CHAPTER 3 PART 1 MATLAB FUNDAMENTALS

Transcription

1 CHAPTER 3 PART 1 MATLAB FUNDAMENTALS

2 Chapter 3 : TOPIC COVERS (PROGRAMMING with MATLAB) MATLAB Environment Use of Built-In Functions and Graphics

3 LEARNING OUTCOMES INTRODUCTION It is expected that students will be able to: Gaining general understanding of MATLAB programming language Use built-in function to solve equation and plot graph

4 3.1 MATLAB Environment MATLAB is a computer program with convenient environment for performing many types of calculations especially for implement NM First part of Chapter 3 will introduce with calculator/interactive commands mode operations and creating plots. Second part of Chapter 3 will show how such commands can be used to create MATLAB programs. The best way to follow this chapter is by implement the commands on MATLAB/computer as we through the exercises/examples.

5 There are three primary windows in MATLAB programs: 1. Command window (to enter commands and data): 1 st part 2. Graphics window (to display plots and graphs): 1 st part 3. Edit window (to create and edit M-files): 2 nd part Starting MATLAB with command window prompt/quick being displayed as: >> Calculator mode of MATLAB operates in a sequential/in order fashion as you type in commands line by line and will give you results as punch enter. For an example: >> MATLAB will display the result: 20

6 We also can assign the answer (ans) as a variable: >> ans + 11 The result will be : >> = 31

7 3.2 ASSIGNMENT Assignment : to assigning values to variable names Scalars Assignment of values to scalar variables is similar to other computer languages: >> a = 4 Echo/repeat prints (produces) to confirm what you have done: a = 4 Echo/repeat printing is a characteristic of MATLAB. Echo printing can be terminated by the command line with the semicolon (;) character.

8 Then, try typing: >> a = 4, A = 6; x = 1; a = 4 MATLAB treats names in a case-sensitive manner : the small a is not the same as the capital A. Try typing : >> a and then enter; >> A They are difference (between a and A) names!!

9 MATLAB can assign complex values to variables. The unit imaginary number is pre-assigned to the variable i. So, a complex value can be assigned simply as : >> x = 2 + i*4 x = i MATLAB is also allows the symbol j to be used to represent the unit imaginary number for input, however, it always uses/shows an i for display. For example; >> x = 2 + j*4 x = i can assign complex values to variables.

10 MATLAB has several predefined variables, for example pi. >> pi See how MATLAB display four decimal places. However, if you desire additional precision, enter the following; >> format long Now when pi is entered the result is displayed to 15 significant figures: >> pi

11 To return to the four decimal version, type: >> format short >> pi Then enter;

12 The following table shows summary of the format commands you will employ routinely in engineering and scientific calculations. They all have the syntax: format type type Results Example short Scale fixed-point format with 5 digits long Scale fixed-point format with 15 digits for double and 7 digits for single short e Floating-point format with 5 digits e+000 long e Floating-point format with 15 digits for double and 7 digits for single e+000 short g Best of fixed- or floating-point with 5 digits long g Best of fixed- or floating-point with 15 digits for double and 7 digits for single short eng Engineering format with at least 5 digits and a power that is a multiple of e+000 long eng Engineering format with exactly 16 significant digits and a power that is a multiple of 3 bank Fixed dollars and cents e+000

13 3.2.2 Arrays/Arrangements, Vectors and Matrices Arrays: is a collections/selection of values that are represented by a single variable name One-dimensional arrays are called Vector and two-dimensional arrays are called Matrices. (The Scalar used in section are actually a matrix with one row and one column) Brackets are used to enter arrays in the command mode. For example, a row vector can be assigned as follows: >> a = [ ] >> a = Note that this assignment overrides the previous assignment of a = 4

14 However, in practice row vectors are rarely/seldom used to solve mathematical problems. If we speak about vectors, we usually refer to column vectors, which are more commonly used. A column vector can be entered in several ways. For examples: >> b = [ 2 ; 4 ; 6 ; 8 ; 10 ] or >> b = [ 2 ; 4 ; 6; 8; 10 ] or, by transposing a row vector with the operator (prime symbol); >> b = [ ]

15 A result in all three cases will be : >> b = Thus, an example of matrix (3x3) can be assigned as follows: >> A = [ ; ; ] A =

16 In addition, the Enter key (carriage return) can be used to separate the row. For example, in the following case, the Enter key would be struck after 3, the 6 and the ] to assign the matrix as: >> A = [ ] At any point in a session, a list of all current variables can be obtained by entering the who command: >> who Your variables are : A a ans b x Or, with more detail, enter the whos command: >> whos

