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<div class="title">Solving Sparse Linear Systems<div class="ingroups"><a class="el" href="group__Sparse__chapter.html">Sparse linear algebra</a></div></div> </div>
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<p>In <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, there are several methods available to solve linear systems when the coefficient matrix is sparse. Because of the special representation of this class of matrices, special care should be taken in order to get a good performance. See <a class="el" href="group__TutorialSparse.html">Sparse matrix manipulations</a> for a detailed introduction about sparse matrices in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>. This page lists the sparse solvers available in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>. The main steps that are common to all these linear solvers are introduced as well. Depending on the properties of the matrix, the desired accuracy, the end-user is able to tune those steps in order to improve the performance of its code. Note that it is not required to know deeply what's hiding behind these steps: the last section presents a benchmark routine that can be easily used to get an insight on the performance of all the available solvers.</p>
<h1><a class="anchor" id="TutorialSparseDirectSolvers"></a>
Sparse solvers</h1>
<p>Eigen currently provides a limited set of built-in solvers, as well as wrappers to external solver libraries. They are summarized in the following table:</p>
<table class="manual">
<tr>
<th>Class</th><th>Module</th><th>Solver kind</th><th><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> kind</th><th>Features related to performance </th><th>Dependencies,License</th><th class="width20em"><p class="starttd"></p>
<p>Notes</p>
<p class="endtd"></p>
</th></tr>
<tr>
<td><a class="el" href="classEigen_1_1SimplicialLLT.html" title="A direct sparse LLT Cholesky factorizations. ">SimplicialLLT</a> </td><td><a class="el" href="group__SparseCholesky__Module.html">SparseCholesky </a></td><td>Direct LLt factorization</td><td>SPD</td><td>Fill-in reducing </td><td>built-in, LGPL </td><td><a class="el" href="classEigen_1_1SimplicialLDLT.html" title="A direct sparse LDLT Cholesky factorizations without square root. ">SimplicialLDLT</a> is often preferable </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1SimplicialLDLT.html" title="A direct sparse LDLT Cholesky factorizations without square root. ">SimplicialLDLT</a> </td><td><a class="el" href="group__SparseCholesky__Module.html">SparseCholesky </a></td><td>Direct LDLt factorization</td><td>SPD</td><td>Fill-in reducing </td><td>built-in, LGPL </td><td>Recommended for very sparse and not too large problems (e.g., 2D Poisson eq.) </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1ConjugateGradient.html" title="A conjugate gradient solver for sparse self-adjoint problems. ">ConjugateGradient</a></td><td><a class="el" href="group__IterativeLinearSolvers__Module.html">IterativeLinearSolvers </a></td><td>Classic iterative CG</td><td>SPD</td><td>Preconditionning </td><td>built-in, MPL2 </td><td>Recommended for large symmetric problems (e.g., 3D Poisson eq.) </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1BiCGSTAB.html" title="A bi conjugate gradient stabilized solver for sparse square problems. ">BiCGSTAB</a></td><td><a class="el" href="group__IterativeLinearSolvers__Module.html">IterativeLinearSolvers </a></td><td>Iterative stabilized bi-conjugate gradient</td><td>Square</td><td>Preconditionning </td><td>built-in, MPL2 </td><td>To speedup the convergence, try it with the <a class="el" href="classEigen_1_1IncompleteLUT.html">IncompleteLUT</a> preconditioner. </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1SparseLU.html" title="Sparse supernodal LU factorization for general matrices. ">SparseLU</a> </td><td><a class="el" href="group__SparseLU__Module.html">SparseLU </a> </td><td>LU factorization </td><td>Square </td><td>Fill-in reducing, Leverage fast dense algebra </td><td>built-in, MPL2 </td><td>optimized for small and large problems with irregular patterns </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1SparseQR.html" title="Sparse left-looking rank-revealing QR factorization. ">SparseQR</a> </td><td><a class="el" href="group__SparseQR__Module.html">SparseQR </a> </td><td>QR factorization </td><td>Any, rectangular</td><td>Fill-in reducing </td><td>built-in, MPL2</td><td>recommended for least-square problems, has a basic rank-revealing feature </td></tr>
