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 <div class="header">
   <div class="headertitle">
 <div class="title">Sparse Linear Systems<div class="ingroups"><a class="el" href="group__grp__other__functions.html">Utilities</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> <ul>
 <li class="level1">
 <a href="#sls_usage">Solution Function</a> </li>
 <li class="level1">
 <a href="#sls_opt_params">Optimizer Parameters</a> </li>
 <li class="level1">
 <a href="#sls_output">Output Tables</a> </li>
 <li class="level1">
 <a href="#sls_examples">Examples</a> </li>
 <li>
 <a href="related">Related Topics</a> </li>
 </ul>
 </div><p>The sparse linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form: </p><p class="formulaDsp">
 \[ Ax = b \]
 </p>
 <p>where \(x \in \mathbb{R}^{n}\), \(A \in \mathbb{R}^{m \times n} \) and \(b \in \mathbb{R}^{m}\). This module accepts sparse matrix input formats for \(A\) and \(b\). We assume that there are no rows of \(A\) where all elements are zero.</p>
 <dl class="section note"><dt>Note</dt><dd>Algorithms with fail if there is an row of the input matrix containing all zeros.</dd></dl>
 <p>The algorithms implemented in this module can handle large sparse square linear systems. Currently, the algorithms implemented in this module solve the linear system using direct or iterative methods.</p>
 <p><a class="anchor" id="sls_usage"></a></p><dl class="section user"><dt>Sparse Linear Systems Solution Function</dt><dd></dd></dl>
 <pre class="syntax">
 linear_solver_sparse( tbl_source_lhs,
                       tbl_source_rhs,
                       tbl_result,
                       lhs_row_id,
                       lhs_col_id,
                       lhs_value,
                       rhs_row_id,
                       rhs_value,
                       grouping_cols := NULL,
                       optimizer := 'direct',
                       optimizer_params :=
                       'algorithm = llt'
                     )
 </pre><p> <b>Arguments</b> </p><dl class="arglist">
 <dt>tbl_source_lhs </dt>
 <dd><p class="startdd">The name of the table containing the left hand side matrix. For the LHS matrix, the input data is expected to be of the following form: </p><pre>
 {TABLE|VIEW} <em>sourceName</em> (
     ...
     <em>row_id</em> FLOAT8,
     <em>col_id</em> FLOAT8,
     <em>value</em> FLOAT8,
     ...
 )</pre><p> Each row represents a single equation. The <em>rhs</em> columns refer to the right hand side of the equations and the <em>lhs</em> columns refer to the multipliers on the variables on the left hand side of the same equations. </p>
 <p class="enddd"></p>
 </dd>
 <dt>tbl_source_rhs </dt>
 <dd><p class="startdd">TEXT. The name of the table containing the right hand side vector. For the RHS matrix, the input data is expected to be of the following form: </p><pre class="fragment">{TABLE|VIEW} &lt;em&gt;sourceName&lt;/em&gt; (
     ...
     &lt;em&gt;row_id&lt;/em&gt; FLOAT8,
     &lt;em&gt;value&lt;/em&gt; FLOAT8
     ...
