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 <span id="projectnumber">1.7.1</span>
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<div id="projectbrief">Free open-source GPU-accelerated linear algebra and solver library.</div>
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<div class="title">least-squares.cpp</div> </div>
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<p>This tutorial shows how least Squares problems for matrices from ViennaCL or Boost.uBLAS can be solved solved.</p>
<p>We start with including the respective header files: </p>
<div class="fragment"><div class="line"><span class="comment">// activate ublas support in ViennaCL</span></div>
<div class="line"><span class="preprocessor">#define VIENNACL_WITH_UBLAS</span></div>
<div class="line"></div>
<div class="line"><span class="comment">//</span></div>
<div class="line"><span class="comment">// include necessary system headers</span></div>
<div class="line"><span class="comment">//</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="comment">// Boost headers</span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/triangular.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/vector.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/vector_proxy.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/matrix.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/matrix_proxy.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/lu.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/io.hpp></span></div>
<div class="line"></div>
<div class="line"></div>
<div class="line"><span class="comment">// ViennaCL headers</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="qr_8hpp.html">viennacl/linalg/qr.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="lu_8hpp.html">viennacl/linalg/lu.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="direct__solve_8hpp.html">viennacl/linalg/direct_solve.hpp</a>"</span></div>
</div><!-- fragment --><p> The minimization problem of finding x such that <img class="formulaInl" alt="$ \Vert Ax - b \Vert $" src="form_112.png"/> is solved as follows:</p>
<ul>
<li>Compute the QR-factorization of A = QR.</li>
<li>Compute <img class="formulaInl" alt="$ b' = Q^{\mathrm{T}} b $" src="form_113.png"/> for the equivalent minimization problem <img class="formulaInl" alt="$ \Vert Rx - Q^{\mathrm{T}} b $" src="form_114.png"/>.</li>
<li>Solve the triangular system <img class="formulaInl" alt="$ \tilde{R} x = b' $" src="form_115.png"/>, where <img class="formulaInl" alt="$ \tilde{R} $" src="form_116.png"/> is the upper square matrix of R.</li>
</ul>
<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> (<span class="keywordtype">int</span>, <span class="keyword">const</span> <span class="keywordtype">char</span> **)</div>
<div class="line">{</div>
<div class="line"> <span class="keyword">typedef</span> <span class="keywordtype">float</span> <a name="a1"></a><a class="code" href="fft__1d_8cpp.html#ad5c19ca4f47d3f8ec21232a5af2624e5">ScalarType</a>; <span class="comment">//feel free to change this to 'double' if supported by your hardware</span></div>
<div class="line"></div>
<div class="line"> <span class="keyword">typedef</span> boost::numeric::ublas::matrix<ScalarType> MatrixType;</div>
<div class="line"> <span class="keyword">typedef</span> boost::numeric::ublas::vector<ScalarType> VectorType;</div>
<div class="line"> <span class="keyword">typedef</span> <a name="_a2"></a><a class="code" href="classviennacl_1_1matrix.html">viennacl::matrix<ScalarType, viennacl::column_major></a> VCLMatrixType;</div>
<div class="line"> <span class="keyword">typedef</span> <a name="_a3"></a><a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> VCLVectorType;</div>
</div><!-- fragment --><p> Create vectors and matrices with data: </p>
<div class="fragment"><div class="line">VectorType ublas_b(4);</div>
<div class="line">ublas_b(0) = -4;</div>
<div class="line">ublas_b(1) = 2;</div>
<div class="line">ublas_b(2) = 5;</div>
<div class="line">ublas_b(3) = -1;</div>
<div class="line"></div>
<div class="line">MatrixType ublas_A(4, 3);</div>
<div class="line"></div>
<div class="line">ublas_A(0, 0) = 2; ublas_A(0, 1) = -1; ublas_A(0, 2) = 1;</div>
<div class="line">ublas_A(1, 0) = 1; ublas_A(1, 1) = -5; ublas_A(1, 2) = 2;</div>
<div class="line">ublas_A(2, 0) = -3; ublas_A(2, 1) = 1; ublas_A(2, 2) = -4;</div>
<div class="line">ublas_A(3, 0) = 1; ublas_A(3, 1) = -1; ublas_A(3, 2) = 1;</div>
</div><!-- fragment --><p> Setup the matrix and vector with ViennaCL objects and copy the data from the uBLAS objects: </p>
<div class="fragment"><div class="line">VCLVectorType <a name="a4"></a><a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>(ublas_b.size());</div>
<div class="line">VCLMatrixType vcl_A(ublas_A.size1(), ublas_A.size2());</div>
<div class="line"></div>
<div class="line"><a name="a5"></a><a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(ublas_b, <a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>);</div>
