docs/v1.9.1/group__grp__dense__linear__solver.html - madlib-site - Git at Google

 <!-- HTML header for doxygen 1.8.4-->
 <!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>
 <meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
 <meta http-equiv="X-UA-Compatible" content="IE=9"/>
 <meta name="generator" content="Doxygen 1.8.10"/>
 <meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
 <title>MADlib: Dense Linear Systems</title>
 <link href="tabs.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="jquery.js"></script>
 <script type="text/javascript" src="dynsections.js"></script>
 <link href="navtree.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="resize.js"></script>
 <script type="text/javascript" src="navtreedata.js"></script>
 <script type="text/javascript" src="navtree.js"></script>
 <script type="text/javascript">
   $(document).ready(initResizable);
   $(window).load(resizeHeight);
 </script>
 <link href="search/search.css" rel="stylesheet" type="text/css"/>
 <script type="text/javascript" src="search/searchdata.js"></script>
 <script type="text/javascript" src="search/search.js"></script>
 <script type="text/javascript">
   $(document).ready(function() { init_search(); });
 </script>
 <!-- hack in the navigation tree -->
 <script type="text/javascript" src="eigen_navtree_hacks.js"></script>
 <link href="doxygen.css" rel="stylesheet" type="text/css" />
 <link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
 <!-- google analytics -->
 <script>
   (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
   (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
   m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
   })(window,document,'script','//www.google-analytics.com/analytics.js','ga');
   ga('create', 'UA-45382226-1', 'madlib.net');
   ga('send', 'pageview');
 </script>
 </head>
 <body>
 <div id="top"><!-- do not remove this div, it is closed by doxygen! -->
 <div id="titlearea">
 <table cellspacing="0" cellpadding="0">
  <tbody>
  <tr style="height: 56px;">
   <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
   <td style="padding-left: 0.5em;">
    <div id="projectname">
    <span id="projectnumber">1.9.1</span>
    </div>
    <div id="projectbrief">User Documentation for MADlib</div>
   </td>
    <td>        <div id="MSearchBox" class="MSearchBoxInactive">
         <span class="left">
           <img id="MSearchSelect" src="search/mag_sel.png"
                onmouseover="return searchBox.OnSearchSelectShow()"
                onmouseout="return searchBox.OnSearchSelectHide()"
                alt=""/>
           <input type="text" id="MSearchField" value="Search" accesskey="S"
                onfocus="searchBox.OnSearchFieldFocus(true)"
                onblur="searchBox.OnSearchFieldFocus(false)"
                onkeyup="searchBox.OnSearchFieldChange(event)"/>
           </span><span class="right">
             <a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
           </span>
         </div>
 </td>
  </tr>
  </tbody>
 </table>
 </div>
 <!-- end header part -->
 <!-- Generated by Doxygen 1.8.10 -->
 <script type="text/javascript">
 var searchBox = new SearchBox("searchBox", "search",false,'Search');
 </script>
 </div><!-- top -->
 <div id="side-nav" class="ui-resizable side-nav-resizable">
   <div id="nav-tree">
     <div id="nav-tree-contents">
       <div id="nav-sync" class="sync"></div>
     </div>
   </div>
   <div id="splitbar" style="-moz-user-select:none;"
        class="ui-resizable-handle">
   </div>
 </div>
 <script type="text/javascript">
 $(document).ready(function(){initNavTree('group__grp__dense__linear__solver.html','');});
 </script>
 <div id="doc-content">
 <!-- window showing the filter options -->
 <div id="MSearchSelectWindow"
      onmouseover="return searchBox.OnSearchSelectShow()"
      onmouseout="return searchBox.OnSearchSelectHide()"
      onkeydown="return searchBox.OnSearchSelectKey(event)">
 </div>

