8 - Chapman MATLAB Programming for Engineers

A column vector, having one column and multiple rows, can be created by specifying it element by element, separating element values with semicolons:

>> c = [1;2;3;4;5] c =

1

2

3

4

5

Thus, separating elements by spaces or commas specifies elements in di erent columns, whereas separating elements by semicolons specifies elements in di erent rows.

The transpose operator (’) is used to transpose a row vector into a column vector

>> b = a’ b =

1

2

3

4

5

For a complex vector, the transpose operator (’) produces the complex conjugate transpose (i.e. the sign on the imaginary part is changed as part of the transpose operation). The dot-transpose operator (.’) transposes the vector but does not conjugate it. The two transpose operators are identical for real data.

>> a = [1+j 2-3j -3+4j] a =

>>b = a’

b =

1.0000- 1.0000i

2.0000+ 3.0000i -3.0000- 4.0000i

>>c = a.’

c =

1.0000+ 1.0000i

2.0000- 3.0000i -3.0000+ 4.0000i

Vectors containing zeros and ones

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Two special functions in Matlab can be used to generate new vectors containing all zeros or all ones.

zeros(m,1) Returns an m-element column vector of zeros zeros(1,n) Returns an n-element row vector of zeros ones(m,1) Returns an m-element column vector of ones ones(1,n) Returns an n-element row vector of ones

Examples:

>>x = zeros(3,1)

x =

0

0

0

>>y = ones(1,3)

y =

1 1 1

5.2Matrix Arrays

A matrix array is two-dimensional, having both multiple rows and multiple columns. Creation of two-dimensional arrays follows that of row and column vectors:

•Begin with [, end with ]

•Spaces or commas are used to separate elements in a row.

•A semicolon or Enter is used to separate rows.

>>f = [1 2 3; 4 5 6]

14

25

36

>>h = [1 2 3 4 5 6 7 8 9]

h =

88

>> k = [1 2;3 4 5]

??? Number of elements in each row must be the same.

Here f is an array having 2 rows and 3 columns, called a 2 by 3 matrix. Applying the transpose operator to f produces g, a 3 by 2 matrix. Array h is 3 by 3, created using Enter to start new rows. The incorrect attempt to construct k shows that all rows must contain the same number of columns.

The transpose operator can be used to generate tables from vectors. For example, generate two vectors, x and y, then display the values such that x(1) and y(1) are on the same line, and so on, as follows:

>>x = 0:4;

>>y = 5:5:25;

>>A = [x’ y’]

A =

05

110

215

320

425

Note that the matrix A has been created from the vectors x and y.

Matrix Addressing

To refer to the element in row 2 and column 3 of matrix f:

>> f(2,3) ans =

6

Thus, the first number in parentheses addresses the row and the second number (following a comma) addresses the column.

Colon in place of a row or column reference represents the entire row or column. To create a column vector from the first column of the array h above:

>> h_1 = h(:,1) h_1 =

1

4

7

Submatrix creation: To create a submatrix using the colon operator, consider:

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>> c = [1,0,0; 1,1,0; 1,-1,0; 0,0,2] c =

>> c_1 = c(:,2:3) c_1 =

00

10

-1 0

02

>>c_2 = c(3:4,1:2) c_2 =

1-1

0 0

Thus, c_1 contains all rows and columns 2 and 3 of c and c_2 contains rows 3 and 4 of columns 1 and 2 of c.

Matrix size

To determine the size of a matrix array:

>> s = size(A) s =

23

>>[r,c] = size(A)

r =

c =

90

3

>>r = size(A,1)

r =

2

>>c = size(A,2)

c =

3

>>length(A)

Grand total is 11 elements using 88 bytes

Matrices containing zeros and ones

Two special functions in Matlab can be used to generate new matrices containing all zeros or all ones.

Examples of the use of zeros:

>> A = zeros(3) A =

00

>>C =[1 2 3; 4 2 5]

C =

12 3

91

4 2 5

>> D = zeros(size(C)) D =

5.2.1Array Operations

Scalar-Array Mathematics

Addition, subtraction, multiplication, and division of an array by a scalar simply apply the operation to all elements of the array.

>> f = [1 2 3; 4 5 6] f =

Element-by-Element Array-Array Mathematics

When two arrays have the same dimensions, addition, subtraction, multiplication, and division apply on an element-by-element basis. The notation for some operations is somewhat unconventional.