<div class="line"><a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(ublas_A, vcl_A);</div>
</div><!-- fragment --> <h2>Option 1: Using Boost.uBLAS</h2>
<p>The implementation in ViennaCL accepts both uBLAS and ViennaCL types. We start with a single-threaded implementation using Boost.uBLAS. </p>
<div class="fragment"><div class="line">std::cout << <span class="stringliteral">"--- Boost.uBLAS ---"</span> << std::endl;</div>
</div><!-- fragment --><p> The first (and computationally most expensive) step is to compute the QR factorization of A. Since we do not need A later, we directly overwrite A with the householder reflectors and the upper triangular matrix R. The returned vector holds the scalar coefficients (betas) for the Householder reflections <img class="formulaInl" alt="$ I - \beta v v^{\mathrm{T}} $" src="form_117.png"/> </p>
<div class="fragment"><div class="line">std::vector<ScalarType> ublas_betas = <a name="a6"></a><a class="code" href="namespaceviennacl_1_1linalg.html#a83f573386903bae3a5379ab4cbfc5e2d">viennacl::linalg::inplace_qr</a>(ublas_A);</div>
</div><!-- fragment --><p> Compute the modified RHS of the minimization problem from the QR factorization, but do not form <img class="formulaInl" alt="$ Q^{\mathrm{T}} $" src="form_118.png"/> explicitly: b' := Q^T b </p>
<div class="fragment"><div class="line"><a name="a7"></a><a class="code" href="namespaceviennacl_1_1linalg.html#adf77bb0fe92ef09522d0458ee9df03db">viennacl::linalg::inplace_qr_apply_trans_Q</a>(ublas_A, ublas_betas, ublas_b);</div>
</div><!-- fragment --><p> Final step: triangular solve: Rx = b'', where b'' are the first three entries in b' We only need the upper left square part of A, which defines the upper triangular matrix R </p>
<div class="fragment"><div class="line"><a name="a8"></a><a class="code" href="namespaceviennacl.html#ae92c62d9fd59870c1f6b881e391d32aa">boost::numeric::ublas::range</a> ublas_range(0, 3);</div>
<div class="line">boost::numeric::ublas::matrix_range<MatrixType> ublas_R(ublas_A, ublas_range, ublas_range);</div>
<div class="line">boost::numeric::ublas::vector_range<VectorType> ublas_b2(ublas_b, ublas_range);</div>
<div class="line"><a name="a9"></a><a class="code" href="namespaceviennacl_1_1linalg_1_1cuda.html#acb0b34e3ac22c72081eceaabeab80f22">boost::numeric::ublas::inplace_solve</a>(ublas_R, ublas_b2, boost::numeric::ublas::upper_tag());</div>
<div class="line"></div>
<div class="line">std::cout << <span class="stringliteral">"Result: "</span> << ublas_b2 << std::endl;</div>
</div><!-- fragment --> <h2>Option 2: Use ViennaCL types</h2>
<p>ViennaCL is used for the computationally intensive BLAS 3 computations. Boost.uBLAS is used for the panel factorization on the host (CPU).</p>
<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">// activate ublas support in ViennaCL</span></div>
<div class="line"><span class="preprocessor">#define VIENNACL_WITH_UBLAS</span></div>
<div class="line"></div>
<div class="line"><span class="comment">//</span></div>
<div class="line"><span class="comment">// include necessary system headers</span></div>
<div class="line"><span class="comment">//</span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="comment">// Boost headers</span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/triangular.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/vector.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/vector_proxy.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/matrix.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/matrix_proxy.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/lu.hpp></span></div>
<div class="line"><span class="preprocessor">#include <boost/numeric/ublas/io.hpp></span></div>
<div class="line"></div>
<div class="line"></div>
<div class="line"><span class="comment">// ViennaCL headers</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="qr_8hpp.html">viennacl/linalg/qr.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="lu_8hpp.html">viennacl/linalg/lu.hpp</a>"</span></div>
<div class="line"><span class="preprocessor">#include "<a class="code" href="direct__solve_8hpp.html">viennacl/linalg/direct_solve.hpp</a>"</span></div>
<div class="line"></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> (<span class="keywordtype">int</span>, <span class="keyword">const</span> <span class="keywordtype">char</span> **)</div>
<div class="line">{</div>
<div class="line"> <span class="keyword">typedef</span> <span class="keywordtype">float</span> <a class="code" href="fft__1d_8cpp.html#ad5c19ca4f47d3f8ec21232a5af2624e5">ScalarType</a>; <span class="comment">//feel free to change this to 'double' if supported by your hardware</span></div>
<div class="line"></div>
<div class="line"> <span class="keyword">typedef</span> boost::numeric::ublas::matrix<ScalarType> MatrixType;</div>