<tr>
<th colspan="7">Wrappers to external solvers </th></tr>
<tr>
<td><a class="el" href="classEigen_1_1PastixLLT.html" title="A sparse direct supernodal Cholesky (LLT) factorization and solver based on the PaStiX library...">PastixLLT</a> <br/>
<a class="el" href="classEigen_1_1PastixLDLT.html" title="A sparse direct supernodal Cholesky (LLT) factorization and solver based on the PaStiX library...">PastixLDLT</a> <br/>
<a class="el" href="classEigen_1_1PastixLU.html" title="Interface to the PaStix solver. ">PastixLU</a></td><td><a class="el" href="group__PaStiXSupport__Module.html">PaStiXSupport </a></td><td>Direct LLt, LDLt, LU factorizations</td><td>SPD <br/>
SPD <br/>
Square</td><td>Fill-in reducing, Leverage fast dense algebra, Multithreading </td><td>Requires the <a href="http://pastix.gforge.inria.fr">PaStiX</a> package, <b>CeCILL-C</b> </td><td>optimized for tough problems and symmetric patterns </td></tr>
<tr>
<td><a class="el" href="classEigen_1_1CholmodSupernodalLLT.html" title="A supernodal Cholesky (LLT) factorization and solver based on Cholmod. ">CholmodSupernodalLLT</a></td><td><a class="el" href="group__CholmodSupport__Module.html">CholmodSupport </a></td><td>Direct LLt factorization</td><td>SPD</td><td>Fill-in reducing, Leverage fast dense algebra </td><td>Requires the <a href="http://www.cise.ufl.edu/research/sparse/SuiteSparse/">SuiteSparse</a> package, <b>GPL</b> </td><td></td></tr>
<tr>
<td><a class="el" href="classEigen_1_1UmfPackLU.html" title="A sparse LU factorization and solver based on UmfPack. ">UmfPackLU</a></td><td><a class="el" href="group__UmfPackSupport__Module.html">UmfPackSupport </a></td><td>Direct LU factorization</td><td>Square</td><td>Fill-in reducing, Leverage fast dense algebra </td><td>Requires the <a href="http://www.cise.ufl.edu/research/sparse/SuiteSparse/">SuiteSparse</a> package, <b>GPL</b> </td><td></td></tr>
<tr>
<td><a class="el" href="classEigen_1_1SuperLU.html" title="A sparse direct LU factorization and solver based on the SuperLU library. ">SuperLU</a></td><td><a class="el" href="group__SuperLUSupport__Module.html">SuperLUSupport </a></td><td>Direct LU factorization</td><td>Square</td><td>Fill-in reducing, Leverage fast dense algebra </td><td>Requires the <a href="http://crd-legacy.lbl.gov/~xiaoye/SuperLU/">SuperLU</a> library, (BSD-like) </td><td></td></tr>
<tr>
<td><a class="el" href="classEigen_1_1SPQR.html" title="Sparse QR factorization based on SuiteSparseQR library. ">SPQR</a></td><td><a class="el" href="group__SPQRSupport__Module.html">SPQRSupport </a> </td><td>QR factorization </td><td>Any, rectangular</td><td>fill-in reducing, multithreaded, fast dense algebra </td><td>requires the <a href="http://www.cise.ufl.edu/research/sparse/SuiteSparse/">SuiteSparse</a> package, <b>GPL</b> </td><td>recommended for linear least-squares problems, has a rank-revealing feature </td></tr>
</table>
<p>Here <code>SPD</code> means symmetric positive definite.</p>
<p>All these solvers follow the same general concept. Here is a typical and general example: </p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/RequiredModuleName></span></div>
<div class="line"><span class="comment">// ...</span></div>
<div class="line">SparseMatrix<double> A;</div>
<div class="line"><span class="comment">// fill A</span></div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga3da45e59796fbacf67fa568297927bd1">VectorXd</a> b, x;</div>
<div class="line"><span class="comment">// fill b</span></div>
<div class="line"><span class="comment">// solve Ax = b</span></div>
<div class="line">SolverClassName<SparseMatrix<double> > solver;</div>
<div class="line">solver.compute(A);</div>
<div class="line"><span class="keywordflow">if</span>(solver.info()!=<a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Success</a>) {</div>
<div class="line"> <span class="comment">// decomposition failed</span></div>
