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<div class="title">lanczos.cpp</div> </div>
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<p>This tutorial shows how to calculate the largest eigenvalues of a matrix using Lanczos' method.</p>
<p>The Lanczos method is particularly attractive for use with large, sparse matrices, since the only requirement on the matrix is to provide a matrix-vector product. Although less common, the method is sometimes also used with dense matrices.</p>
<p>We start with including the necessary headers: </p>
<div class="fragment"><div class="line"><span class="comment">// include necessary system headers</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="comment">//include basic scalar and vector types of ViennaCL</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="scalar_8hpp.html">viennacl/scalar.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="vector_8hpp.html">viennacl/vector.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix_8hpp.html">viennacl/matrix.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix__proxy_8hpp.html">viennacl/matrix_proxy.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="compressed__matrix_8hpp.html">viennacl/compressed_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="lanczos_8hpp.html">viennacl/linalg/lanczos.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix__market_8hpp.html">viennacl/io/matrix_market.hpp</a>"</span></div>
<div class="line"></div>
<div class="line"><span class="comment">// Some helper functions for this tutorial:</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <string></span></div>
<div class="line"><span class="preprocessor">#include <iomanip></span></div>
</div><!-- fragment --><p> We read a sparse matrix from a matrix-market file, then run the Lanczos method. Finally, the computed eigenvalues are printed. </p>
<div class="fragment"><div class="line"><span class="keywordtype">int</span> <a name="a0"></a><a class="code" href="tests_2src_2bisect_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a>()</div>
<div class="line">{</div>
<div class="line"> <span class="comment">// If you GPU does not support double precision, use `float` instead of `double`:</span></div>
<div class="line"> <span class="keyword">typedef</span> <span class="keywordtype">double</span> <a name="a1"></a><a class="code" href="fft__1d_8cpp.html#ad5c19ca4f47d3f8ec21232a5af2624e5">ScalarType</a>;</div>
</div><!-- fragment --><p> Create the sparse matrix and read data from a Matrix-Market file: </p>
<div class="fragment"><div class="line">std::vector< std::map<unsigned int, ScalarType> > host_A;</div>
<div class="line"><span class="keywordflow">if</span> (!<a name="a2"></a><a class="code" href="namespaceviennacl_1_1io.html#ad934125ed3dbe661e264bcd7d62b1048">viennacl::io::read_matrix_market_file</a>(host_A, <span class="stringliteral">"../examples/testdata/mat65k.mtx"</span>))</div>
<div class="line">{</div>
<div class="line"> std::cout << <span class="stringliteral">"Error reading Matrix file"</span> << std::endl;</div>
<div class="line"> <span class="keywordflow">return</span> EXIT_FAILURE;</div>
<div class="line">}</div>
<div class="line"></div>
<div class="line"><a name="_a3"></a><a class="code" href="classviennacl_1_1compressed__matrix.html">viennacl::compressed_matrix<ScalarType></a> A;</div>
<div class="line"><a name="a4"></a><a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(host_A, A);</div>
</div><!-- fragment --><p> Create the configuration for the Lanczos method. All constructor arguments are optional, so feel free to use default settings. </p>
<div class="fragment"><div class="line"><a name="_a5"></a><a class="code" href="classviennacl_1_1linalg_1_1lanczos__tag.html">viennacl::linalg::lanczos_tag</a> ltag(0.75, <span class="comment">// Select a power of 0.75 as the tolerance for the machine precision.</span></div>
<div class="line"> 10, <span class="comment">// Compute (approximations to) the 10 largest eigenvalues</span></div>
<div class="line"> <a name="a6"></a><a class="code" href="classviennacl_1_1linalg_1_1lanczos__tag.html#a8a743b9f474124a0f779ed35c6f6a684a937ee8052e51309672b2bcdba4bb015e">viennacl::linalg::lanczos_tag::partial_reorthogonalization</a>, <span class="comment">// use partial reorthogonalization</span></div>
<div class="line"> 30); <span class="comment">// Maximum size of the Krylov space</span></div>
</div><!-- fragment --><p> Run the Lanczos method for computing eigenvalues by passing the tag to the routine <a class="el" href="namespaceviennacl_1_1linalg.html#af5002a88fa4cc3fbe65a904797a0ecba" title="Implementation of the calculation of eigenvalues using lanczos (with and without reorthogonalization)...">viennacl::linalg::eig()</a>. </p>
<div class="fragment"><div class="line">std::cout << <span class="stringliteral">"Running Lanczos algorithm (eigenvalues only). This might take a while..."</span> << std::endl;</div>
