Vectorized functions and operators; more on graphs

Matlab has many commands to create special matrices; the following command creates a row vector whose components increase arithmetically:

>> t = 1:5
t =
     1     2     3     4     5

The components can change by non-unit steps:

>> x = 0:.1:1
x =
  Columns 1 through 7 
         0    0.1000    0.2000    0.3000    0.4000    0.5000    0.6000
  Columns 8 through 11 
    0.7000    0.8000    0.9000    1.0000

A negative step is also allowed. The command linspace has similar results; it creates a vector with linearly spaced entries. Specifically, linspace(a,b,n) creates a vector of length n with entries :

>> linspace(0,1,11)
ans =
  Columns 1 through 7 
         0    0.1000    0.2000    0.3000    0.4000    0.5000    0.6000
  Columns 8 through 11 
    0.7000    0.8000    0.9000    1.0000

There is a similar command logspace for creating vectors with logarithmically spaced entries:

>> logspace(0,1,11)
ans =
  Columns 1 through 7 
    1.0000    1.2589    1.5849    1.9953    2.5119    3.1623    3.9811
  Columns 8 through 11 
    5.0119    6.3096    7.9433   10.0000

See help logspace for details.

A vector with linearly spaced entries can be regarded as defining a one-dimensional grid, which is useful for graphing functions. To create a graph of y = f(x) (or, to be precise, to graph points of the form (x,f(x)) and connect them with line segments), one can create a grid in the vector x and then create a vector y with the corresponding function values.

It is easy to create the needed vectors to graph a built-in function, since Matlab functions are vectorized. This means that if a built-in function such as sine is applied to a array, the effect is to create a new array of the same size whose entries are the function values of the entries of the original array. For example (see Figure 3):

>> x = (0:.1:2*pi);
>> y = sin(x);
>> plot(x,y)

  
Figure 3: Graph of y = sin(x)

Matlab also provides vectorized arithmetic operators, which are the same as the ordinary operators, preceded by ``.''. For example, to graph :

>> x = (-5:.1:5); 
>> y = x./(1+x.^2);
>> plot(x,y)

(the graph is not shown). Thus x.^2 squares each component of x, and x./z divides each component of x by the corresponding component of z. Addition and subtraction are performed component-wise by definition, so there are no ``.+'' or ``.-'' operators. Note the difference between A^2 and A.^2. The first is only defined if A is a square matrix, while the second is defined for any n-dimensional array A.