Mercurial > hg > de.mpg.mpiwg.itgroup.digilib.plugin
diff libs/commons-math-2.1/docs/userguide/linear.html @ 10:5f2c5fb36e93
commons-math-2.1 added
| author | dwinter |
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| date | Tue, 04 Jan 2011 10:00:53 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/libs/commons-math-2.1/docs/userguide/linear.html Tue Jan 04 10:00:53 2011 +0100 @@ -0,0 +1,362 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + + + + + + + + + + + +<html xmlns="http://www.w3.org/1999/xhtml"> + <head> + <title>Math - The Commons Math User Guide - Linear Algebra</title> + <style type="text/css" media="all"> + @import url("../css/maven-base.css"); + @import url("../css/maven-theme.css"); + @import url("../css/site.css"); + </style> + <link rel="stylesheet" href="../css/print.css" type="text/css" media="print" /> + <meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1" /> + </head> + <body class="composite"> + <div id="banner"> + <span id="bannerLeft"> + + Commons Math User Guide + + </span> + <div class="clear"> + <hr/> + </div> + </div> + <div id="breadcrumbs"> + + + + + + + + + <div class="xright"> + + + + + + + + </div> + <div class="clear"> + <hr/> + </div> + </div> + <div id="leftColumn"> + <div id="navcolumn"> + + + + + + + + + <h5>User Guide</h5> + <ul> + + <li class="none"> + <a href="../userguide/index.html">Contents</a> + </li> + + <li class="none"> + <a href="../userguide/overview.html">Overview</a> + </li> + + <li class="none"> + <a href="../userguide/stat.html">Statistics</a> + </li> + + <li class="none"> + <a href="../userguide/random.html">Data Generation</a> + </li> + + <li class="none"> + <strong>Linear Algebra</strong> + </li> + + <li class="none"> + <a href="../userguide/analysis.html">Numerical Analysis</a> + </li> + + <li class="none"> + <a href="../userguide/special.html">Special Functions</a> + </li> + + <li class="none"> + <a href="../userguide/utilities.html">Utilities</a> + </li> + + <li class="none"> + <a href="../userguide/complex.html">Complex Numbers</a> + </li> + + <li class="none"> + <a href="../userguide/distribution.html">Distributions</a> + </li> + + <li class="none"> + <a href="../userguide/fraction.html">Fractions</a> + </li> + + <li class="none"> + <a href="../userguide/transform.html">Transform Methods</a> + </li> + + <li class="none"> + <a href="../userguide/geometry.html">3D Geometry</a> + </li> + + <li class="none"> + <a href="../userguide/optimization.html">Optimization</a> + </li> + + <li class="none"> + <a href="../userguide/ode.html">Ordinary Differential Equations</a> + </li> + + <li class="none"> + <a href="../userguide/genetics.html">Genetic Algorithms</a> + </li> + </ul> + <a href="http://maven.apache.org/" title="Built by Maven" class="poweredBy"> + <img alt="Built by Maven" src="../images/logos/maven-feather.png"></img> + </a> + + + + + + + + + </div> + </div> + <div id="bodyColumn"> + <div id="contentBox"> + <div class="section"><h2><a name="a3_Linear_Algebra"></a>3 Linear Algebra</h2> +<div class="section"><h3><a name="a3.1_Overview"></a>3.1 Overview</h3> +<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> +</div> +<div class="section"><h3><a name="a3.2_Real_matrices"></a>3.2 Real matrices</h3> +<p> + The <a href="../apidocs/org/apache/commons/math/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> +<li>Scalar addition and multiplication</li> +<li>transpose</li> +<li>Norm and Trace</li> +<li>Operation on a vector</li> +</ul> +</p> +<p> + Example: + <div class="source"><pre> +// Create a real matrix with two rows and three columns +double[][] matrixData = { {1d,2d,3d}, {2d,5d,3d}}; +RealMatrix m = new Array2DRowRealMatrix(matrixData); + +// One more with three rows, two columns +double[][] matrixData2 = { {1d,2d}, {2d,5d}, {1d, 7d}}; +RealMatrix n = new Array2DRowRealMatrix(matrixData2); + +// Note: The constructor copies the input double[][] array. + +// 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 LUDecompositionImpl(p).getSolver().getInverse(); + </pre> +</div> +</p> +<p> + The three main implementations of the interface are <a href="../apidocs/org/apache/commons/math/linear/Array2DRowRealMatrix.html"> + Array2DRowRealMatrix</a> and <a href="../apidocs/org/apache/commons/math/linear/BlockRealMatrix.html"> + BlockRealMatrix</a> for dense matrices (the second one being more suited to + dimensions above 50 or 100) and <a href="../apidocs/org/apache/commons/math/linear/SparseRealMatrix.html"> + SparseRealMatrix</a> for sparse matrices. + </p> +</div> +<div class="section"><h3><a name="a3.3_Real_vectors"></a>3.3 Real vectors</h3> +<p> + The <a href="../apidocs/org/apache/commons/math/linear/RealVector.html"> + RealVector</a> interface represents a vector with real numbers as + entries. The following basic matrix operations are supported: + <ul><li>Vector addition, subtraction</li> +<li>Element by element multiplication, division</li> +<li>Scalar addition, subtraction, multiplication, division and power</li> +<li>Mapping of mathematical functions (cos, sin ...)