You need to find the values of x, y, and z that satisfy the system of linear equations shown in Figure 8-2.

Figure 8-2. A system of linear equations

Use the RealMatrix and RealMatrixImpl from Commons Math. Represent this system of linear equations as matrices in the Ax=B form, as shown in Figure 8-3. Place the coefficients of A in a RealMatrix, and put B in a double[]. Call the solve( ) method on RealMatrix to retrieve a double[] of values for x, y, and z that satisfy this system of equations.

Figure 8-3. System of linear equations in Ax=B form

The following example takes the coefficients and constants from Figure 8-3 and uses a RealMatrix to solve this system:

import org.apache.commons.math.linear.RealMatrix;
import org.apache.commons.math.linear.RealMatrixImpl;
import org.apache.commons.lang.ArrayUtils;
double[][] coefficients = { { 3.0, 20.0, 89.0 },
                            { 4.0, 40.0, 298.0 },
                            { 7.0, 21.0, 0.42 } };
double[] values = { 1324, 2999, 2039 };
RealMatrix matrix = new RealMatrixImpl( );
matrix.setData( coefficients );
        
double[] answers = matrix.solve( values );
System.out.println( "Answers: " + ArrayUtils.toString( answers ) );

This example solves this system of equations and prints out the values of x, y, and z using Commons Lang ArrayUtils to print a double[]:

Answers: {400.4839095455532,-36.59139305646149,9.599731825759218}

Using Commons Math, we find that the following values satisfy this system of equations: x = 400, y = -36, and z = 9.6.

To solve these equations, a double[][] of coefficients is created to represent a 3 3 matrix, and a double[] of constants is created. The RealMatrix interface is implemented by RealMatrixImpl, which stores a matrix as a double[][]; to populate this double[][], pass the double[][], coefficients, to the setData( ) method of RealMatrixImpl. To solve the system, the values double[] is passed to matrix.solve( ), and a double[] containing x, y, and z is returned.

This method will not work for every matrix; there are systems of linear equations that are unsolvable. For example, if one attempts to find values for the system of equations from Figure 8-4, an InvalidMatrixException will be thrown stating that the matrix is singular. Additionally, if the number of rows in B does not equal the number of columns in A, solve() will throw an InvalidMatrixException.

Figure 8-4. An unsolvable system of equations

For more information about solving systems of linear equations (or, for that matter, information about anything), see Wikipedia (http://en.wikipedia.org/wiki/System_of_linear_equations). RealMatixImpl uses a process known as LU decomposition to solve this system of equations. For more information about LU decomposition, see the JavaDoc for the org.apache.commons.math.linear package (http://commons.apache.org/math/apidocs/index.html).