 <!-- iframe showing the search results (closed by default) -->
 <div id="MSearchResultsWindow">
 <iframe src="javascript:void(0)" frameborder="0"
         name="MSearchResults" id="MSearchResults">
 </iframe>
 </div>

 <div class="header">
   <div class="headertitle">
 <div class="title">Dense Linear Systems<div class="ingroups"><a class="el" href="group__grp__utility__functions.html">Utility Functions</a> &raquo; <a class="el" href="group__grp__linear__solver.html">Linear Solvers</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b> </p><ul>
 <li class="level1">
 <a href="#dls_usage">Solution Function</a> </li>
 <li class="level1">
 <a href="#dls_opt_params">Optimizer Parameters</a> </li>
 <li class="level1">
 <a href="#dls_examples">Examples</a> </li>
 <li class="level1">
 <a href="#related">Related Topics</a> </li>
 </ul>
 </div><p>The linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form: </p><p class="formulaDsp">
 <img class="formulaDsp" alt="\[ Ax = b \]" src="form_212.png"/>
 </p>
 <p>where <img class="formulaInl" alt="$x \in \mathbb{R}^{n}$" src="form_213.png"/>, <img class="formulaInl" alt="$A \in \mathbb{R}^{m \times n} $" src="form_214.png"/> and <img class="formulaInl" alt="$b \in \mathbb{R}^{m}$" src="form_215.png"/>. We assume that there are no rows of <img class="formulaInl" alt="$A$" src="form_41.png"/> where all elements are zero. The algorithms implemented in this module can handle large dense linear systems. Currently, the algorithms implemented in this module solve the linear system by a direct decomposition. Hence, these methods are known as <em>direct method</em>.</p>
 <p><a class="anchor" id="dls_usage"></a></p><dl class="section user"><dt>Solution Function</dt><dd><pre class="syntax">
 linear_solver_dense( tbl_source,
                      tbl_result,
                      row_id,
                      LHS,
                      RHS,
                      grouping_col,
                      optimizer,
                      optimizer_params
                    )
 </pre> <b>Arguments</b> <dl class="arglist">
 <dt>tbl_source </dt>
 <dd><p class="startdd">TEXT. The name of the table containing the training data. The input data is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>sourceName</em> (
     ...
     <em>row_id</em>          FLOAT8,
     <em>left_hand_side</em>  FLOAT8[],
     <em>right_hand_side</em> FLOAT8,
     ...
 )</pre><p>Each row represents a single equation. The <em>right_hand_side</em> column refers to the right hand side of the equations while the <em>left_hand_side</em> column refers to the multipliers on the variables on the left hand side of the same equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>tbl_result </dt>
 <dd><p class="startdd">TEXT. The name of the table where the output is saved. The output is stored in the table named by the <em>tbl_result</em> argument. It contains the following columns: </p><table  class="output">
 <tr>
 <th>solution </th><td>FLOAT8[]. The solution variables in the same order as that provided as input in the 'left_hand_side' column name of the <em>source_table</em>  </td></tr>
 <tr>
 <th>residual_norm </th><td>FLOAT8. The scaled residual norm, defined as <img class="formulaInl" alt="$ \frac{|Ax - b|}{|b|} $" src="form_216.png"/>. This value is an indication of the accuracy of the solution.  </td></tr>
 <tr>
 <th>iters </th><td>INTEGER. Number of iterations required by the algorithm (only applicable for iterative algorithms). The output is NULL for 'direct' methods.   </td></tr>
 </table>
 <p class="enddd"></p>
 </dd>
 <dt>row_id </dt>
 <dd><p class="startdd">TEXT. The name of the column storing the 'row id' of the equations.</p>
 <p>For a system with N equations, the row_id's must be a continuous range of integers from <img class="formulaInl" alt="$ 0 \ldots n-1 $" src="form_217.png"/>. </p>
 <p class="enddd"></p>
 </dd>
 <dt>LHS </dt>
 <dd><p class="startdd">TEXT. The name of the column storing the 'left hand side' of the equations, stored as an array.</p>