17 >> whos Name Size Bytes Class A 3x3 72 double array a 1x5 40 double array ans 1x1 8 double array b 5x1 40 double array x 1x1 16 double array (complex) Grand total is 21 elements using 176 bytes Subscript notation can be used to access an individual element of any array. For example, the 4 th element of the column vector b can be displayed as: >> b (4) 8

18 For any array, A(m,n) selects the element in m th row and the n th column. For example, >> A (2, 3) 6 They are several built-in functions that can be used to create matrices. For example, the one and zeros functions create vectors or matrices filled with ones and zeros, respectively. Both have two arguments/influence, the first for the number of rows and the second for the number of columns. For example, to create a 2x3 matrix of zeros: >> E = zeros (2, 3) E =

19 Similarly, the ones function can be used to create a row vector of ones: >> u = ones (1, 3) u = 1 1 1

20 3.2.3 Colon ( : ) Operator Colon operator: is a powerful tool for crating and manipulating arrays (collections/arrangements of values). If a colon is used to separate two number, MATLAB generates the numbers between them using increment of one, for example: >> t = 1 : 5 t = If colons are used to separate three numbers, MATLAB generates the numbers between the first and third numbers using an increment equal to the second number: >> t = 1 : 0.5 : 3 t =

21 For negative increments: >> t = 10 : -1 : 5 t = Aside from creating series of numbers, the colon can also be used as a wildcards/randomly to select the individual rows and columns of a matrix. In here, the colon represents the entire row and column. For example, a second row of the matrix A can be selected as: >> A (2, : ) 4 5 6

22 The colon notation/entry is used to selectively extract a series of elements from within an array. For example, based on the previous definition of the vector t : >> t (2 : 4) In here, the second through the fourth elements are returned/came back.

23 3.2.4 The linspace and logspace Functions They are used to generate vectors of spaced points. The linespace function generates a row vector of equally spaced points. For example: >> linspace (x1, x2, n) which generates n points between x1 and x2. For example : >> linspace (0, 1, 6) Ans = If the n is omitted/excluded, the function automatically generates 100 points.

24 The logspace function generates a row vector that is logarithmically equally spaced. It has the form : >> logspace (x1, x2, n) which generates n logarithmically equal spaced points between decades 10 x1 and 10 x2. For example : >> logspace (-1, 2, 4) If the n is omitted/excluded, it automatically generates 50 points.

25 3.3 MATHEMATICAL OPERATIONS Operation with scalar quantities are handled in a straightforward manner, similar to other computer languages. The common operators, in order of priority, are : Symbol Explanation ^ Exponentiation - Negation * / Multiplication and division \ Left division Addition and subtraction

26 These operators will work in calculator fashion. For example : >> 2 * pi Also, scalar real variables can be included : For example : >> y = pi/4; >> y ^ This result of calculation can be assigned to a variable.

27 The priority order can be overridden/overrule with parentheses/addition comment. For example, because exponentiation (^) has higher priority then negation (-), the following result would be obtained : >> y = - 4 ^ 2 y = - 16 Thus, 4 is first squared and then negated. Parentheses can be used to overridden the priorities as : >> y = (- 4) ^ 2 y = 16

28 Calculations can also involve complex quantities. Here are some examples that values of x (2 + 4i) and y (16) defined previously >> 3 * x i >> 1 / x i >> x ^ i >> x + y i

29 The real power of MATLAB is in ability to carry out vector-matrix calculations. Most of the calculation will be discussed in Chapter 5 (Linear Algebraic Equations). However, some of vector-matrix calculation will be shown, for example: the inner product of two vectors (dot product) can be calculated using the * operator. >> a * b 110 and likewise, the outer product: >> b * a

30 To further illustrate vector-matrix multiplication, first redefine a and b: >> a = [1 2 3]; and >> b = [4 5 6] now, try; >> a * A or >> A * b

31 Matrices cannot be multiplied if the inner dimensions are unequal. Here what happens when the dimensions are unequal: >> A * a MATLAB automatically display the error message:?? Error using ==> mtimes Inner matrix dimensions must agree Matrix-matrix multiplication is carried out in likewise fashion: >> A * A

32 Mixed operations with scalars are also possible, for example: >> A / pi Example for calculations items by items in a matrix or vector: >> A ^ The result (above) after matrix multiplication of A with itself.

