Example: Solving a System of Linear Equations

Because the syntax of the SAS/IML language is similar to the notation used in linear algebra, it is often possible to directly translate mathematical methods from matrix-algebraic expressions into executable SAS/IML statements. For example, consider the problem of solving three simultaneous equations:

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

These equations can be written in matrix form as

\[  \left[ \begin{array}{rrr} 3 &  -1 &  2 \\ 2 &  -2 &  3 \\ 4 &  1 &  -4 \\ \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ \end{array} \right] ~  = ~  \left[ \begin{array}{c} 8 \\ 2 \\ 9 \\ \end{array} \right]  \]

and can be expressed symbolically as

\[  \mb {Ax} = \mb {c}  \]

where $\mb {A}$ is the matrix of coefficients for the linear system. Because $\mb {A}$ is nonsingular, the system has a solution given by

\[  \mb {x} = \mb {A}^{-1}\mb {c} ~   \]

This example solves this linear system of equations.

  1. Define the matrices $\mb {A}$ and $\mb {c}$. Both of these matrices are input as matrix literals; that is, you type the row and column values as discussed in Chapter 3: Understanding the SAS/IML Language.

    proc iml;
    a = {3  -1  2,
         2  -2  3,
         4   1 -4};
    c = {8, 2, 9};
    
  2. Solve the equation by using the built-in INV function and the matrix multiplication operator. The INV function returns the inverse of a square matrix and * is the operator for matrix multiplication. Consequently, the solution is computed as follows:

    x = inv(a) * c;
    print x;
    

    Figure 4.1: The Solution of a Linear System of Equations

    x
    3
    5
    2


  3. Equivalently, you can solve the linear system by using the more efficient SOLVE function, as shown in the following statement:

    x = solve(a, c);
    

After SAS/IML executes the statements, the rows of the vector x contain the $x_1, x_2$, and $x_3$ values that solve the linear system.

You can end PROC IML by using the QUIT statement:

quit;