 <p class="enddd"></p>
 </dd>
 <dt>RHS </dt>
 <dd><p class="startdd">TEXT. The name of the column storing the 'right hand side' of the equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>grouping_cols (optional)  </dt>
 <dd>TEXT, default: NULL. Group by column names. <em>Not currently implemented. Any non-NULL value is ignored.</em> </dd>
 <dt>optimizer (optional)  </dt>
 <dd><p class="startdd">TEXT, default: 'direct'. The type of optimizer.</p>
 <p class="enddd"></p>
 </dd>
 <dt>optimizer_params (optional)  </dt>
 <dd>TEXT, default: NULL. Optimizer specific parameters. </dd>
 </dl>
 </dd></dl>
 <p><a class="anchor" id="dls_opt_params"></a></p><dl class="section user"><dt>Optimizer Parameters</dt><dd></dd></dl>
 <p>For each optimizer, there are specific parameters that can be tuned for better performance.</p>
 <dl class="arglist">
 <dt>algorithm (default: householderqr) </dt>
 <dd><p class="startdd">There are several algorithms that can be classified as 'direct' methods of solving linear systems. MADlib dense linear system solvers provide various algorithmic options for users.</p>
 <p>The following table provides a guideline on the choice of algorithm based on conditions on the A matrix, speed of the algorithms and numerical stability. </p><pre class="fragment"> Algorithm            | Conditions on A  | Speed | Accuracy
  ----------------------------------------------------------
  householderqr        | None             |  ++   |  +
  partialpivlu         | Invertable       |  ++   |  +
  fullpivlu            | None             |  -    |  +++
  colpivhouseholderqr  | None             |  +    |  ++
  fullpivhouseholderqr | None             |  -    |  +++
  llt                  | Pos. Definite    |  +++  |  +
  ldlt                 | Pos. or Neg Def  |  +++  |  ++
 </pre><p>For speed '++' is faster than '+', which is faster than '-'. For accuracy '+++' is better than '++'.</p>
 <p class="enddd">More details about the individual algorithms can be found in the <a href="http://eigen.tuxfamily.org/dox-devel/group__TutorialLinearAlgebra.html">Eigen documentation</a>. Eigen is an open source library for linear algebra.  </p>
 </dd>
 </dl>
 <p><a class="anchor" id="dls_examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
 <ol type="1">
 <li>View online help for the linear systems solver function. <pre class="example">
 SELECT madlib.linear_solver_dense();
 </pre></li>
 <li>Create the sample data set. <pre class="example">
 CREATE TABLE linear_systems_test_data( id INTEGER NOT NULL,
                                        lhs DOUBLE PRECISION[],
                                        rhs DOUBLE PRECISION
                                      );
 INSERT INTO linear_systems_test_data(id, lhs, rhs)
        VALUES
         (0, ARRAY[1,0,0], 20),
         (1, ARRAY[0,1,0], 15),
         (2, ARRAY[0,0,1], 20);
 </pre></li>
 <li>Solve the linear systems with default parameters. <pre class="example">
 SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
                                    'output_table',
                                    'id',
                                    'lhs',
                                    'rhs'
                                  );
 </pre></li>
 <li>Obtain the output from the output table. <pre class="example">
 \x on
 SELECT * FROM output_table;
 </pre> Result: <pre class="result">
 --------------------+-------------------------------------
 solution            | {20,15,20}
 residual_norm       | 0
 iters               | NULL
 </pre></li>
 <li>Choose an algorithm different than the default. <pre class="example">
 DROP TABLE IF EXISTS result_table;
 SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
                                    'result_table',
                                    'id',
                                    'lhs',
                                    'rhs',
                                    NULL,
                                    'direct',