>>A = [2 5 6];

>>B = [2 3 5];

>>C = A.*B

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>> F = 3.0.^A F =

9 243 729

Array-array mathematics on other than an element-by-element basis will be discussed later.

5.3Array Plotting Capabilities

The plot command introduced previously can be used to produce an xy plot of vector x against vector y, as linear plots with the x and y axes divided into equally spaced intervals. Other plots use a logarithmic scale (base 10) when a variable ranges over many orders of magnitude because the wide range of values can be graphed without compressing the smaller values.

plot(x,y) Generates a linear plot of the values of x (horizontal axis) and y (vertical axis).

semilogx(x,y) Generates a plot of the values of x and y using a logarithmic scale for x and a linear scale for y.

semilogy(x,y) Generates a plot of the values of x and y using a linear scale for x and a logarithmic scale for y.

loglog(x,y) Generates a plot of the values of x and y using logarithmic scales for both x and y.

Note that the logarithm of a negative value or of zero does not exist and if the data contains negative or zero values, a warning message will be printed by Matlab informing you that these data points have been omitted from the data plotted. Examples are shown in Figure 5.4 below.

Plot Linestyles

An optional argument to the various plot commands above controls the linestyle. This argument is a character string consisting of one of the characters shown below. This character can be combined with the previously described characters that control colors and markers.

−−dashed line

Example 5.1 Vertical motion under gravity

An object is thrown vertically upward with an initial speed v, under acceleration of gravity g. Neglecting air resistance, determine and plot the height of the object as a function of time, from time zero when the object is thrown until it returns to the ground.

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The height of the object at time t is

h(t) = vt − 12gt2

The object returns to the ground at time tg , when

h(tg ) = vtg − 12gt2g = 0

or

2v tg = g

If the initial velocity v is 60 m/s, the following script will compute and plot the vertical height as a function of the time of flight.

%Vertical motion under gravity

%Define input values

The resulting plot is shown in Figure 5.1. Note that the option ’:’ in the plot command produces a dotted line in the plot.

Multiple Curves

Multiple curves can be plotted on the same graph by using multiple arguments in a plot command, as in plot(x,y,w,z), where the variables x, y, w and z are vectors. The curve of y versus x will be plotted, and then the curve corresponding to z versus w will be plotted on the same graph.

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Vertical mtion under gravity

time (s)

Figure 5.1: Height-time graph of vertical motion under gravity

Another way to generate multiple curves on one plot is to use a single matrix with multiple columns as the second argument of the plot command. Thus, the command plot(x,F), where x is a vector and F is a matrix, will produce a graph in which each column of F is a curve plotted against x.

To distinguish between plots on the graph, the legend command is used. This command has the form:

legend(’string1’,’string2’,’string3’,...)

It places a box on the plot, with the curve type for first curve labeled with the text string ’string1’, the second curved labeled with the text string ’string2’, and so on. The position of the curve can be controlled by adding a final integer position argument pos to the legend command. Values of pos and the corresponding positions are:

0 = Automatic “best” placement (least conflict with data)

1 = Upper right-hand corner (default)

2 = Upper left-hand corner

3 = Lower left-hand corner

4 = Lower right-hand corner -1 = To the right of the plot

Example 5.2 Multiple plot of polynomial curves

Consider plotting the two polynomial functions:

f1 = x2 − 3x + 2

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Multiple Function Plot

Figure 5.2: Plot with two polynomial curves

f2 = 2x2 + x − 3

% Generate plots of polynomials

%

x = 0:0.1:5;

f(:,1) = x’.^2 -3*x’ + 2; f(:,2) = 2*x’.^2 +x’ - 3;

plot(x,f),title(’Multiple Function Plot’),...

xlabel(’x’), grid, legend(’f1’,’f2’)

The resulting plot is shown in Figure 5.2 (the line colors di er, allowing the curves to be distinguished from one another).

To plot two curves with separate y-axis scaling and labeling, the plotyy function can be used. This command is apparently not well developed, as it does not accept line style options and the legend command causes one of the curves to disappear.

Example 5.3 Multiple plot of polynomial curves with separate y axes

Again consider plotting the two polynomial functions from the previous example, but with separate y axes.

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