Prof. Manoochehr Shirzaei. RaTlab.asu.edu

Transcription

1 RaTlab.asu.edu

2 Introduction To MATLAB

3 Introduction To MATLAB This lecture is an introduction of the basic MATLAB commands. We learn; Functions Procedures for naming and saving the user generated files Comment lines Access to MATLAB s Editor/Debugger Finding the roots of a polynomial, Making plots Several examples are provided with detailed explanations.

4 Introduction To MATLAB

5 To use the Editor/Debugger From the File menu on the toolbar, choose New and click on script(m File). This takes you to the Editor Window where you can type your script (list of statements) for a new file. You must save your program with a file name which starts with a letter. Important! MATLAB is case sensitive. Files are saved with the file name and the extension.m; for example, myfile01.m. To execute a program, type the file name without the.m extension at the prompt in command window and press <enter>

6 Note If you have saved your file in drive a or any other drive, you must make sure that it is added it to the desired directory in MATLAB s search path. To get help with MATLAB topics, in command window you can type help followed by any topic. When you are done and want to leave MATLAB, type quit or exit. If you want to clear all previous values, variables, and equations without exiting, use the clear command. The clc command clears the screen but MATLAB still remembers all values. All text after the % (percent) symbol is interpreted by MATLAB as a comment line and thus it is ignored during the execution of a program. Example >> conv(p,q) % performs multiplication of polynomials p and q >> % The next statement performs partial fraction expansion of p(x) / q(x)

7 Roots of Polynomials Example 1 A root of the polynomial p is a solution of the equation p = 0 In MATLAB, a polynomial is expressed as a row vector of the form The elements of this vector are the coefficients of the polynomial in descending order. You must include terms whose coefficients are zero. You can find the roots of any polynomial with the roots(p) function where p is a row vector containing the polynomial coefficients in descending order.

8 Example 2: Find the roots of the polynomial Solution: The roots are found with the following two statements. We have denoted the polynomial as p1, and the roots as roots_ p1. Note that MATLAB displays the polynomial coefficients as a row vector, and the roots as a column vector.

9 Example 3: Find the roots of the polynomial Solution: There is no cube term; therefore, we must enter zero as its coefficient. The roots are found with the statements below where we have defined the polynomial as p2, and the roots of this polynomial as roots_ p2. Note: The result indicates that this polynomial has three real roots, and two complex roots.

10 Evaluation of a Polynomial at Specified Values The polyval(p,x) function evaluates a polynomial at some specified value of the independent variable. Example 4: Evaluate the polynomial Solution:

11 Using MATLAB to Make Plots Quite often, we want to plot a set of ordered pairs. This is done with the MATLAB plot(x,y) command which plots Y versus X. X is the horizontal axis (abscissa) and Y is the vertical axis (ordinate). Example 5: Given W and Z, plot Z versus W. >> plot(w, z)

12 How to make a plot more presentable? You can make a plot, more presentable with the following commands: grid on: This command adds grid lines to the plot grid off: command removes the grid Note: The command grid toggles them, that is, changes from off to on or vice versa. The default is off. box off: This command removes the box (the solid lines which enclose the plot) title( string ): This command adds a line of the text string (label) at the top of the plot xlabel( string ) and ylabel( string ) are used to label the x and y axis respectively

13 Other Plotting Functions semilogx(x,y) command is similar to the plot(x,y) command, except that the x axis is represented as a log scale, and the y axis as a linear scale semilogy(x,y) command is similar to the plot(x,y) command, except that the y axis is represented as a log scale, and the x axis as a linear scale. loglog(x,y) command uses logarithmic scales for both axes in MATLAB log(x) is the natural logarithm log10(x) is the common logarithm log2(x) is logarithm to the base 2

14 Example 6: Redraw the plot with the above options

15 gtext vs text The command gtext( string ) switches to the current Figure Window, and displays a cross hair which can be moved around with the mouse. For instance, we can use the command gtext( I love crustal deformation ), and this will place a cross hair in the Figure window. Then, using the mouse, we can move the cross hair to the position where we want our label to begin, and we press <enter>. The command text(x,y, string ) is similar to gtext( string ). It places a label on a plot in some specific location specified by x and y, and string is the label which we want to place at that location. I will illustrate its use with the following example which plots a 3 phase sinusoidal waveform

16 Example 7: The first line of the script below has the form linspace(first_value, last_value, number_of_values) This command specifies the number of data points but not the increments between data points. An alternate command uses the colon notation and has the format x=first: increment: last This format specifies the increments between points but not the number of data points. The script for the 3 phase plot is as follows:

17 Styles, colors, and markets used in MATLAB In our previous examples, we did not specify line styles, markers, and colors for plots. However, MATLAB allows to specify various line types, plot symbols, and colors. These, or a combination of these, can be added with the plot(x,y,s) command, where s is a character string containing one or more characters shown on the three columns of Table below. Command plot(x,y,'m*:') plots a magenta dotted line with a star at each data point Command plot(x,y,'rs') plots a red square at each data point, but does not draw any line because no line was selected. If we want to connect the data points with a solid line, you must type plot(x,y,'rs ')

18 3D plots The plots which we have discussed thus far are two dimensional. MATLAB has also a three dimensional (three axes) capability. The command plot3(x,y,z) plots a line in 3 space through the points whose coordinates are the elements of X, Y, and Z, where X, Y, and Z are three vectors of the same length. The general format is plot3(x1,y1,z1,s1,x2,y2,z2,s2,x3,y3,z3,s3,...) where xn, yn, and zn are vectors or matrices, and sn are strings specifying color, marker symbol, or line style. These strings are the same as those of the two dimensional plots.