 )</pre><p> Each row represents a single equation. The <em>rhs</em> columns refer to the right hand side of the equations while the <em>lhs</em> columns 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. Output is stored in the tabled named by the <em>tbl_result</em> argument. The table contains the following columns. The output contains the following columns: </p><table class="output">
 <tr>
 <th>solution </th><td>FLOAT8[]. The solution is an array with the variables in the same order as that provided as input in the 'left_hand_side' column name of the 'source_table'   </td></tr>
 <tr>
 <th>residual_norm </th><td>FLOAT8. Scaled residual norm, defined as \( \frac{|Ax - b|}{|b|} \). 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>lhs_row_id </dt>
 <dd>TEXT. The name of the column storing the 'row id' of the equations. <dl class="section note"><dt>Note</dt><dd>For a system with N equations, the row_id's must be a continuous range of integers from \( 0 \ldots n-1 \).</dd></dl>
 </dd>
 <dt>lhs_col_id </dt>
 <dd><p class="startdd">TEXT. The name of the column (in tbl_source_lhs) storing the 'col id' of the equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>lhs_value </dt>
 <dd><p class="startdd">TEXT. The name of the column (in tbl_source_lhs) storing the 'value' of the equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>rhs_row_id </dt>
 <dd><p class="startdd">TEXT. The name of the column (in tbl_source_rhs) storing the 'col id' of the equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>rhs_value </dt>
 <dd><p class="startdd">TEXT. The name of the column (in tbl_source_rhs) storing the 'value' of the equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>num_vars </dt>
 <dd><p class="startdd">INTEGER. The number of variables in the linear system equations.</p>
 <p class="enddd"></p>
 </dd>
 <dt>grouping_col (optional)  </dt>
 <dd>TEXT, default: NULL. Group by column names. <dl class="section note"><dt>Note</dt><dd>The grouping feature is currently not implemented and this parameter is only a placeholder.</dd></dl>
 </dd>
 <dt>optimizer (optional)  </dt>
 <dd><p class="startdd">TEXT, default: 'direct'. Type of optimizer.</p>
 <p class="enddd"></p>
 </dd>
 <dt>optimizer_params (optional) </dt>
 <dd>TEXT, default: NULL. Optimizer specific parameters. </dd>
 </dl>
 <p><a class="anchor" id="sls_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: ldlt) </dt>
 <dd><p class="startdd"></p>
 <p>There are several algorithms that can be classified as 'direct' methods of solving linear systems. Madlib functions provide various algorithmic options available 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 | Memory
     ----------------------------------------------------------
     llt                | Sym. Pos Def     |  ++   |  ---
     ldlt               | Sym. Pos Def     |  ++   |  ---

     For speed '++' is faster than '+', which is faster than '-'.
     For accuracy '+++' is better than '++'.
     For memory, '-' uses less memory than '--'.

     Note: ldlt is often preferred over llt
 </pre><p>There are several algorithms that can be classified as 'iterative' methods of solving linear systems. Madlib functions provide various algorithmic options available 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 | Memory | Convergence
     ----------------------------------------------------------------------
     cg-mem               | Sym. Pos Def     |  +++  |   -    |    ++
     bicgstab-mem         | Square           |  ++   |   -    |    +
     precond-cg-mem       | Sym. Pos Def     |  ++   |   -    |    +++
     precond-bicgstab-mem | Square           |  +    |   -    |    ++

     For memory, '-' uses less memory than '--'.
     For speed, '++' is faster than '+'.
 </pre><p>Algorithm Details: </p><table class="output">
 <tr>
 <th>cg-mem</th><td>In memory conjugate gradient with diagonal preconditioners.  </td></tr>
 <tr>
 <th>bicgstab-mem</th><td>Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners.  </td></tr>
 <tr>
 <th>precond-cg-mem</th><td>In memory conjugate gradient with diagonal preconditioners.  </td></tr>
 <tr>
 <th>bicgstab-mem</th><td>Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners.  </td></tr>
 </table>
 <p class="enddd"></p>
 </dd>
 <dt>toler (default: 1e-5) </dt>
 <dd><p class="startdd">Termination tolerance (applicable only for iterative methods) which determines the stopping criterion (with respect to residual norm) for iterative methods. </p>
 <p class="enddd"></p>
 </dd>
 </dl>
 <p><a class="anchor" id="sls_examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
 <ol type="1">
 <li>View online help for the sparse linear systems solver function. <pre class="example">
 SELECT madlib.linear_solver_sparse();
 </pre></li>
 <li>Create the sample data set. <pre class="example">
 DROP TABLE IF EXISTS sparse_linear_systems_lhs;
 CREATE TABLE sparse_linear_systems_lhs (
     rid INTEGER NOT NULL,