<div class="line"> <span class="keywordflow">return</span>;</div>
<div class="line">}</div>
<div class="line">x = solver.solve(b);</div>
<div class="line"><span class="keywordflow">if</span>(solver.info()!=<a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Success</a>) {</div>
<div class="line"> <span class="comment">// solving failed</span></div>
<div class="line"> <span class="keywordflow">return</span>;</div>
<div class="line">}</div>
<div class="line"><span class="comment">// solve for another right hand side:</span></div>
<div class="line">x1 = solver.solve(b1);</div>
</div><!-- fragment --><p>For <code>SPD</code> solvers, a second optional template argument allows to specify which triangular part have to be used, e.g.:</p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/IterativeLinearSolvers></span></div>
<div class="line"></div>
<div class="line">ConjugateGradient<SparseMatrix<double>, <a class="code" href="group__enums.html#ggab59c1bec446b10af208f977a871d910bae70afef0d3ff7aca74e17e85ff6c9f2e">Eigen::Upper</a>> solver;</div>
<div class="line">x = solver.compute(A).solve(b);</div>
</div><!-- fragment --><p> In the above example, only the upper triangular part of the input matrix A is considered for solving. The opposite triangle might either be empty or contain arbitrary values.</p>
<p>In the case where multiple problems with the same sparsity pattern have to be solved, then the "compute" step can be decomposed as follow: </p>
<div class="fragment"><div class="line">SolverClassName<SparseMatrix<double> > solver;</div>
<div class="line">solver.analyzePattern(A); <span class="comment">// for this step the numerical values of A are not used</span></div>
<div class="line">solver.factorize(A);</div>
<div class="line">x1 = solver.solve(b1);</div>
<div class="line">x2 = solver.solve(b2);</div>
<div class="line">...</div>
<div class="line">A = ...; <span class="comment">// modify the values of the nonzeros of A, the nonzeros pattern must stay unchanged</span></div>
<div class="line">solver.factorize(A);</div>
<div class="line">x1 = solver.solve(b1);</div>
<div class="line">x2 = solver.solve(b2);</div>
<div class="line">...</div>
</div><!-- fragment --><p> The compute() method is equivalent to calling both analyzePattern() and factorize().</p>
<p>Finally, each solver provides some specific features, such as determinant, access to the factors, controls of the iterations, and so on. More details are availble in the documentations of the respective classes.</p>
<h1><a class="anchor" id="TheSparseCompute"></a>
The Compute Step</h1>
<p>In the compute() function, the matrix is generally factorized: <a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> for self-adjoint matrices, <a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> for general hermitian matrices, LU for non hermitian matrices and QR for rectangular matrices. These are the results of using direct solvers. For this class of solvers precisely, the compute step is further subdivided into analyzePattern() and factorize().</p>
<p>The goal of analyzePattern() is to reorder the nonzero elements of the matrix, such that the factorization step creates less fill-in. This step exploits only the structure of the matrix. Hence, the results of this step can be used for other linear systems where the matrix has the same structure. Note however that sometimes, some external solvers (like <a class="el" href="classEigen_1_1SuperLU.html" title="A sparse direct LU factorization and solver based on the SuperLU library. ">SuperLU</a>) require that the values of the matrix are set in this step, for instance to equilibrate the rows and columns of the matrix. In this situation, the results of this step should not be used with other matrices.</p>
<p><a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> provides a limited set of methods to reorder the matrix in this step, either built-in (COLAMD, AMD) or external (METIS). These methods are set in template parameter list of the solver : </p>
<div class="fragment"><div class="line">DirectSolverClassName<SparseMatrix<double>, OrderingMethod<IndexType> > solver;</div>
</div><!-- fragment --><p>See the <a class="el" href="group__OrderingMethods__Module.html">OrderingMethods module </a> for the list of available methods and the associated options.</p>
