Tutorial: A Module for Linear Regression: Solving a System of Equations

Tutorial: A Module for Linear Regression

Solving a System of Equations

Because IML is built around traditional matrix algebra notation, it is often possible to directly translate mathematical methods from matrix algebraic expressions into executable IML statements. For example, consider the problem of solving three simultaneous equations:

$3x_1 - x_2 + 2x_3 & = & 8 \ 2x_1 - 2x_2 + 3x_3 & = & 2 \ 4x_1 + x_2 - 4x_3 & = & 9 \$

These equations can be written in matrix form as

$[ 3 & -1 & 2 \ 2 & -2 & 3 \ 4 & 1 & -4 \ ] [ x_1 \ x_2 \ x_3 \ ] = [ 8 \ 2 \ 9 \ ]$

and can be expressed symbolically as

Because

is nonsingular, the system has a solution given by

$x = a^{-1}c$

In the following example, you solve this system of equations by using an interactive session. Submit the PROC IML statement as follows to begin the procedure. (Throughout this chapter, the right angle brackets (>) indicate statements you submit; responses from IML follow.)

  
    > proc iml; 
  
      IML Ready

Enter this statement:

  
    > reset print;

The PRINT option of the RESET command causes automatic printing of results. Notice that as you submit each statement, it is executed and the results are displayed. While you are learning IML or developing modules, it is a good idea to have all results printed automatically. When you are familiar with SAS/IML software, you will not need to use automatic printing.

Next, set up the matrices and . Both of these matrices are input as matrix literals; that is, input the row and column values as discussed in Chapter 2.

  
    > a={3  -1  2, 
    >    2  -2  3, 
    >    4   1 -4}; 
  
                A             3 rows      3 cols    (numeric) 
  
                                 3        -1         2 
                                 2        -2         3 
                                 4         1        -4 
  
    > c={8, 2, 9}; 
  
                 C             3 rows      1 col     (numeric) 
  
                                           8 
                                           2 
                                           9

Now write the solution equation, $x = a^{-1}c$ , as an IML statement, as follows. The appropriate statement is an assignment statement that uses a built-in function and an operator (INV is a built-in function that takes the inverse of a square matrix, and * is the operator for matrix multiplication).

  
    > x=inv(a)*c; 
  
               X             3 rows      1 col     (numeric) 
  
                                           3 
                                           5 
                                           2

After IML executes the statement, the first row of matrix contains the value for which you are solving, the second row contains the value, and the third row contains the value.

Now end the session by entering the QUIT command:

  
    >   quit; 
  
        Exiting IML

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