19 Example 8: Plot the function Solution: We arbitrarily choose the interval (length) shown with the script below.

20 Note The command plot3(x,y,z,'bd ') will display the plot in blue diamonds, connected with a solid line. In a three dimensional plot, we can use the zlabel( string ) command in addition to the xlabel( string ) and ylabel( string ). In a two dimensional plot, we can set the limits of the x and y axes with the axis([xmin xmax ymin ymax]) command. Likewise, in a three dimensional plot we can set the limits of all three axes with the axis([xmin xmax ymin ymax zmin zmax]) command. It must be placed after the plot(x,y) or plot3(x,y,z) commands, or on the same line without first executing the plot command. This must be done for each plot. The three dimensional text(x,y,z, string ) command will place string beginning at the co ordinate (x, y, z) on the plot. For three dimensional plots, grid on and box off are the default states.

21 Mesh mesh(x, y, z) command displays a three dimensional plot. contour(x, y, z, n) draws contour lines for n levels. Length(x) = n Length(y) = m [m, n] = size(z) To produce a mesh plot of a function of two variables, say z = f(x, y), we must first generate the x and y matrices which consist of repeated rows and columns over the range of the variables x and y. We can generate the matrices X and Y with the [X,Y]=meshgrid(x,y) function which creates the matrix X whose rows are copies of the vector x, and the matrix Y whose columns are copies of the vector y.

22 Example 9: The volume V of a right circular cone of radius r and height h is given by Plot the volume of the cone as r and h vary on the intervals 0 r 4 and 0 h 6 meters. Solution: The volume of the cone is a function of both the radius and the height, that is, V = f(r, h) The three dimensional plot is created with the following MATLAB script where, in the second line we have used the dot multiplication, division, and exponentiation.

23 Continued The three dimensional plot shows how the volume of the cone increases as the radius and height are increased

24 Multiplication, Division and Exponentiation MATLAB recognizes two types of multiplication, division, and exponentiation. These are; The matrix multiplication, division, and exponentiation, The element by element multiplication, division, and exponentiation. Note: the arrays [a b c ] such a those that contained the coefficients of polynomials, consisted of one row and multiple columns, and thus are called row vectors. If an array has one column and multiple rows, it is called a column vector. Note: the elements of a row vector are separated by spaces. Note: To distinguish between row and column vectors, the elements of a column vector must be separated by semicolons. Note: An easier way to construct a column vector, is to write it first as a row vector, and then transpose it into a column vector. MATLAB uses the single quotation character ( ) to transpose a vector.

25 Matrix Multiplication Given A is defined as a row vector whereas B is defined as a column vector. Multiplication of the row vector A by the column vector B, is performed with the matrix multiplication operator (*). Example this is verified with the MATLAB script

26 Note Suppose that both A and B are row vectors, and we attempt to perform a row by row multiplication with the following MATLAB statements. When these statements are executed, MATLAB displays the following message: Here, because we have used the matrix multiplication operator (*) in A*B, MATLAB expects vector B to be a column vector, not a row vector. It recognizes that B is a row vector, and warns us that we cannot perform this multiplication using the matrix multiplication operator (*). Accordingly, we must perform this type of multiplication with a different operator. This operator is defined below.

27 Element by Element Multiplication Given Multiplication of the row vector C by the row vector D is performed with the dot multiplication operator (.*). There is no space between the dot and the multiplication symbol. Thus, Example this is verified with the MATLAB script

28 Note division (/) and exponentiation (^) operators, are used for matrix division and exponentiation, whereas dot division (./) and dot exponentiation (.^) are used for element by element division and exponentiation remember that no space is allowed between the dot (.) and the multiplication (*), division ( /), and exponentiation (^) operators. Note: A dot (.) is never required with the plus (+) and minus ( ) operators.

29 Example: Write the MATLAB script that produces a simple plot for the waveform defined as in the 0 t 5 seconds interval. Solution:

30 Display Formats MATLAB displays the results on the screen in integer format without decimals if the result is an integer number, short floating point format with four decimals if it a fractional number The output format can changed with the format command FORMAT Default. Same as SHORT. FORMAT SHORT Scaled fixed point format with 5 digits. FORMAT LONG Scaled fixed point format with 15 digits. FORMAT SHORT E Floating point format with 5 digits. FORMAT LONG E Floating point format with 15 digits. FORMAT SHORT G Best of fixed or floating point format with 5 digits. FORMAT LONG G Best of fixed or floating point format with 15 digits.

31 Continued FORMAT HEX Hexadecimal format. FORMAT BANK Fixed format for dollars and cents. FORMAT RAT Approximation by ratio of small integers. Spacing: FORMAT COMPACT Suppress extra line-feeds. FORMAT LOOSE Puts the extra line-feeds back in.

32 Example format short Four decimal digits (default) format long digits format short e e+01 Four decimal digits plus exponent format short g Better of format short or format short e format bank two decimal digits format + only + or or zero are printed format rat 100/3 rational approximation

33 References Matlab Online Documentation - MathWorks