33 To make square each element of A, for example : Note: the. preceding the ^ operator signifies that the operation is to be carried out element by element. MATLAB manuals calls these as array operations or element-byelement operations >> A.^

34 MATLAB can obtain shortcut for viewing the calculation/operation that you ve already done. Just press the up-arrow key, you should get back the last line you type in. For example, previously: >> A.^ 2 Then, by pressing Enter will perform the calculation again. But you can also edit this line. For example, change it to the line below and then press Enter: >> A.^

35 3.4 BUILT-IN/FITTED FUNCTIONS (in MATLAB program) MATLAB and its Toolboxes have a rich collection of built-in functions. Use online help to find out more. For example, if you want to learn about the log function, type in : >> help log LOG Natural logarithm LOG (X) is the natural logarithm of the elements of x. Complex results are produced if X is not positive. See also LOG2, LOG10, EXP, LOGM. For a list of all the elementary functions, type : >> help elfun

36 One of important properties of MATLAB s built-in functions is that they will operate directly on vector and matrix quantities. For example : >> log (A) The result is that the natural logarithm function is applied in array/dispaly style, element by element, to the matrix A. Most functions, such as sqrt, abs, sin, acos, tanh, and exp, operate in array fashion.

37 For certain functions, such as exponential (exp) and square root (sqrt), have matrix definition as well. MATLAB will evaluate the matrix version when the letter m is appended to the function name. For example : >> sqrtm (A) i i i i i i i i i There are several functions for rounding. The round function rounds the elements of E to the nearest integer/digit. For example : >> E = [ ] >> round (E)

38 The ceil (short for ceiling) function round to the nearest integers toward infinity : >> ceil (E) The floor function rounds down to the nearest integers toward minus infinity : >> floor (E)

39 For the function with special actions on the elements of matrices and arrays. For example, the sum function returns the sum of the elements : >> F = [ ] >> sum (F) 19 Try others commands as well : >> min (F), max (F), mean (F), prod (F), sort (F) Prod. of : 1 x 3 x 4 x 5 x 6 = 360

40 A common use of functions is to evaluate a formula for a series of arguments. For example a free-falling bungee jumper can be compute as (equation 1.9) : v gm c d tanh gc m d t Where, v = velocity (m/s) g = acceleration due to gravity (9.81 m/s 2 ) m = mass (kg) t = time (s) c d = drag coefficient (kg/m)

41 Question: Create a column vector t that contains values from 0 to 20 in steps of 2? Answer: >> t = [ 0 : 2 : 20 ] t =

42 Check the number of items in the t array with the length function : >> length (t) Ans = 11 Then assign values to the parameters/limits : >> g = 9.81; m = 68.1; cd = 0.25; MATLAB allows you to evaluate such as v = f (t), where the formula is computed for each value of the t array, and the result is assigned to a corresponding position in the v array. For our case : >> v = sqrt (g * m/cd) * tanh (sqrt (g * cd/m) * t)

43 v =

44 3.5 GRAPHICS/GRAPHS/FIGURES MATLAB allows graphs to be created quickly and conveniently. For example, to create a graph of the t and v arrays from the data previously, enter : >> plot (t, v) The graph appears in the graphic window and can be printed or transferred via the clipboard to other program. (Note: refers to your text book page 33 for illustrates the graph) You can customize/modify the graph a bit with commands such as : >> title ( plot of v versus t ) >> xlabel ( Values of t ) >> ylabel ( Values of v ) >> grid (Note: refers to your text book page 33 for illustrates your customized graph)

45 Table 2.2: Specifiers for colors, symbols and line types Colors Symbols Line Types Blue b Point. Solid - Green g Circle ο Dotted : Red r X-mark x Dashdot -. Cyan c Plus + Dashed - - Magenta m Star * Yellow y Square s Black k Diamond d Triangle (down) ᴠ Triangle (up) ᴧ Triangle (left) < Triangle (right) > Pentagon p Hexagram h

46 The plot command displays a solid line by default. If you want to plot each point with a symbol, you can include a specifier enclosed (Table 2.2) in single quotes in the plot function. For example; to plot with open cycle enter: >> plot (t, v, o ) MATLAB allows you to display more than one data set on the same plot. For example, to connect each date marker with a straight line: >> plot (t, v, t, v, o )

47 Note: by default, previous plots are erased every time the plot command is implemented. The hold on command holds the current plot and all axis properties so that additional graphing commands can be added to the existing plot. The hold off command returns to the default mode. For example, if we had typed the following commands, the final plot would only display symbols: >> plot (t, v,) >> plot (t, v, o ) In contrast, the following commands would result in both lines and symbols being displayed: >> plot (t, v,) >> hold on >> plot (t, v, o ) >> hold off

48 In addition to hold, another handy function is subplot, which allows you to split the graph window into subwindows or panels. Try to enter: >> subplot (m, n, p) This command break the graph window into an m-by-n matrix of small axes, and selects the p-th axes for the current plot.