<div class="line"> <span class="keyword">typedef</span> boost::numeric::ublas::vector<ScalarType> VectorType;</div>
<div class="line"> <span class="keyword">typedef</span> <a class="code" href="classviennacl_1_1matrix.html">viennacl::matrix<ScalarType, viennacl::column_major></a> VCLMatrixType;</div>
<div class="line"> <span class="keyword">typedef</span> <a class="code" href="classviennacl_1_1vector.html">viennacl::vector<ScalarType></a> VCLVectorType;</div>
<div class="line"></div>
<div class="line"> VectorType ublas_b(4);</div>
<div class="line"> ublas_b(0) = -4;</div>
<div class="line"> ublas_b(1) = 2;</div>
<div class="line"> ublas_b(2) = 5;</div>
<div class="line"> ublas_b(3) = -1;</div>
<div class="line"></div>
<div class="line"> MatrixType ublas_A(4, 3);</div>
<div class="line"></div>
<div class="line"> ublas_A(0, 0) = 2; ublas_A(0, 1) = -1; ublas_A(0, 2) = 1;</div>
<div class="line"> ublas_A(1, 0) = 1; ublas_A(1, 1) = -5; ublas_A(1, 2) = 2;</div>
<div class="line"> ublas_A(2, 0) = -3; ublas_A(2, 1) = 1; ublas_A(2, 2) = -4;</div>
<div class="line"> ublas_A(3, 0) = 1; ublas_A(3, 1) = -1; ublas_A(3, 2) = 1;</div>
<div class="line"></div>
<div class="line"> VCLVectorType <a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>(ublas_b.size());</div>
<div class="line"> VCLMatrixType vcl_A(ublas_A.size1(), ublas_A.size2());</div>
<div class="line"></div>
<div class="line"> <a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(ublas_b, <a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>);</div>
<div class="line"> <a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(ublas_A, vcl_A);</div>
<div class="line"></div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"--- Boost.uBLAS ---"</span> << std::endl;</div>
<div class="line"> std::vector<ScalarType> ublas_betas = <a class="code" href="namespaceviennacl_1_1linalg.html#a83f573386903bae3a5379ab4cbfc5e2d">viennacl::linalg::inplace_qr</a>(ublas_A);</div>
<div class="line"></div>
<div class="line"> <a class="code" href="namespaceviennacl_1_1linalg.html#adf77bb0fe92ef09522d0458ee9df03db">viennacl::linalg::inplace_qr_apply_trans_Q</a>(ublas_A, ublas_betas, ublas_b);</div>
<div class="line"></div>
<div class="line"> <a class="code" href="namespaceviennacl.html#ae92c62d9fd59870c1f6b881e391d32aa">boost::numeric::ublas::range</a> ublas_range(0, 3);</div>
<div class="line"> boost::numeric::ublas::matrix_range<MatrixType> ublas_R(ublas_A, ublas_range, ublas_range);</div>
<div class="line"> boost::numeric::ublas::vector_range<VectorType> ublas_b2(ublas_b, ublas_range);</div>
<div class="line"> <a class="code" href="namespaceviennacl_1_1linalg_1_1cuda.html#acb0b34e3ac22c72081eceaabeab80f22">boost::numeric::ublas::inplace_solve</a>(ublas_R, ublas_b2, boost::numeric::ublas::upper_tag());</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"Result: "</span> << ublas_b2 << std::endl;</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"--- ViennaCL (hybrid implementation) ---"</span> << std::endl;</div>
<div class="line"> std::vector<ScalarType> <a name="a10"></a><a class="code" href="least-squares_8cpp.html#ac887df28e24b43a0bda14bd218659c78">hybrid_betas</a> = <a class="code" href="namespaceviennacl_1_1linalg.html#a83f573386903bae3a5379ab4cbfc5e2d">viennacl::linalg::inplace_qr</a>(vcl_A);</div>
<div class="line"></div>
<div class="line"> <a class="code" href="namespaceviennacl_1_1linalg.html#adf77bb0fe92ef09522d0458ee9df03db">viennacl::linalg::inplace_qr_apply_trans_Q</a>(vcl_A, hybrid_betas, <a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>);</div>
<div class="line"></div>
<div class="line"> <a name="_a11"></a><a class="code" href="classviennacl_1_1basic__range.html">viennacl::range</a> vcl_range(0, 3);</div>
<div class="line"> <a name="_a12"></a><a class="code" href="classviennacl_1_1matrix__range.html">viennacl::matrix_range<VCLMatrixType></a> <a name="a13"></a><a class="code" href="least-squares_8cpp.html#aba586554adcf743c2cd6b09d207957f5">vcl_R</a>(vcl_A, vcl_range, vcl_range);</div>
<div class="line"> <a name="_a14"></a><a class="code" href="classviennacl_1_1vector__range.html">viennacl::vector_range<VCLVectorType></a> <a name="a15"></a><a class="code" href="least-squares_8cpp.html#aa87c3ccaa3cee96a4db6f62dff634562">vcl_b2</a>(<a class="code" href="least-squares_8cpp.html#a0aff2af0ec74ea3ddb0129b72d9b921f">vcl_b</a>, vcl_range);</div>
<div class="line"> <a name="a16"></a><a class="code" href="namespaceviennacl_1_1linalg.html#adf06f61be9418e5817eed66004f9dd2b">viennacl::linalg::inplace_solve</a>(vcl_R, vcl_b2, <a name="_a17"></a><a class="code" href="structviennacl_1_1linalg_1_1upper__tag.html">viennacl::linalg::upper_tag</a>());</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"Result: "</span> << vcl_b2 << std::endl;</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"!!!! TUTORIAL COMPLETED SUCCESSFULLY !!!!"</span> << std::endl;</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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