<p>In factorize(), the factors of the coefficient matrix are computed. This step should be called each time the values of the matrix change. However, the structural pattern of the matrix should not change between multiple calls.</p>
<p>For iterative solvers, the compute step is used to eventually setup a preconditioner. For instance, with the ILUT preconditioner, the incomplete factors L and U are computed in this step. Remember that, basically, the goal of the preconditioner is to speedup the convergence of an iterative method by solving a modified linear system where the coefficient matrix has more clustered eigenvalues. For real problems, an iterative solver should always be used with a preconditioner. In <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, a preconditioner is selected by simply adding it as a template parameter to the iterative solver object. </p>
<div class="fragment"><div class="line">IterativeSolverClassName<SparseMatrix<double>, PreconditionerName<SparseMatrix<double> > solver; </div>
</div><!-- fragment --><p> The member function preconditioner() returns a read-write reference to the preconditioner to directly interact with it. See the <a class="el" href="group__IterativeLinearSolvers__Module.html">Iterative solvers module </a> and the documentation of each class for the list of available methods.</p>
<h1><a class="anchor" id="TheSparseSolve"></a>
The Solve step</h1>
<p>The solve() function computes the solution of the linear systems with one or many right hand sides. </p>
<div class="fragment"><div class="line">X = solver.solve(B);</div>
</div><!-- fragment --><p> Here, B can be a vector or a matrix where the columns form the different right hand sides. The solve() function can be called several times as well, for instance when all the right hand sides are not available at once. </p>
<div class="fragment"><div class="line">x1 = solver.solve(b1);</div>
<div class="line"><span class="comment">// Get the second right hand side b2</span></div>
<div class="line">x2 = solver.solve(b2); </div>
<div class="line"><span class="comment">// ...</span></div>
</div><!-- fragment --><p> For direct methods, the solution are computed at the machine precision. Sometimes, the solution need not be too accurate. In this case, the iterative methods are more suitable and the desired accuracy can be set before the solve step using <b>setTolerance()</b>. For all the available functions, please, refer to the documentation of the <a class="el" href="group__IterativeLinearSolvers__Module.html">Iterative solvers module </a>.</p>
<h1><a class="anchor" id="BenchmarkRoutine"></a>
BenchmarkRoutine</h1>
<p>Most of the time, all you need is to know how much time it will take to qolve your system, and hopefully, what is the most suitable solver. In <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, we provide a benchmark routine that can be used for this purpose. It is very easy to use. In the build directory, navigate to bench/spbench and compile the routine by typing <b>make</b> <em>spbenchsolver</em>. Run it with –help option to get the list of all available options. Basically, the matrices to test should be in <a href="http://math.nist.gov/MatrixMarket/formats.html">MatrixMarket Coordinate format</a>, and the routine returns the statistics from all available solvers in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>.</p>
<p>The following table gives an example of XML statistics from several <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> built-in and external solvers. </p>
<table border="1">
<tr>
<th><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> </th><th>N </th><th>NNZ </th><th></th><th>UMFPACK </th><th>SUPERLU </th><th>PASTIX LU </th><th><a class="el" href="classEigen_1_1BiCGSTAB.html" title="A bi conjugate gradient stabilized solver for sparse square problems. ">BiCGSTAB</a> </th><th>BiCGSTAB+ILUT </th><th>GMRES+ILUT</th><th><a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> </th><th>CHOLMOD <a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> </th><th>PASTIX <a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> </th><th><a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> </th><th>CHOLMOD SP <a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> </th><th>CHOLMOD <a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> </th><th>PASTIX <a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> </th><th>CG </th></tr>