     cid  INTEGER,
     val DOUBLE PRECISION
 );
 DROP TABLE IF EXISTS sparse_linear_systems_rhs;
 CREATE TABLE sparse_linear_systems_rhs (
     rid INTEGER NOT NULL,
     val DOUBLE PRECISION
 );
 INSERT INTO sparse_linear_systems_lhs(rid, cid, val) VALUES
 (0, 0, 1),
 (1, 1, 1),
 (2, 2, 1),
 (3, 3, 1);
 INSERT INTO sparse_linear_systems_rhs(rid, val) VALUES
 (0, 10),
 (1, 20),
 (2, 30);
 </pre></li>
 <li>Solve the linear systems with default parameters. <pre class="example">
 SELECT madlib.linear_solver_sparse( 'sparse_linear_systems_lhs',
                                     'sparse_linear_systems_rhs',
                                     'output_table',
                                     'rid',
                                     'cid',
                                     'val',
                                     'rid',
                                     'val',
                                     4
                                   );
 </pre></li>
 <li>View the contents of the output table. <pre class="example">
 \x on
 SELECT * FROM output_table;
 </pre> Result: <pre class="result">
 --------------------+-------------------------------------
 solution            | {10,20,30,0}
 residual_norm       | 0
 iters               | NULL
 </pre></li>
 <li>Choose a different algorithm than the default algorithm. <pre class="example">
 DROP TABLE IF EXISTS output_table;
 SELECT madlib.linear_solver_sparse( 'sparse_linear_systems_lhs',
                                     'sparse_linear_systems_rhs',
                                     'output_table',
                                     'rid',
                                     'cid',
                                     'val',
                                     'rid',
                                     'val',
                                     4,
                                     NULL,
                                     'direct',
                                     'algorithm=llt'
                                   );
 </pre></li>
 <li>Choose a different algorithm than the default algorithm. <pre class="example">
 DROP TABLE IF EXISTS output_table;
 SELECT madlib.linear_solver_sparse(
                                     'sparse_linear_systems_lhs',
                                     'sparse_linear_systems_rhs',
                                     'output_table',
                                     'rid',
                                     'cid',
                                     'val',
                                     'rid',
                                     'val',
                                     4,
                                     NULL,
                                     'iterative',
                                     'algorithm=cg-mem, toler=1e-5'
                                   );
 </pre></li>
 </ol>
 <p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File sparse_linear_sytems.sql_in documenting the SQL functions.</dd></dl>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Sparse Linear Systems<div class="ingroups"><a class="el" href="group__grp__other__functions.html">Utilities</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> <ul>
	<li class="level1">
	<a href="#sls_usage">Solution Function</a> </li>
	<li class="level1">
	<a href="#sls_opt_params">Optimizer Parameters</a> </li>
	<li class="level1">
	<a href="#sls_output">Output Tables</a> </li>
	<li class="level1">
	<a href="#sls_examples">Examples</a> </li>
	<li>
	<a href="related">Related Topics</a> </li>
	</ul>
	</div><p>The sparse linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form: </p><p class="formulaDsp">
	\[ Ax = b \]
	</p>
	<p>where \(x \in \mathbb{R}^{n}\), \(A \in \mathbb{R}^{m \times n} \) and \(b \in \mathbb{R}^{m}\). This module accepts sparse matrix input formats for \(A\) and \(b\). We assume that there are no rows of \(A\) where all elements are zero.</p>
	<dl class="section note"><dt>Note</dt><dd>Algorithms with fail if there is an row of the input matrix containing all zeros.</dd></dl>
	<p>The algorithms implemented in this module can handle large sparse square linear systems. Currently, the algorithms implemented in this module solve the linear system using direct or iterative methods.</p>
	<p><a class="anchor" id="sls_usage"></a></p><dl class="section user"><dt>Sparse Linear Systems Solution Function</dt><dd></dd></dl>
	<pre class="syntax">
	linear_solver_sparse( tbl_source_lhs,
	tbl_source_rhs,
	tbl_result,
	lhs_row_id,
	lhs_col_id,
	lhs_value,
	rhs_row_id,
	rhs_value,
	grouping_cols := NULL,
	optimizer := 'direct',
	optimizer_params :=
	'algorithm = llt'
	)
	</pre><p> <b>Arguments</b> </p><dl class="arglist">
	<dt>tbl_source_lhs </dt>
	<dd><p class="startdd">The name of the table containing the left hand side matrix. For the LHS matrix, the input data is expected to be of the following form: </p><pre>
	{TABLE\|VIEW} <em>sourceName</em> (
	...