</li> +<li>Dot product, outer product</li> +<li>Distance and norm according to norms L1, L2 and Linf</li> +</ul> +</p> +<p> + The <a href="../apidocs/org/apache/commons/math/linear/RealVectorFormat.html"> + RealVectorFormat</a> class handles input/output of vectors in a customizable + textual format. + </p> +</div> +<div class="section"><h3><a name="a3.4_Solving_linear_systems"></a>3.4 Solving linear systems</h3> +<p> + The <code>solve()</code> methods of the <a href="../apidocs/org/apache/commons/math/linear/DecompositionSolver.html">DecompositionSolver</a> + interface support solving linear systems of equations of the form AX=B, either + in linear sense or in least square sense. A <code>RealMatrix</code> 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 + <div class="source"><pre> +RealMatrix coefficients = + new Array2DRowRealMatrix(new double[][] { { 2, 3, -2 }, { -1, 7, 6 }, { 4, -3, -5 } }, + false); +DecompositionSolver solver = new LUDecompositionImpl(coefficients).getSolver(); + </pre> +</div> + + Next create a <code>RealVector</code> array to represent the constant + vector B and use <code>solve(RealVector)</code> to solve the system + <div class="source"><pre> +RealVector constants = new RealVectorImpl(new double[] { 1, -2, 1 }, false); +RealVector solution = solver.solve(constants); + </pre> +</div> + + The <code>solution</code> vector will contain values for x + (<code>solution.getEntry(0)</code>), y (<code>solution.getEntry(1)</code>), + and z (<code>solution.getEntry(2)</code>) 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. 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. + </p> +<p><table class="bodyTable"><tr class="a"><td><font size="+2">Decomposition algorithms</font></td> +</tr> +<tr class="b"><font size="+1"><td>Name</td> +<td>coefficients matrix</td> +<td>problem type</td> +</font></tr> +<tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/LUDecomposition.html">LU</a></td> +<td>square</td> +<td>exact solution only</td> +</tr> +<tr class="b"><td><a href="../apidocs/org/apache/commons/math/linear/CholeskyDecomposition.html">Cholesky</a></td> +<td>symmetric positive definite</td> +<td>exact solution only</td> +</tr> +<tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/QRDecomposition.html">QR</a></td> +<td>any</td> +<td>least squares solution</td> +</tr> +<tr class="b"><td><a href="../apidocs/org/apache/commons/math/linear/EigenDecomposition.html">eigen decomposition</a></td> +<td>square</td> +<td>exact solution only</td> +</tr> +<tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/SingularValueDecomposition.html">SVD</a></td> +<td>any</td> +<td>least squares solution</td> +</tr> +</table> +</p> +<p> + It is possible to use a simple array of double instead of a <code>RealVector</code>. + In this case, the solution will be provided also as an array of double. + </p> +<p> + It is possible to solve multiple systems with the same coefficient matrix + in one method call. To do this, create a matrix whose column vectors correspond + to the constant vectors for the systems to be solved and use <code>solve(RealMatrix),</code> + which returns a matrix with column vectors representing the solutions. + </p> +</div> +<div class="section"><h3><a name="a3.5_Eigenvalueseigenvectors_and_singular_valuessingular_vectors"></a>3.5 Eigenvalues/eigenvectors and singular values/singular vectors</h3> +<p> + Decomposition algorithms may be used for themselves and not only for linear system solving. + This is of prime interest with eigen decomposition and singular value decomposition. + </p> +<p> + The <code>getEigenvalue()</code>, <code>getEigenvalues()</code>, <code>getEigenVector()</code>, + <code>getV()</code>, <code>getD()</code> and <code>getVT()</code> methods of the + <code>EigenDecomposition</code> interface support solving eigenproblems of the form + AX = lambda X where lambda is a real scalar. + </p> +<p>The <code>getSingularValues()</code>, <code>getU()</code>, <code>getS()</code> and + <code>getV()</code> methods of the <code>SingularValueDecomposition</code> interface + allow to solve singular values problems of the form AXi = lambda Yi where lambda is a + real scalar, and where the Xi and Yi vectors form orthogonal bases of their respective + vector spaces (which may have different dimensions). + </p> +</div> +<div class="section"><h3><a name="a3.6_Non-real_fields_complex_fractions_..."></a>3.6 Non-real fields (complex, fractions ...)</h3> +<p> + In addition to the real field, matrices and vectors using non-real <a href="../apidocs/org/apache/commons/math/FieldElement.html">field elements</a> can be used. + The fields already supported by the library are: + <ul><li><a href="../apidocs/org/apache/commons/math/complex/Complex.html">Complex</a></li> +<li><a href="../apidocs/org/apache/commons/math/fraction/Fraction.html">Fraction</a></li> +<li><a href="../apidocs/org/apache/commons/math/fraction/BigFraction.html">BigFraction</a></li> +<li><a href="../apidocs/org/apache/commons/math/util/BigReal.html">BigReal</a></li> +</ul> +</p> +</div> +</div> + + </div> + </div> + <div class="clear"> + <hr/> + </div> + <div id="footer"> + <div class="xright">© + 2003-2010 + + + + + + + + + + </div> + <div class="clear"> + <hr/> + </div> + </div> + </body> +</html>