<tr>
<th rowspan="4">vector_graphics </th><td rowspan="4">12855 </td><td rowspan="4">72069 </td><th>Compute Time </th><td>0.0254549</td><td>0.0215677</td><td>0.0701827</td><td>0.000153388</td><td>0.0140107</td><td>0.0153709</td><td>0.0101601</td><td style="background-color:red">0.00930502</td><td>0.0649689 </td></tr>
<tr>
<th>Solve Time </th><td>0.00337835</td><td>0.000951826</td><td>0.00484373</td><td>0.0374886</td><td>0.0046445</td><td>0.00847754</td><td>0.000541813</td><td style="background-color:red">0.000293696</td><td>0.00485376 </td></tr>
<tr>
<th>Total Time </th><td>0.0288333</td><td>0.0225195</td><td>0.0750265</td><td>0.037642</td><td>0.0186552</td><td>0.0238484</td><td>0.0107019</td><td style="background-color:red">0.00959871</td><td>0.0698227 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.299e-16 </td><td>2.04207e-16 </td><td>4.83393e-15 </td><td>3.94856e-11 (80) </td><td>1.03861e-12 (3) </td><td>5.81088e-14 (6) </td><td>1.97578e-16 </td><td>1.83927e-16 </td><td>4.24115e-15 </td></tr>
<tr>
<th rowspan="4">poisson_SPD </th><td rowspan="4">19788 </td><td rowspan="4">308232 </td><th>Compute Time </th><td>0.425026</td><td>1.82378</td><td>0.617367</td><td>0.000478921</td><td>1.34001</td><td>1.33471</td><td>0.796419</td><td>0.857573</td><td>0.473007</td><td>0.814826</td><td style="background-color:red">0.184719</td><td>0.861555</td><td>0.470559</td><td>0.000458188 </td></tr>
<tr>
<th>Solve Time </th><td>0.0280053</td><td>0.0194402</td><td>0.0268747</td><td>0.249437</td><td>0.0548444</td><td>0.0926991</td><td>0.00850204</td><td>0.0053171</td><td>0.0258932</td><td>0.00874603</td><td style="background-color:red">0.00578155</td><td>0.00530361</td><td>0.0248942</td><td>0.239093 </td></tr>
<tr>
<th>Total Time </th><td>0.453031</td><td>1.84322</td><td>0.644241</td><td>0.249916</td><td>1.39486</td><td>1.42741</td><td>0.804921</td><td>0.862891</td><td>0.4989</td><td>0.823572</td><td style="background-color:red">0.190501</td><td>0.866859</td><td>0.495453</td><td>0.239551 </td></tr>
<tr>
<th>Error(Iter) </th><td>4.67146e-16 </td><td>1.068e-15 </td><td>1.3397e-15 </td><td>6.29233e-11 (201) </td><td>3.68527e-11 (6) </td><td>3.3168e-15 (16) </td><td>1.86376e-15 </td><td>1.31518e-16 </td><td>1.42593e-15 </td><td>3.45361e-15 </td><td>3.14575e-16 </td><td>2.21723e-15 </td><td>7.21058e-16 </td><td>9.06435e-12 (261) </td></tr>
<tr>
<th rowspan="4">sherman2 </th><td rowspan="4">1080 </td><td rowspan="4">23094 </td><th>Compute Time </th><td style="background-color:red">0.00631754</td><td>0.015052</td><td>0.0247514 </td><td>-</td><td>0.0214425</td><td>0.0217988 </td></tr>
<tr>
<th>Solve Time </th><td style="background-color:red">0.000478424</td><td>0.000337998</td><td>0.0010291 </td><td>-</td><td>0.00243152</td><td>0.00246152 </td></tr>
<tr>
<th>Total Time </th><td style="background-color:red">0.00679597</td><td>0.01539</td><td>0.0257805 </td><td>-</td><td>0.023874</td><td>0.0242603 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.83099e-15 </td><td>8.19351e-15 </td><td>2.625e-14 </td><td>1.3678e+69 (1080) </td><td>4.1911e-12 (7) </td><td>5.0299e-13 (12) </td></tr>
<tr>
<th rowspan="4">bcsstk01_SPD </th><td rowspan="4">48 </td><td rowspan="4">400 </td><th>Compute Time </th><td>0.000169079</td><td>0.00010789</td><td>0.000572538</td><td>1.425e-06</td><td>9.1612e-05</td><td>8.3985e-05</td><td style="background-color:red">5.6489e-05</td><td>7.0913e-05</td><td>0.000468251</td><td>5.7389e-05</td><td>8.0212e-05</td><td>5.8394e-05</td><td>0.000463017</td><td>1.333e-06 </td></tr>
<tr>
<th>Solve Time </th><td>1.2288e-05</td><td>1.1124e-05</td><td>0.000286387</td><td>8.5896e-05</td><td>1.6381e-05</td><td>1.6984e-05</td><td style="background-color:red">3.095e-06</td><td>4.115e-06</td><td>0.000325438</td><td>3.504e-06</td><td>7.369e-06</td><td>3.454e-06</td><td>0.000294095</td><td>6.0516e-05 </td></tr>
<tr>