	<em>row_id</em> FLOAT8,
	<em>col_id</em> FLOAT8,
	<em>value</em> FLOAT8,
	...
	)</pre><p> Each row represents a single equation. The <em>rhs</em> columns refer to the right hand side of the equations and the <em>lhs</em> columns refer to the multipliers on the variables on the left hand side of the same equations. </p>
	<p class="enddd"></p>
	</dd>
	<dt>tbl_source_rhs </dt>
	<dd><p class="startdd">TEXT. The name of the table containing the right hand side vector. For the RHS matrix, the input data is expected to be of the following form: </p><pre class="fragment">{TABLE\|VIEW} <em>sourceName</em> (
	...
	<em>row_id</em> FLOAT8,
	<em>value</em> FLOAT8
	...
	)</pre><p> Each row represents a single equation. The <em>rhs</em> columns refer to the right hand side of the equations while the <em>lhs</em> columns 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. Output is stored in the tabled named by the <em>tbl_result</em> argument. The table contains the following columns. The output contains the following columns: </p><table class="output">
	<tr>
	<th>solution </th><td>FLOAT8[]. The solution is an array with the variables in the same order as that provided as input in the 'left_hand_side' column name of the 'source_table' </td></tr>
	<tr>
	<th>residual_norm </th><td>FLOAT8. Scaled residual norm, defined as \( \frac{\|Ax - b\|}{\|b\|} \). 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>lhs_row_id </dt>
	<dd>TEXT. The name of the column storing the 'row id' of the equations. <dl class="section note"><dt>Note</dt><dd>For a system with N equations, the row_id's must be a continuous range of integers from \( 0 \ldots n-1 \).</dd></dl>
	</dd>
	<dt>lhs_col_id </dt>
	<dd><p class="startdd">TEXT. The name of the column (in tbl_source_lhs) storing the 'col id' of the equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>lhs_value </dt>
	<dd><p class="startdd">TEXT. The name of the column (in tbl_source_lhs) storing the 'value' of the equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>rhs_row_id </dt>
	<dd><p class="startdd">TEXT. The name of the column (in tbl_source_rhs) storing the 'col id' of the equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>rhs_value </dt>
	<dd><p class="startdd">TEXT. The name of the column (in tbl_source_rhs) storing the 'value' of the equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>num_vars </dt>
	<dd><p class="startdd">INTEGER. The number of variables in the linear system equations.</p>
	<p class="enddd"></p>
	</dd>
	<dt>grouping_col (optional) </dt>
	<dd>TEXT, default: NULL. Group by column names. <dl class="section note"><dt>Note</dt><dd>The grouping feature is currently not implemented and this parameter is only a placeholder.</dd></dl>
	</dd>
	<dt>optimizer (optional) </dt>
	<dd><p class="startdd">TEXT, default: 'direct'. Type of optimizer.</p>
	<p class="enddd"></p>
	</dd>
	<dt>optimizer_params (optional) </dt>
	<dd>TEXT, default: NULL. Optimizer specific parameters. </dd>
	</dl>
	<p><a class="anchor" id="sls_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: ldlt) </dt>
	<dd><p class="startdd"></p>
	<p>There are several algorithms that can be classified as 'direct' methods of solving linear systems. Madlib functions provide various algorithmic options available 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 \| Memory
	----------------------------------------------------------
	llt \| Sym. Pos Def \| ++ \| ---
	ldlt \| Sym. Pos Def \| ++ \| ---

	For speed '++' is faster than '+', which is faster than '-'.
	For accuracy '+++' is better than '++'.
	For memory, '-' uses less memory than '--'.