                                    'algorithm=llt'
                                  );
 </pre></li>
 </ol>
 <p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="dense__linear__systems_8sql__in.html" title="SQL functions for linear systems. ">dense_linear_systems.sql_in</a> documenting the SQL functions</dd></dl>
 </div><!-- contents -->
 </div><!-- doc-content -->
 <!-- start footer part -->
 <div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
   <ul>
     <li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
     <a href="http://www.doxygen.org/index.html">
     <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
   </ul>
 </div>
 </body>
 </html>
	<!-- HTML header for doxygen 1.8.4-->
	<!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>
	<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
	<meta http-equiv="X-UA-Compatible" content="IE=9"/>
	<meta name="generator" content="Doxygen 1.8.10"/>
	<meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
	<title>MADlib: Dense Linear Systems</title>
	<link href="tabs.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="jquery.js"></script>
	<script type="text/javascript" src="dynsections.js"></script>
	<link href="navtree.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="resize.js"></script>
	<script type="text/javascript" src="navtreedata.js"></script>
	<script type="text/javascript" src="navtree.js"></script>
	<script type="text/javascript">
	$(document).ready(initResizable);
	$(window).load(resizeHeight);
	</script>
	<link href="search/search.css" rel="stylesheet" type="text/css"/>
	<script type="text/javascript" src="search/searchdata.js"></script>
	<script type="text/javascript" src="search/search.js"></script>
	<script type="text/javascript">
	$(document).ready(function() { init_search(); });
	</script>
	<!-- hack in the navigation tree -->
	<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
	<link href="doxygen.css" rel="stylesheet" type="text/css" />
	<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
	<!-- google analytics -->
	<script>
	(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]\|\|function(){
	(i[r].q=i[r].q\|\|[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
	m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
	})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
	ga('create', 'UA-45382226-1', 'madlib.net');
	ga('send', 'pageview');
	</script>
	</head>
	<body>
	<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
	<div id="titlearea">
	<table cellspacing="0" cellpadding="0">
	<tbody>
	<tr style="height: 56px;">
	<td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
	<td style="padding-left: 0.5em;">
	<div id="projectname">
	<span id="projectnumber">1.9.1</span>
	</div>
	<div id="projectbrief">User Documentation for MADlib</div>
	</td>
	<td> <div id="MSearchBox" class="MSearchBoxInactive">
	<span class="left">
	<img id="MSearchSelect" src="search/mag_sel.png"
	onmouseover="return searchBox.OnSearchSelectShow()"
	onmouseout="return searchBox.OnSearchSelectHide()"
	alt=""/>
	<input type="text" id="MSearchField" value="Search" accesskey="S"
	onfocus="searchBox.OnSearchFieldFocus(true)"
	onblur="searchBox.OnSearchFieldFocus(false)"
	onkeyup="searchBox.OnSearchFieldChange(event)"/>
	</span><span class="right">
	<a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
	</span>
	</div>
	</td>
	</tr>
	</tbody>
	</table>
	</div>
	<!-- end header part -->
	<!-- Generated by Doxygen 1.8.10 -->
	<script type="text/javascript">
	var searchBox = new SearchBox("searchBox", "search",false,'Search');
	</script>
	</div><!-- top -->
	<div id="side-nav" class="ui-resizable side-nav-resizable">
	<div id="nav-tree">
	<div id="nav-tree-contents">
	<div id="nav-sync" class="sync"></div>
	</div>
	</div>
	<div id="splitbar" style="-moz-user-select:none;"
	class="ui-resizable-handle">
	</div>
	</div>
	<script type="text/javascript">
	$(document).ready(function(){initNavTree('group__grp__dense__linear__solver.html','');});