<th>Total Time </th><td>0.000181367</td><td>0.000119014</td><td>0.000858925</td><td>8.7321e-05</td><td>0.000107993</td><td>0.000100969</td><td style="background-color:red">5.9584e-05</td><td>7.5028e-05</td><td>0.000793689</td><td>6.0893e-05</td><td>8.7581e-05</td><td>6.1848e-05</td><td>0.000757112</td><td>6.1849e-05 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.03474e-16 </td><td>2.23046e-16 </td><td>2.01273e-16 </td><td>4.87455e-07 (48) </td><td>1.03553e-16 (2) </td><td>3.55965e-16 (2) </td><td>2.48189e-16 </td><td>1.88808e-16 </td><td>1.97976e-16 </td><td>2.37248e-16 </td><td>1.82701e-16 </td><td>2.71474e-16 </td><td>2.11322e-16 </td><td>3.547e-09 (48) </td></tr>
<tr>
<th rowspan="4">sherman1 </th><td rowspan="4">1000 </td><td rowspan="4">3750 </td><th>Compute Time </th><td>0.00228805</td><td>0.00209231</td><td>0.00528268</td><td>9.846e-06</td><td>0.00163522</td><td>0.00162155</td><td>0.000789259</td><td style="background-color:red">0.000804495</td><td>0.00438269 </td></tr>
<tr>
<th>Solve Time </th><td>0.000213788</td><td>9.7983e-05</td><td>0.000938831</td><td>0.00629835</td><td>0.000361764</td><td>0.00078794</td><td>4.3989e-05</td><td style="background-color:red">2.5331e-05</td><td>0.000917166 </td></tr>
<tr>
<th>Total Time </th><td>0.00250184</td><td>0.00219029</td><td>0.00622151</td><td>0.0063082</td><td>0.00199698</td><td>0.00240949</td><td>0.000833248</td><td style="background-color:red">0.000829826</td><td>0.00529986 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.16839e-16 </td><td>2.25968e-16 </td><td>2.59116e-16 </td><td>3.76779e-11 (248) </td><td>4.13343e-11 (4) </td><td>2.22347e-14 (10) </td><td>2.05861e-16 </td><td>1.83555e-16 </td><td>1.02917e-15 </td></tr>
<tr>
<th rowspan="4">young1c </th><td rowspan="4">841 </td><td rowspan="4">4089 </td><th>Compute Time </th><td>0.00235843</td><td style="background-color:red">0.00217228</td><td>0.00568075</td><td>1.2735e-05</td><td>0.00264866</td><td>0.00258236 </td></tr>
<tr>
<th>Solve Time </th><td>0.000329599</td><td style="background-color:red">0.000168634</td><td>0.00080118</td><td>0.0534738</td><td>0.00187193</td><td>0.00450211 </td></tr>
<tr>
<th>Total Time </th><td>0.00268803</td><td style="background-color:red">0.00234091</td><td>0.00648193</td><td>0.0534865</td><td>0.00452059</td><td>0.00708447 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.27029e-16 </td><td>2.81321e-16 </td><td>5.0492e-15 </td><td>8.0507e-11 (706) </td><td>3.00447e-12 (8) </td><td>1.46532e-12 (16) </td></tr>
<tr>
<th rowspan="4">mhd1280b </th><td rowspan="4">1280 </td><td rowspan="4">22778 </td><th>Compute Time </th><td>0.00234898</td><td>0.00207079</td><td>0.00570918</td><td>2.5976e-05</td><td>0.00302563</td><td>0.00298036</td><td>0.00144525</td><td style="background-color:red">0.000919922</td><td>0.00426444 </td></tr>
<tr>
<th>Solve Time </th><td>0.00103392</td><td>0.000211911</td><td>0.00105</td><td>0.0110432</td><td>0.000628287</td><td>0.00392089</td><td>0.000138303</td><td style="background-color:red">6.2446e-05</td><td>0.00097564 </td></tr>
<tr>
<th>Total Time </th><td>0.0033829</td><td>0.0022827</td><td>0.00675918</td><td>0.0110692</td><td>0.00365392</td><td>0.00690124</td><td>0.00158355</td><td style="background-color:red">0.000982368</td><td>0.00524008 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.32953e-16 </td><td>3.08646e-16 </td><td>6.734e-16 </td><td>8.83132e-11 (40) </td><td>1.51153e-16 (1) </td><td>6.08556e-16 (8) </td><td>1.89264e-16 </td><td>1.97477e-16 </td><td>6.68126e-09 </td></tr>
<tr>
<th rowspan="4">crashbasis </th><td rowspan="4">160000 </td><td rowspan="4">1750416 </td><th>Compute Time </th><td>3.2019</td><td>5.7892</td><td>15.7573</td><td style="background-color:red">0.00383515</td><td>3.1006</td><td>3.09921 </td></tr>
<tr>
<th>Solve Time </th><td>0.261915</td><td>0.106225</td><td>0.402141</td><td style="background-color:red">1.49089</td><td>0.24888</td><td>0.443673 </td></tr>
<tr>
<th>Total Time </th><td>3.46381</td><td>5.89542</td><td>16.1594</td><td style="background-color:red">1.49473</td><td>3.34948</td><td>3.54288 </td></tr>
<tr>
<th>Error(Iter) </th><td>1.76348e-16 </td><td>4.58395e-16 </td><td>1.67982e-14 </td><td>8.64144e-11 (61) </td><td>8.5996e-12 (2) </td><td><p class="starttd">6.04042e-14 (5)</p>
<p class="endtd"></p>
</td></tr>
</table>
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