/*
 Copyright (c) 2011, Intel Corporation. All rights reserved.
 Copyright (C) 2011-2016 Gael Guennebaud <gael.guennebaud@inria.fr>

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 ********************************************************************************
 *   Content : Documentation on the use of BLAS/LAPACK libraries through Eigen
 ********************************************************************************
*/

namespace Eigen {

/** \page TopicUsingBlasLapack Using BLAS/LAPACK from %Eigen


Since %Eigen version 3.3 and later, any F77 compatible BLAS or LAPACK libraries can be used as backends for dense matrix products and dense matrix decompositions.
For instance, one can use <a href="http://eigen.tuxfamily.org/Counter/redirect_to_mkl.php">Intel® MKL</a>, Apple's Accelerate framework on OSX, <a href="http://www.openblas.net/">OpenBLAS</a>, <a href="http://www.netlib.org/lapack">Netlib LAPACK</a>, etc.

Do not miss this \link TopicUsingIntelMKL page \endlink for further discussions on the specific use of Intel® MKL (also includes VML, PARDISO, etc.)

In order to use an external BLAS and/or LAPACK library, you must link you own application to the respective libraries and their dependencies.
For LAPACK, you must also link to the standard <a href="http://www.netlib.org/lapack/lapacke.html">Lapacke</a> library, which is used as a convenient think layer between %Eigen's C++ code and LAPACK F77 interface. Then you must activate their usage by defining one or multiple of the following macros (\b before including any %Eigen's header):

\note For Mac users, in order to use the lapack version shipped with the Accelerate framework, you also need the lapacke library.
Using <a href="https://www.macports.org/">MacPorts</a>, this is as easy as:
\code
sudo port install lapack
\endcode
and then use the following link flags: \c -framework \c Accelerate \c /opt/local/lib/lapack/liblapacke.dylib

<table class="manual">
<tr><td>\c EIGEN_USE_BLAS </td><td>Enables the use of external BLAS level 2 and 3 routines (compatible with any F77 BLAS interface)</td></tr>
<tr class="alt"><td>\c EIGEN_USE_LAPACKE </td><td>Enables the use of external Lapack routines via the <a href="http://www.netlib.org/lapack/lapacke.html">Lapacke</a> C interface to Lapack (compatible with any F77 LAPACK interface)</td></tr>
<tr><td>\c EIGEN_USE_LAPACKE_STRICT </td><td>Same as \c EIGEN_USE_LAPACKE but algorithms of lower numerical robustness are disabled. \n This currently concerns only JacobiSVD which otherwise would be replaced by \c gesvd that is less robust than Jacobi rotations.</td></tr>
</table>

When doing so, a number of %Eigen's algorithms are silently substituted with calls to BLAS or LAPACK routines.
These substitutions apply only for \b Dynamic \b or \b large enough objects with one of the following four standard scalar types: \c float, \c double, \c complex<float>, and \c complex<double>.
Operations on other scalar types or mixing reals and complexes will continue to use the built-in algorithms.

The breadth of %Eigen functionality that can be substituted is listed in the table below.
<table class="manual">
<tr><th>Functional domain</th><th>Code example</th><th>BLAS/LAPACK routines</th></tr>
<tr><td>Matrix-matrix operations \n \c EIGEN_USE_BLAS </td><td>\code
m1*m2.transpose();
m1.selfadjointView<Lower>()*m2;
m1*m2.triangularView<Upper>();
m1.selfadjointView<Lower>().rankUpdate(m2,1.0);
\endcode</td><td>\code
?gemm
?symm/?hemm
?trmm
dsyrk/ssyrk
\endcode</td></tr>
<tr class="alt"><td>Matrix-vector operations \n \c EIGEN_USE_BLAS </td><td>\code
m1.adjoint()*b;
m1.selfadjointView<Lower>()*b;
m1.triangularView<Upper>()*b;
\endcode</td><td>\code
?gemv
?symv/?hemv
?trmv
\endcode</td></tr>
<tr><td>LU decomposition \n \c EIGEN_USE_LAPACKE \n \c EIGEN_USE_LAPACKE_STRICT </td><td>\code
v1 = m1.lu().solve(v2);
\endcode</td><td>\code
?getrf
\endcode</td></tr>
<tr class="alt"><td>Cholesky decomposition \n \c EIGEN_USE_LAPACKE \n \c EIGEN_USE_LAPACKE_STRICT </td><td>\code
v1 = m2.selfadjointView<Upper>().llt().solve(v2);
\endcode</td><td>\code
?potrf
\endcode</td></tr>
<tr><td>QR decomposition \n \c EIGEN_USE_LAPACKE \n \c EIGEN_USE_LAPACKE_STRICT </td><td>\code
m1.householderQr();
m1.colPivHouseholderQr();
\endcode</td><td>\code
?geqrf
?geqp3
\endcode</td></tr>
<tr class="alt"><td>Singular value decomposition \n \c EIGEN_USE_LAPACKE </td><td>\code
JacobiSVD<MatrixXd> svd;
svd.compute(m1, ComputeThinV);
\endcode</td><td>\code
?gesvd
\endcode</td></tr>
<tr><td>Eigen-value decompositions \n \c EIGEN_USE_LAPACKE \n \c EIGEN_USE_LAPACKE_STRICT </td><td>\code
EigenSolver<MatrixXd> es(m1);
ComplexEigenSolver<MatrixXcd> ces(m1);
SelfAdjointEigenSolver<MatrixXd> saes(m1+m1.transpose());
GeneralizedSelfAdjointEigenSolver<MatrixXd>
    gsaes(m1+m1.transpose(),m2+m2.transpose());
\endcode</td><td>\code
?gees
?gees
?syev/?heev
?syev/?heev,
?potrf
\endcode</td></tr>
<tr class="alt"><td>Schur decomposition \n \c EIGEN_USE_LAPACKE \n \c EIGEN_USE_LAPACKE_STRICT </td><td>\code
RealSchur<MatrixXd> schurR(m1);
ComplexSchur<MatrixXcd> schurC(m1);
\endcode</td><td>\code
?gees
\endcode</td></tr>
</table>
In the examples, m1 and m2 are dense matrices and v1 and v2 are dense vectors.

*/

}