	</script>
	<div id="doc-content">
	<!-- window showing the filter options -->
	<div id="MSearchSelectWindow"
	onmouseover="return searchBox.OnSearchSelectShow()"
	onmouseout="return searchBox.OnSearchSelectHide()"
	onkeydown="return searchBox.OnSearchSelectKey(event)">
	</div>

	<!-- iframe showing the search results (closed by default) -->
	<div id="MSearchResultsWindow">
	<iframe src="javascript:void(0)" frameborder="0"
	name="MSearchResults" id="MSearchResults">
	</iframe>
	</div>

	<div class="header">
	<div class="headertitle">
	<div class="title">Dense Linear Systems<div class="ingroups"><a class="el" href="group__grp__utility__functions.html">Utility Functions</a> » <a class="el" href="group__grp__linear__solver.html">Linear Solvers</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b> </p><ul>
	<li class="level1">
	<a href="#dls_usage">Solution Function</a> </li>
	<li class="level1">
	<a href="#dls_opt_params">Optimizer Parameters</a> </li>
	<li class="level1">
	<a href="#dls_examples">Examples</a> </li>
	<li class="level1">
	<a href="#related">Related Topics</a> </li>
	</ul>
	</div><p>The linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form: </p><p class="formulaDsp">
	<img class="formulaDsp" alt="\[ Ax = b \]" src="form_212.png"/>
	</p>
	<p>where <img class="formulaInl" alt="$x \in \mathbb{R}^{n}$" src="form_213.png"/>, <img class="formulaInl" alt="$A \in \mathbb{R}^{m \times n} $" src="form_214.png"/> and <img class="formulaInl" alt="$b \in \mathbb{R}^{m}$" src="form_215.png"/>. We assume that there are no rows of <img class="formulaInl" alt="$A$" src="form_41.png"/> where all elements are zero. The algorithms implemented in this module can handle large dense linear systems. Currently, the algorithms implemented in this module solve the linear system by a direct decomposition. Hence, these methods are known as <em>direct method</em>.</p>
	<p><a class="anchor" id="dls_usage"></a></p><dl class="section user"><dt>Solution Function</dt><dd><pre class="syntax">
	linear_solver_dense( tbl_source,
	tbl_result,
	row_id,
	LHS,
	RHS,
	grouping_col,
	optimizer,
	optimizer_params
	)
	</pre> <b>Arguments</b> <dl class="arglist">
	<dt>tbl_source </dt>
	<dd><p class="startdd">TEXT. The name of the table containing the training data. The input data is expected to be of the following form: </p><pre>{TABLE\|VIEW} <em>sourceName</em> (
	...
	<em>row_id</em> FLOAT8,
	<em>left_hand_side</em> FLOAT8[],
	<em>right_hand_side</em> FLOAT8,
	...
	)</pre><p>Each row represents a single equation. The <em>right_hand_side</em> column refers to the right hand side of the equations while the <em>left_hand_side</em> column refers to the multipliers on the variables on the left hand side of the same equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>tbl_result </dt>
	<dd><p class="startdd">TEXT. The name of the table where the output is saved. The output is stored in the table named by the <em>tbl_result</em> argument. It contains the following columns: </p><table class="output">
	<tr>
	<th>solution </th><td>FLOAT8[]. The solution variables in the same order as that provided as input in the 'left_hand_side' column name of the <em>source_table</em> </td></tr>
	<tr>
	<th>residual_norm </th><td>FLOAT8. The scaled residual norm, defined as <img class="formulaInl" alt="$ \frac{\|Ax - b\|}{\|b\|} $" src="form_216.png"/>. This value is an indication of the accuracy of the solution. </td></tr>
	<tr>
	<th>iters </th><td>INTEGER. Number of iterations required by the algorithm (only applicable for iterative algorithms). The output is NULL for 'direct' methods. </td></tr>
	</table>
	<p class="enddd"></p>
	</dd>
	<dt>row_id </dt>
	<dd><p class="startdd">TEXT. The name of the column storing the 'row id' of the equations.</p>
	<p>For a system with N equations, the row_id's must be a continuous range of integers from <img class="formulaInl" alt="$ 0 \ldots n-1 $" src="form_217.png"/>. </p>
	<p class="enddd"></p>
	</dd>
