Tutorial: A Module for Linear Regression

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:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

These equations can be written in matrix form as

$\text{[math]}$

and can be expressed symbolically as

$\text{[math]}$

where $\text{[math]}$ is the matrix of coefficients for the linear system. Because $\text{[math]}$ is nonsingular, the system has a solution given by

$\text{[math]}$

This example solves this linear system of equations.

Define the matrices $\text{[math]}$ and $\text{[math]}$ . 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};

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;o
print x;

Figure 4.1 The Solution of a Linear System of Equations

x
3
5
2

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 $\text{[math]}$ , and $\text{[math]}$ values that solve the linear system.

You can end PROC IML by using the QUIT statement:

quit;

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