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<title>MADlib: Sparse Linear Systems</title>
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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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