<div class="line">std::vector<ScalarType> lanczos_eigenvalues = <a name="a7"></a><a class="code" href="namespaceviennacl_1_1linalg.html#af5002a88fa4cc3fbe65a904797a0ecba">viennacl::linalg::eig</a>(A, ltag);</div>
</div><!-- fragment --><p> Re-run the Lanczos method, this time also computing eigenvectors. To do so, we pass a dense matrix for which each column will ultimately hold the computed eigenvectors. </p>
<div class="fragment"><div class="line">std::cout << <span class="stringliteral">"Running Lanczos algorithm (with eigenvectors). This might take a while..."</span> << std::endl;</div>
<div class="line"><a name="_a8"></a><a class="code" href="classviennacl_1_1matrix.html">viennacl::matrix<ScalarType></a> approx_eigenvectors_A(A.<a name="a9"></a><a class="code" href="classviennacl_1_1compressed__matrix.html#a463cf1739f9cdd387aa185cb574db183">size1</a>(), ltag.num_eigenvalues());</div>
<div class="line">lanczos_eigenvalues = <a class="code" href="namespaceviennacl_1_1linalg.html#af5002a88fa4cc3fbe65a904797a0ecba">viennacl::linalg::eig</a>(A, approx_eigenvectors_A, ltag);</div>
</div><!-- fragment --><p> Print the computed eigenvalues and exit: </p>
<div class="fragment"><div class="line"> <span class="keywordflow">for</span> (std::size_t i = 0; i< lanczos_eigenvalues.size(); i++)</div>
<div class="line"> {</div>
<div class="line"> std::cout << <span class="stringliteral">"Approx. eigenvalue "</span> << std::setprecision(7) << lanczos_eigenvalues[i];</div>
<div class="line"></div>
<div class="line"> <span class="comment">// test approximated eigenvector by computing A*v:</span></div>
<div class="line"> <a name="_a10"></a><a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> approx_eigenvector = <a name="a11"></a><a class="code" href="namespaceviennacl.html#a7fca08f4a83edffe7f47666d298ca87d">viennacl::column</a>(approx_eigenvectors_A, static_cast<unsigned int>(i));</div>
<div class="line"> <a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> Aq = <a name="a12"></a><a class="code" href="namespaceviennacl_1_1linalg.html#aa18d10f8a90e38bd9ff43c650fc670ef">viennacl::linalg::prod</a>(A, approx_eigenvector);</div>
<div class="line"> std::cout << <span class="stringliteral">" ("</span> << <a name="a13"></a><a class="code" href="namespaceviennacl_1_1linalg.html#ab35950c4374eb3be08a03d852508c01a">viennacl::linalg::inner_prod</a>(Aq, approx_eigenvector) << <span class="stringliteral">" for <Av,v> with approx. eigenvector v)"</span> << std::endl;</div>
<div class="line"> }</div>
<div class="line"></div>
<div class="line"> <span class="keywordflow">return</span> EXIT_SUCCESS;</div>
<div class="line">}</div>
</div><!-- fragment --> <h2>Full Example Code</h2>
<div class="fragment"><div class="line"><span class="comment">/* =========================================================================</span></div>
<div class="line"><span class="comment"> Copyright (c) 2010-2016, Institute for Microelectronics,</span></div>
<div class="line"><span class="comment"> Institute for Analysis and Scientific Computing,</span></div>
<div class="line"><span class="comment"> TU Wien.</span></div>
<div class="line"><span class="comment"> Portions of this software are copyright by UChicago Argonne, LLC.</span></div>
<div class="line"><span class="comment"></span></div>
<div class="line"><span class="comment"> -----------------</span></div>
<div class="line"><span class="comment"> ViennaCL - The Vienna Computing Library</span></div>
<div class="line"><span class="comment"> -----------------</span></div>
<div class="line"><span class="comment"></span></div>
<div class="line"><span class="comment"> Project Head: Karl Rupp rupp@iue.tuwien.ac.at</span></div>
<div class="line"><span class="comment"></span></div>
<div class="line"><span class="comment"> (A list of authors and contributors can be found in the PDF manual)</span></div>
<div class="line"><span class="comment"></span></div>
<div class="line"><span class="comment"> License: MIT (X11), see file LICENSE in the base directory</span></div>
<div class="line"><span class="comment">============================================================================= */</span></div>
<div class="line"></div>
<div class="line"><span class="comment">// include necessary system headers</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="comment">//include basic scalar and vector types of ViennaCL</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="scalar_8hpp.html">viennacl/scalar.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="vector_8hpp.html">viennacl/vector.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix_8hpp.html">viennacl/matrix.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix__proxy_8hpp.html">viennacl/matrix_proxy.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="compressed__matrix_8hpp.html">viennacl/compressed_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="lanczos_8hpp.html">viennacl/linalg/lanczos.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="matrix__market_8hpp.html">viennacl/io/matrix_market.hpp</a>"</span></div>