	Note: ldlt is often preferred over llt
	</pre><p>There are several algorithms that can be classified as 'iterative' methods of solving linear systems. Madlib functions provide various algorithmic options available 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 \| Memory \| Convergence
	----------------------------------------------------------------------
	cg-mem \| Sym. Pos Def \| +++ \| - \| ++
	bicgstab-mem \| Square \| ++ \| - \| +
	precond-cg-mem \| Sym. Pos Def \| ++ \| - \| +++
	precond-bicgstab-mem \| Square \| + \| - \| ++

	For memory, '-' uses less memory than '--'.
	For speed, '++' is faster than '+'.
	</pre><p>Algorithm Details: </p><table class="output">
	<tr>
	<th>cg-mem</th><td>In memory conjugate gradient with diagonal preconditioners. </td></tr>
	<tr>
	<th>bicgstab-mem</th><td>Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners. </td></tr>
	<tr>
	<th>precond-cg-mem</th><td>In memory conjugate gradient with diagonal preconditioners. </td></tr>
	<tr>
	<th>bicgstab-mem</th><td>Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners. </td></tr>
	</table>
	<p class="enddd"></p>
	</dd>
	<dt>toler (default: 1e-5) </dt>
	<dd><p class="startdd">Termination tolerance (applicable only for iterative methods) which determines the stopping criterion (with respect to residual norm) for iterative methods. </p>
	<p class="enddd"></p>
	</dd>
	</dl>
	<p><a class="anchor" id="sls_examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
	<ol type="1">
	<li>View online help for the sparse linear systems solver function. <pre class="example">
	SELECT madlib.linear_solver_sparse();
	</pre></li>
	<li>Create the sample data set. <pre class="example">
	DROP TABLE IF EXISTS sparse_linear_systems_lhs;
	CREATE TABLE sparse_linear_systems_lhs (
	rid INTEGER NOT NULL,
	cid INTEGER,
	val DOUBLE PRECISION
	);
	DROP TABLE IF EXISTS sparse_linear_systems_rhs;
	CREATE TABLE sparse_linear_systems_rhs (
	rid INTEGER NOT NULL,
	val DOUBLE PRECISION
	);
	INSERT INTO sparse_linear_systems_lhs(rid, cid, val) VALUES
	(0, 0, 1),
	(1, 1, 1),
	(2, 2, 1),
	(3, 3, 1);
	INSERT INTO sparse_linear_systems_rhs(rid, val) VALUES
	(0, 10),
	(1, 20),
	(2, 30);
	</pre></li>
	<li>Solve the linear systems with default parameters. <pre class="example">
	SELECT madlib.linear_solver_sparse( 'sparse_linear_systems_lhs',
	'sparse_linear_systems_rhs',
	'output_table',
	'rid',
	'cid',
	'val',
	'rid',
	'val',
	4
	);
	</pre></li>
	<li>View the contents of the output table. <pre class="example">
	\x on
	SELECT * FROM output_table;
	</pre> Result: <pre class="result">
	--------------------+-------------------------------------
	solution \| {10,20,30,0}
	residual_norm \| 0
	iters \| NULL
	</pre></li>
	<li>Choose a different algorithm than the default algorithm. <pre class="example">
	DROP TABLE IF EXISTS output_table;
	SELECT madlib.linear_solver_sparse( 'sparse_linear_systems_lhs',
	'sparse_linear_systems_rhs',
	'output_table',
	'rid',
	'cid',
	'val',
	'rid',
	'val',
	4,
	NULL,
	'direct',
	'algorithm=llt'
	);
	</pre></li>
	<li>Choose a different algorithm than the default algorithm. <pre class="example">
	DROP TABLE IF EXISTS output_table;
	SELECT madlib.linear_solver_sparse(
	'sparse_linear_systems_lhs',
	'sparse_linear_systems_rhs',
	'output_table',
	'rid',
	'cid',
	'val',
	'rid',
	'val',
	4,
	NULL,
	'iterative',
	'algorithm=cg-mem, toler=1e-5'
	);
	</pre></li>
	</ol>
	<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File sparse_linear_sytems.sql_in documenting the SQL functions.</dd></dl>
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