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The linear.xml Java example source code
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<?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
<document url="linear.html">
<properties>
<title>The Commons Math User Guide - Linear Algebra
</properties>
<body>
<section name="3 Linear Algebra">
<subsection name="3.1 Overview" href="overview">
<p>
Linear algebra support in commons-math provides operations on real matrices
(both dense and sparse matrices are supported) and vectors. It features basic
operations (addition, subtraction ...) and decomposition algorithms that can
be used to solve linear systems either in exact sense and in least squares sense.
</p>
</subsection>
<subsection name="3.2 Real matrices" href="real_matrices">
<p>
The <a href="../apidocs/org/apache/commons/math3/linear/RealMatrix.html">
RealMatrix</a> interface represents a matrix with real numbers as
entries. The following basic matrix operations are supported:
<ul>
<li>Matrix addition, subtraction, multiplication
<li>Scalar addition and multiplication
<li>transpose
<li>Norm and Trace
<li>Operation on a vector
</ul>
</p>
<p>
Example:
<source>
// Create a real matrix with two rows and three columns, using a factory
// method that selects the implementation class for us.
double[][] matrixData = { {1d,2d,3d}, {2d,5d,3d}};
RealMatrix m = MatrixUtils.createRealMatrix(matrixData);
// One more with three rows, two columns, this time instantiating the
// RealMatrix implementation class directly.
double[][] matrixData2 = { {1d,2d}, {2d,5d}, {1d, 7d}};
RealMatrix n = new Array2DRowRealMatrix(matrixData2);
// Note: The constructor copies the input double[][] array in both cases.
// Now multiply m by n
RealMatrix p = m.multiply(n);
System.out.println(p.getRowDimension()); // 2
System.out.println(p.getColumnDimension()); // 2
// Invert p, using LU decomposition
RealMatrix pInverse = new LUDecomposition(p).getSolver().getInverse();
</source>
</p>
<p>
The three main implementations of the interface are <a
href="../apidocs/org/apache/commons/math3/linear/Array2DRowRealMatrix.html">
Array2DRowRealMatrix</a> and
<li>Element by element multiplication, division
<li>Scalar addition, subtraction, multiplication, division and power
<li>Mapping of mathematical functions (cos, sin ...)
<li>Dot product, outer product
<li>Distance and norm according to norms L1, L2 and Linf
</ul>
</p>
<p>
The <a href="../apidocs/org/apache/commons/math3/linear/RealVectorFormat.html">
RealVectorFormat</a> class handles input/output of vectors in a customizable
textual format.
</p>
</subsection>
<subsection name="3.4 Solving linear systems" href="solve">
<p>
The <code>solve() methods of the
interface support solving linear systems of equations of the form AX=B, either
in linear sense or in least square sense. A <code>RealMatrix instance is
used to represent the coefficient matrix of the system. Solving the system is a
two phases process: first the coefficient matrix is decomposed in some way and
then a solver built from the decomposition solves the system. This allows to
compute the decomposition and build the solver only once if several systems have
to be solved with the same coefficient matrix.
</p>
<p>
For example, to solve the linear system
<pre>
2x + 3y - 2z = 1
-x + 7y + 6x = -2
4x - 3y - 5z = 1
</pre>
Start by decomposing the coefficient matrix A (in this case using LU decomposition)
and build a solver
<source>
RealMatrix coefficients =
new Array2DRowRealMatrix(new double[][] { { 2, 3, -2 }, { -1, 7, 6 }, { 4, -3, -5 } },
false);
DecompositionSolver solver = new LUDecomposition(coefficients).getSolver();
</source>
Next create a <code>RealVector array to represent the constant
vector B and use <code>solve(RealVector) to solve the system
<source>
RealVector constants = new ArrayRealVector(new double[] { 1, -2, 1 }, false);
RealVector solution = solver.solve(constants);
</source>
The <code>solution vector will contain values for x
(<code>solution.getEntry(0)), y (solution.getEntry(1)),
and z (<code>solution.getEntry(2)) that solve the system.
</p>
<p>
Each type of decomposition has its specific semantics and constraints on
the coefficient matrix as shown in the following table. For algorithms that
solve AX=B in least squares sense the value returned for X is such that the
residual AX-B has minimal norm. Least Square sense means a solver can be computed
for an overdetermined system, (i.e. a system with more equations than unknowns,
which corresponds to a tall A matrix with more rows than columns). If an exact
solution exist (i.e. if for some X the residual AX-B is exactly 0), then this
exact solution is also the solution in least square sense. This implies that
algorithms suited for least squares problems can also be used to solve exact
problems, but the reverse is not true. In any case, if the matrix is singular
within the tolerance set at construction, an error will be triggered when
the solve method will be called, both for algorithms that compute exact solutions
and for algorithms that compute least square solutions.
</p>
<p>
<table border="1" align="center">
<tr BGCOLOR="#CCCCFF">Decomposition algorithms | |
<tr BGCOLOR="#EEEEFF">
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