<div class="line"></div>
<div class="line"><span class="comment">// Some helper functions for this tutorial:</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <string></span></div>
<div class="line"><span class="preprocessor">#include <iomanip></span></div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> <a class="code" href="tests_2src_2bisect_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a>()</div>
<div class="line">{</div>
<div class="line"> <span class="comment">// If you GPU does not support double precision, use `float` instead of `double`:</span></div>
<div class="line"> <span class="keyword">typedef</span> <span class="keywordtype">double</span> <a class="code" href="fft__1d_8cpp.html#ad5c19ca4f47d3f8ec21232a5af2624e5">ScalarType</a>;</div>
<div class="line"></div>
<div class="line"> std::vector< std::map<unsigned int, ScalarType> > host_A;</div>
<div class="line"> <span class="keywordflow">if</span> (!<a class="code" href="namespaceviennacl_1_1io.html#ad934125ed3dbe661e264bcd7d62b1048">viennacl::io::read_matrix_market_file</a>(host_A, <span class="stringliteral">"../examples/testdata/mat65k.mtx"</span>))</div>
<div class="line"> {</div>
<div class="line"> std::cout << <span class="stringliteral">"Error reading Matrix file"</span> << std::endl;</div>
<div class="line"> <span class="keywordflow">return</span> EXIT_FAILURE;</div>
<div class="line"> }</div>
<div class="line"></div>
<div class="line"> <a class="code" href="classviennacl_1_1compressed__matrix.html">viennacl::compressed_matrix<ScalarType></a> A;</div>
<div class="line"> <a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(host_A, A);</div>
<div class="line"></div>
<div class="line"> <a class="code" href="classviennacl_1_1linalg_1_1lanczos__tag.html">viennacl::linalg::lanczos_tag</a> ltag(0.75, <span class="comment">// Select a power of 0.75 as the tolerance for the machine precision.</span></div>
<div class="line"> 10, <span class="comment">// Compute (approximations to) the 10 largest eigenvalues</span></div>
<div class="line"> <a class="code" href="classviennacl_1_1linalg_1_1lanczos__tag.html#a8a743b9f474124a0f779ed35c6f6a684a937ee8052e51309672b2bcdba4bb015e">viennacl::linalg::lanczos_tag::partial_reorthogonalization</a>, <span class="comment">// use partial reorthogonalization</span></div>
<div class="line"> 30); <span class="comment">// Maximum size of the Krylov space</span></div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"Running Lanczos algorithm (eigenvalues only). This might take a while..."</span> << std::endl;</div>
<div class="line"> std::vector<ScalarType> lanczos_eigenvalues = <a class="code" href="namespaceviennacl_1_1linalg.html#af5002a88fa4cc3fbe65a904797a0ecba">viennacl::linalg::eig</a>(A, ltag);</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"Running Lanczos algorithm (with eigenvectors). This might take a while..."</span> << std::endl;</div>
<div class="line"> <a class="code" href="classviennacl_1_1matrix.html">viennacl::matrix<ScalarType></a> approx_eigenvectors_A(A.<a class="code" href="classviennacl_1_1compressed__matrix.html#a463cf1739f9cdd387aa185cb574db183">size1</a>(), ltag.<a name="a14"></a><a class="code" href="classviennacl_1_1linalg_1_1lanczos__tag.html#ae744c43467774ee300f5fab0c607aed1">num_eigenvalues</a>());</div>
<div class="line"> lanczos_eigenvalues = <a class="code" href="namespaceviennacl_1_1linalg.html#af5002a88fa4cc3fbe65a904797a0ecba">viennacl::linalg::eig</a>(A, approx_eigenvectors_A, ltag);</div>
<div class="line"></div>
<div class="line"> <span class="keywordflow">for</span> (std::size_t i = 0; i< lanczos_eigenvalues.size(); i++)</div>
<div class="line"> {</div>
<div class="line"> std::cout << <span class="stringliteral">"Approx. eigenvalue "</span> << std::setprecision(7) << lanczos_eigenvalues[i];</div>
<div class="line"></div>
<div class="line"> <span class="comment">// test approximated eigenvector by computing A*v:</span></div>
<div class="line"> <a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> approx_eigenvector = <a class="code" href="namespaceviennacl.html#a7fca08f4a83edffe7f47666d298ca87d">viennacl::column</a>(approx_eigenvectors_A, static_cast<unsigned int>(i));</div>
<div class="line"> <a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> Aq = <a class="code" href="namespaceviennacl_1_1linalg.html#aa18d10f8a90e38bd9ff43c650fc670ef">viennacl::linalg::prod</a>(A, approx_eigenvector);</div>
<div class="line"> std::cout << <span class="stringliteral">" ("</span> << <a class="code" href="namespaceviennacl_1_1linalg.html#ab35950c4374eb3be08a03d852508c01a">viennacl::linalg::inner_prod</a>(Aq, approx_eigenvector) << <span class="stringliteral">" for <Av,v> with approx. eigenvector v)"</span> << std::endl;</div>
<div class="line"> }</div>
<div class="line"></div>
<div class="line"> <span class="keywordflow">return</span> EXIT_SUCCESS;</div>
<div class="line">}</div>
<div class="line"></div>
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