Provided by the Softeware e.g.; max, min, median 2.User-Defined Functions are.m-files that can accept InPut Arguments/Parameters and Return OutPut Values

4. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Array Function Operations  MATLAB will treat a variable as an array automatically.  For example, to compute the square roots of 7, 11 and 3-5j, type >> u = [7, 11, (3-5j)] >> sqrt(u) ans = 2.6458 3.3166 2.1013 - 1.1897i

5. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Expressing Fcn Arguments  We can write sin2 in text, but MATLAB requires parentheses surrounding the 2 The 2 is called the fcn argument or parameter  To evaluate sin2 in MATLAB, type sin(2) The MATLAB function name must be followed by a pair of parentheses that enclose the argument  To express in text the sine of the 2 nd element of the array x, we would type sin[x(2)]. However, in MATLAB you cannot use square brackets or braces in this way, and you must type sin(x(2))

6. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Expressing Fcn Arguments cont  Evaluate sin(x 2 + 5) → type sin(x.^2 + 5)  Evaluate sin(√x+1) → type sin(sqrt(x)+1)  Using a function as an argument of another function is called function composition. Check the order of precedence and the number and placement of parentheses when typing such expressions  Every left-facing parenthesis requires a right-facing mate. However, this condition does NOT guarantee that the expression is CORRECT!

7. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Expressing Fcn Arguments cont  Another common mistake involves expressions like cos 2 y, which means (cosy) 2.  In MATLAB write this expression as (cos(y))^2 or cos(y)^2  Do NOT write the Expression as cos^2(y) cos^2y cos(y^2) >> cos(pi/1.73) ans = -0.2427 >> (cos(pi/1.73))^2 ans = 0.0589

8. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Built-In Trig Functions  MATLAB trig fcns operate in radian mode Thus sin(5) returns the sine of 5 rads, NOT the sine of 5°. CommandConventional Math Function cos(x) Cosine; cos x cot(x) Cotangent; cot x csc(x) Cosecant; csc x sec(x) Secant; sec x sin(x) Sine; sin x tan(x) Tangent; tan x

9. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Built-In Inverse-Trig Functions  MATLAB trig -1 fcns also operate in radian mode i.e., They Return Angles in RADIANS CommandConventional Math Function acos(x) Inverse cosine; arccos x. acot(x) Inverse cotangent; arccot x. acsc(x) Inverse cosecant; arccsc x asec(x) Inverse secant; arcsec x asin(x) Inverse sine; arcsin x atan(x) Inverse tangent; arctan x atan2(y,x) Four-quadrant inverse tangent Two aTan’s?

10. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Why Two aTan’s?  Consider the Situation at Right  Due to aTan range −  /2  y   /2

11. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Why Two aTan’s? cont  Calculator Confusion from  MATLAB Solves This problem with atan2 which allows input of the CoOrds to ID the proper Quadant  MATLAB atan >> q1 = atan(-3/4) q1 = -0.6435 >> q2 = atan(3/-4) q2 = -0.6435  MATLAB atan2 >> q3 = atan2(-3,4) q3 = -0.6435 >> q4 = atan2(3,-4) q4 = 2.4981

12. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods atan2 built into Complex angle  As noted in the Discussion of Complex numbers converting between the RECTANGULAR & POLAR forms require that we note the Range-Limits of the atan function  MATLAB’s angle function uses atan2 >> z1 = 4 - 3j z1 = 4.0000 - 3.0000i >> z2 = -4 + 3j z2 = -4.0000 + 3.0000i >> r1 = abs(z1) r1 = 5 >> r2 = abs(z2) r2 = 5 >> theta1 = atan2(imag(z1),real(z1)) theta1 = -0.6435 >> theta2 = atan2(imag(z2),real(z2)) theta2 = 2.4981 >> z1a = r1 *exp(j *theta1) z1a = 4.0000 - 3.0000i >> z2b = r2 *exp(j *theta2) z2b = -4.0000 + 3.0000i

13. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods “Degree” Trig Functions  MATLAB now has Trig Functions that Operate on angles in DEGREES (º)  Example: Y = cosd(θ) Where is Measured in Degrees  cosd(θ)  sind(θ)  tand(θ)  cotd(θ)  cscd(θ)  secd(θ)  acosd(y)  asind(y)  atand(y)  acotd(y)  acscd(y)  asecd(y) NO atan2d(y) though (we’ll make one)

14. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Built-In Hyperbolic Functions CommandConventional Math Function cosh(x) Hyperbolic cosine. coth(x) Hyperbolic cotangent csch(x) Hyperbolic cosecant sech(x) Hyperbolic secant sinh(x) Hyperbolic sine tanh(x) Hyperbolic tangent

15. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Built-In Inverse-Hyp Functions CommandConventional Math Function acosh(x) Inverse Hyperbolic cosine. acoth(x) Inverse Hyperbolic cotangent acsch(x) Inverse Hyperbolic cosecant asech(x) Inverse Hyperbolic secant asinh(x) Inverse Hyperbolic sine atanh(x) Inverse Hyperbolic tangent >> asinh(37) ans = 4.3042 >> asinh(-37) ans = -4.3042 >> acosh(37) ans = 4.3039 >> acosh(-37) ans = 4.3039 + 3.1416i

16. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods User-Defined Functions (UDFs)  The first line in a function file must begin with a FUNCTION DEFINITION LINE that has a list of inputs & outputs. This line distinguishes a function m-file from a script m-file. The 1 st Line Syntax: function [output variables] = name(input variables) Note that the OUTPUT variables are enclosed in SQUARE BRACKETS, while the input variables must be enclosed with parentheses. The function name (here, name ) should be the same as the file name in which it is saved (with the.m extension).

17. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF Example  The Active m-File Lines function ave3 = avg3(a, b, c) TriTot = a+b+c; ave3 = TriTot/3; Note the use of a semicolon at the end of the lines. This prevents the values of TriTot and ave3 from being displayed

18. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF Example cont  “Call” this Function with its OutPut Agrument >> TriAve = avg3(7,13,-3) TriAve = 5.6667  To Determine TriAve the Function takes a = 7 b = 13 c = −3

19. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF Example cont  Call the avg3 function withOUT its output argument and try to access its internal value, avg3. You will see an error message. >> avg3(-23,19,29) ans = 8.3333 >> ave3 ??? Undefined function or variable 'ave3'.  The variable ave3 is LOCAL to the UDF It’s Value is NOT defined OutSide the UDF

20. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF Example cont  The variables a, b, & c are local to the function avg3, so unless you pass their values by naming them in the command window, their values will not be available in the workspace outside the function.  The variable TriTot is also local to the function. >> a=-37; b=73; c=19; >> s = avg3(a,b,c); >> a a = -37 >> b b = 73 >> c c = 19 >> TriTot ??? Undefined function or variable 'TriTot'.

21. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF – 2 nd Example  New Function  Write Function DecayCos with inputs of t & ω function ecos = DecayCos(t,w) % Calc the decay of a cosine % Bruce Mayer, PE ENGR25 28Jun05 % INPUTS: % t = time (sec) % w = angular frequency (rads/s) % Calculation section: ePart = exp(t/6); ecos =cos(w.*t)./ePart; w & t can be Arrays

22. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF – 2 nd Example cont  Pass Arrays to DecayCos >> p = DecayCos([1:3],[11:13]) p = 0.0037 0.3039 0.1617  Note that in MATLAB [11:13]  [11:1:13] = [ 11, 12, 13]

23. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 23 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF – Multiple Outputs  A function may have more than one output. These are enclosed in square brackets [ ] The OUPput is on the LEFT-hand side of the “=“ sign in the “function definition” line  For example, the function Res computes the Electrical Current, Ires, and Power- Dissipated, Pres, in an Electrical Resistor given its Resistance, R, and Voltage-Drop, v, as input arguments

24. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods UDF – Multiple Outputs cont  Calling the function Res for 4.7Ω and 28 V Res has 2-INputs & 2-OUTputs >> [I1 P1] = res(4.7,28) I1 = 5.9574 P1 = 166.8085

25. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Examples of Fcn Def Lines 1.One input, one output: function [area_square] = square(side) 2.Vector-Brackets are optional for one output: function area_square = square(side) 3.Three inputs, one output (Brackets Optional): function [volume_box] = box(height,width,length) 4.One input, Two outputs (Vector-Brackets Reqd): function [area_circle,circumf] = circle(radius) 5.Output can be a Plot as well as number(s) function sqplot(side)

26. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 26 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Free Fall  Consider an object with arbitrary mass, m, falling downward under the acceleration of gravity with negligible air-resistance (drag) With original velocity v 0  Take DOWN as POSITIVE direction

27. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Free Fall cont  Use Newtonian Mechanics to analyze the Ballistics of the falling Object  The Velocity and Distance-Traveled as a function of time & Original Velocity

28. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 28 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Free Fall/Drop MATLAB Fcn function [dist, vel] = drop(g, v0, t); % Calc Speed and distance-traveled by % dropped object subject to accel of gravity % Bruce Mayer, PE ENGR25 27Jun05 % % Parameter/Argument List % g = accel of gravity % v0 = velocity at drop point (zero-pt) % t = falling time % % Calculation section: vel = g*t + v0; dist = 0.5*g*t.^2 + v0*t;

29. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 29 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop 1.The variable names used in the function definition may, but need not, be used when the function is called: >> a = 32.17; % ft/s^2 >> v_zero = -17; % ft/s >> tf = 4.3; % s >> [dist_ft, spd_fps] = drop(a, v_zero, tf) dist_ft = 224.3117 spd_fps = 121.3310

30. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 30 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop cont 2.The input variables need not be assigned values outside the function prior to the function call: >> [ft_dist, fps_spd] = drop(32.17, -17, 4.3) ft_dist = 224.3117 fps_spd = 121.3310

31. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 31 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop cont 3.The inputs and outputs may be arrays:

32. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 32 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Local Variables  The names of the input variables given in the function definition line are local to that function i.e., the Function sets up a PRIVATE Workspace  This means that other variable names can be used when you call the function  All variables inside a function are erased after the function finishes executing, except when the same variable names appear in the output variable list used in the function call.

33. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 33 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Global Variables  The global command declares certain variables global, and therefore their values are available to the basic workspace and to other functions that declare these variables global  The syntax to declare the variables a, x, and q as GLOBAL is: global a x q  Any assignment to those variables, in any function or in the base workspace, is available to all the other functions declaring them global.

34. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 34 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Private WorkSpace Cmd Window WorkSpace Function-1 WorkSpace Command Window Function 1 INPUTS to Function-1 OUTPUTS from Function-1

35. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 35 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Make “atan2d” Function function Q = atan2d(Y,X); % Bruce Mayer, PE * 8Sep10 % ENGR25 * Fcn Lecture % % Modifies Built-in Function atan2 to return answer in DEGREES % Q = (180/pi)*atan2(Y,X); >> a1 = atan2d(3,-4) a1 = 143.1301 >> a2 = atan2d(-3,4) a2 = -36.8699

36. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 36 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today This workspace for rent

37. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 37 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/Math/Physics 25 Appendix

38. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 38 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop – 1 1.Make as current directory C:\Working_Files_Laptop_0505\BMayer\ Chabot_Engineering\ENGR-25_Comp- Meth\E25_Demo 2.Use FILE => NEW => M-FILE to open m-file editor 3.To Slide-28 and Copy the “drop” Text 4.Paste drop-Text into m-file editor and save-as drop.m

39. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 39 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop - 2 5.Test drop using SI units >> [y_m, s_mps] = drop(9.8, 2.3,.78) y_m = 4.7752 s_mps = 9.9440>>

40. BMayer@ChabotCollege.edu ENGR-25_Functions-2.ppt 40 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Calling Function drop - 3 6.The inputs and outputs may be arrays: >> [ft_fall, fps_vel] = drop(32.17,-17, [0:.5:2.5]) ft_fall = 0 -4.4787 -0.9150 10.6913 30.3400 58.0313 fps_vel = -17.0000 -0.9150 15.1700 31.2550 47.3400 63.4250  This function call produces the arrays ft_fall and fps_vel, each with six values corresponding to the six values of time in the array t.