	<dt>LHS </dt>
	<dd><p class="startdd">TEXT. The name of the column storing the 'left hand side' of the equations, stored as an array.</p>
	<p class="enddd"></p>
	</dd>
	<dt>RHS </dt>
	<dd><p class="startdd">TEXT. The name of the column storing the 'right hand side' of the equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>grouping_cols (optional) </dt>
	<dd>TEXT, default: NULL. Group by column names. <em>Not currently implemented. Any non-NULL value is ignored.</em> </dd>
	<dt>optimizer (optional) </dt>
	<dd><p class="startdd">TEXT, default: 'direct'. The type of optimizer.</p>
	<p class="enddd"></p>
	</dd>
	<dt>optimizer_params (optional) </dt>
	<dd>TEXT, default: NULL. Optimizer specific parameters. </dd>
	</dl>
	</dd></dl>
	<p><a class="anchor" id="dls_opt_params"></a></p><dl class="section user"><dt>Optimizer Parameters</dt><dd></dd></dl>
	<p>For each optimizer, there are specific parameters that can be tuned for better performance.</p>
	<dl class="arglist">
	<dt>algorithm (default: householderqr) </dt>
	<dd><p class="startdd">There are several algorithms that can be classified as 'direct' methods of solving linear systems. MADlib dense linear system solvers provide various algorithmic options for users.</p>
	<p>The following table provides a guideline on the choice of algorithm based on conditions on the A matrix, speed of the algorithms and numerical stability. </p><pre class="fragment"> Algorithm \| Conditions on A \| Speed \| Accuracy
	----------------------------------------------------------
	householderqr \| None \| ++ \| +
	partialpivlu \| Invertable \| ++ \| +
	fullpivlu \| None \| - \| +++
	colpivhouseholderqr \| None \| + \| ++
	fullpivhouseholderqr \| None \| - \| +++
	llt \| Pos. Definite \| +++ \| +
	ldlt \| Pos. or Neg Def \| +++ \| ++
	</pre><p>For speed '++' is faster than '+', which is faster than '-'. For accuracy '+++' is better than '++'.</p>
	<p class="enddd">More details about the individual algorithms can be found in the <a href="http://eigen.tuxfamily.org/dox-devel/group__TutorialLinearAlgebra.html">Eigen documentation</a>. Eigen is an open source library for linear algebra. </p>
	</dd>
	</dl>
	<p><a class="anchor" id="dls_examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
	<ol type="1">
	<li>View online help for the linear systems solver function. <pre class="example">
	SELECT madlib.linear_solver_dense();
	</pre></li>
	<li>Create the sample data set. <pre class="example">
	CREATE TABLE linear_systems_test_data( id INTEGER NOT NULL,
	lhs DOUBLE PRECISION[],
	rhs DOUBLE PRECISION
	);
	INSERT INTO linear_systems_test_data(id, lhs, rhs)
	VALUES
	(0, ARRAY[1,0,0], 20),
	(1, ARRAY[0,1,0], 15),
	(2, ARRAY[0,0,1], 20);
	</pre></li>
	<li>Solve the linear systems with default parameters. <pre class="example">
	SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
	'output_table',
	'id',
	'lhs',
	'rhs'
	);
	</pre></li>
	<li>Obtain the output from the output table. <pre class="example">
	\x on
	SELECT * FROM output_table;
	</pre> Result: <pre class="result">
	--------------------+-------------------------------------
	solution \| {20,15,20}
	residual_norm \| 0
	iters \| NULL
	</pre></li>
	<li>Choose an algorithm different than the default. <pre class="example">
	DROP TABLE IF EXISTS result_table;
	SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
	'result_table',
	'id',
	'lhs',
	'rhs',
	NULL,
	'direct',
	'algorithm=llt'
	);
	</pre></li>
	</ol>
	<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="dense__linear__systems_8sql__in.html" title="SQL functions for linear systems. ">dense_linear_systems.sql_in</a> documenting the SQL functions</dd></dl>
	</div><!-- contents -->
	</div><!-- doc-content -->
	<!-- start footer part -->
	<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
	<ul>
	<li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
	<a href="http://www.doxygen.org/index.html">
	<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
	</ul>
	</div>
	</body>
	</html>