49 We can demonstrate : subplot by examining MATLAB s capability to generate three-dimensional plots. The simplest manifestation of this capability is the plot3 command which has the syntax/sentence structure : >> t = 0 : pi/50 : 10*pi; >> subplot (1, 2, 1) ; plot (sin (t), cos (t)) >> axis square >> title ( (a) ) Fig. 2.1a shows the result is a circle. The circle is distorted if eliminates the axis square command.

50 How about if we want to add the helix shape to the graph s right pane/window. To do this, employ a parametric of, x = sin (t), y = cos (t) and z = 1 >> subplot (1, 2, 1) ; plot3 (sin (t), cos (t), (t)) >> title ( (b) ) Fig. 2.1b shows the result is a helix shape. Can you visualize what s going on? As time evolves, the x and y coordinates sketch out the circumference of the circle in the x-y plane in the same fashion as the two-dimension plot. However, simultaneously, the curve rises vertically as the z coordinate increase linearly with time. Thus, the result is the characteristic spring or spiral staircase shape of the helix.

51 3.6 OTHER RESOURCES 1) Excellent books devoted to MATLAB : Palm, 2005; Hanselman and Littlefield, 2005; Moore 2007) 2) Help facility can be assessed by clicking on the Help menu in the command window. This will provide with a number of different options for exploring and searching through MATLAB s Help material. 3) help command can be assessed by typing help command followed by the name of a command function as described previously. 4) lookfor command to find all functions related. For example to find all function relate to logarithms : >> lookfor logarithm MATLAB will display all references that include the word logarithm.

52 5) For further information click : You will find links product information, newsgroups, books, and technical support.

53 3.7 Case study : Exploratory/Examining Data Analysis We will use MATLAB s graphical capabilities to discern/distinguish patterns in the data for estimating drag coefficient, c d of a free-falling human based (equation 2.1) mg cd (2.1) 2 vt where, v t = terminal velocity (m/s) g = gravitational acceleration (m/s2) m = mass (kg) c d = drag coefficient (kg/m) v t thus gm c d

54 Solution: Input the data from Table 2.1 and gravitational acceleration in MATLAB command window as: >> m = [ ]; >> vt = [ ]; >> g = 9.81; The drag coefficient, c d can be computed with equation (2.1). Because calculation performing element-by-element operation on vector, we have include periods prior to the operation of MATLAB: >> cd = g*m./vt.^2 cd =

55 Then, use some of MATLAB s built-in function to generate some statistics results: >> cdavg = mean (cd), cdmin = min (cd), cdmax = max(cd) adavg = cdmin = cdmax = The result above shows that the average value is for a range from to kg/m Now, try using equation (2.1) to make prediction of the terminal velocity based on the average value of c d : >> vpred = sqrt (g*m/cdavg) Vpred =

56 Notice that no periods prior haven t used to the operation in this formula? Why? Try to plot these values versus the actual measured terminal velocities. Put command of superimpose a line indicating exact predictions (the 1:1 line) to help assess the results. As we need to generate a second plot, input the command subplot : >> subplot (2, 1, 1) ; plot (vt, vpred, o, vt, vt) >> xlabel ( measured ) >> ylabel ( predicted ) >> title ( Plot of predicted versus measured terminal velocities )

57 As in the top plot of Fig. 2.2, because the predictions generally follow the 1:1 line, you might initially conclude that the average drag coefficient yields decent results. However, notice how the mode tends to unpredict the low velocities and overpredict the high. This suggests that rather than being constant, there might be a trend in the drag coefficients. This can be seen by plotting the estimated drag coefficient versus mass as : >> subplot (2, 1, 2) ; plot (m, cd, o ) >> xlabel ( mass (kg) ) >> ylabel ( estimated drag coefficient (kg/m) ) >> title ( Plot of drag coefficient versus mass ) The resulting plot (bottom graph of Fig. 2.2), suggests that rather than being constant, the drag coefficient seems to be increasing as the mass of the jumper increases.

58 Based on this result, it might motivate you to conduct further experiments with a larger number of jumpers to confirm your preliminary/initial findings. Further investigates would discover that the parameter c d is actually a lumped/knob/collide drag coefficient that along with the true drag includes other factors such as the jumper s frontal area and air density, which discovers another formula for c d.

59 END OF CHAPTER 3 PART 1