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<h1>arnoldi.h</h1><div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">/*</span>
<a name="l00002"></a>00002 <span class="comment"> * Copyright 2008-2009 NVIDIA Corporation</span>
<a name="l00003"></a>00003 <span class="comment"> *</span>
<a name="l00004"></a>00004 <span class="comment"> * Licensed under the Apache License, Version 2.0 (the "License");</span>
<a name="l00005"></a>00005 <span class="comment"> * you may not use this file except in compliance with the License.</span>
<a name="l00006"></a>00006 <span class="comment"> * You may obtain a copy of the License at</span>
<a name="l00007"></a>00007 <span class="comment"> *</span>
<a name="l00008"></a>00008 <span class="comment"> * http://www.apache.org/licenses/LICENSE-2.0</span>
<a name="l00009"></a>00009 <span class="comment"> *</span>
<a name="l00010"></a>00010 <span class="comment"> * Unless required by applicable law or agreed to in writing, software</span>
<a name="l00011"></a>00011 <span class="comment"> * distributed under the License is distributed on an "AS IS" BASIS,</span>
<a name="l00012"></a>00012 <span class="comment"> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</span>
<a name="l00013"></a>00013 <span class="comment"> * See the License for the specific language governing permissions and</span>
<a name="l00014"></a>00014 <span class="comment"> * limitations under the License.</span>
<a name="l00015"></a>00015 <span class="comment"> */</span>
<a name="l00016"></a>00016
<a name="l00017"></a>00017
<a name="l00018"></a>00018 <span class="preprocessor">#pragma once</span>
<a name="l00019"></a>00019 <span class="preprocessor"></span>
<a name="l00020"></a>00020 <span class="preprocessor">#include <<a class="code" href="multiply_8h.html" title="Matrix multiplication.">cusp/multiply.h</a>></span>
<a name="l00021"></a>00021 <span class="preprocessor">#include <<a class="code" href="array1d_8h.html" title="One-dimensional array.">cusp/array1d.h</a>></span>
<a name="l00022"></a>00022 <span class="preprocessor">#include <cusp/detail/random.h></span>
<a name="l00023"></a>00023
<a name="l00024"></a>00024 <span class="keyword">namespace </span>cusp
<a name="l00025"></a>00025 {
<a name="l00026"></a>00026 <span class="keyword">namespace </span>krylov
<a name="l00027"></a>00027 {
<a name="l00028"></a>00028
<a name="l00029"></a>00029 <span class="keyword">template</span> <<span class="keyword">typename</span> Matrix, <span class="keyword">typename</span> Array2d>
<a name="l00030"></a>00030 <span class="keywordtype">void</span> lanczos(<span class="keyword">const</span> Matrix& A, Array2d& H, <span class="keywordtype">size_t</span> k = 10)
<a name="l00031"></a>00031 {
<a name="l00032"></a>00032 <span class="keyword">typedef</span> <span class="keyword">typename</span> Matrix::value_type ValueType;
<a name="l00033"></a>00033 <span class="keyword">typedef</span> <span class="keyword">typename</span> Matrix::memory_space MemorySpace;
<a name="l00034"></a>00034
<a name="l00035"></a>00035 <span class="keywordtype">size_t</span> N = A.num_cols;
<a name="l00036"></a>00036 <span class="keywordtype">size_t</span> maxiter = std::min(N, k);
<a name="l00037"></a>00037
<a name="l00038"></a>00038 <span class="comment">// allocate workspace</span>
<a name="l00039"></a>00039 <a class="code" href="classcusp_1_1array1d.html">cusp::array1d<ValueType,MemorySpace></a> v0(N);
<a name="l00040"></a>00040 <a class="code" href="classcusp_1_1array1d.html">cusp::array1d<ValueType,MemorySpace></a> v1(N);
<a name="l00041"></a>00041 <a class="code" href="classcusp_1_1array1d.html">cusp::array1d<ValueType,MemorySpace></a> w(N);
<a name="l00042"></a>00042
<a name="l00043"></a>00043 <span class="comment">// initialize starting vector to random values in [0,1)</span>
<a name="l00044"></a>00044 <a class="code" href="group__algorithms.html#ga74fcfc1486aa8b2311631f60c0bda938">cusp::copy</a>(cusp::detail::random_reals<ValueType>(N), v1);
<a name="l00045"></a>00045
<a name="l00046"></a>00046 <a class="code" href="group__blas.html#ga9ce2f79064a8d258884a33e98c7d79bd">cusp::blas::scal</a>(v1, ValueType(1) / cusp::blas::nrm2(v1));
<a name="l00047"></a>00047
<a name="l00048"></a>00048 Array2d H_(maxiter + 1, maxiter, 0);
<a name="l00049"></a>00049
<a name="l00050"></a>00050 ValueType alpha = 0.0, beta = 0.0;
<a name="l00051"></a>00051
<a name="l00052"></a>00052 <span class="keywordtype">size_t</span> j;
<a name="l00053"></a>00053
<a name="l00054"></a>00054 <span class="keywordflow">for</span>(j = 0; j < maxiter; j++)
<a name="l00055"></a>00055 {
<a name="l00056"></a>00056 <a class="code" href="group__algorithms.html#ga9ebf00a9f894c28ef6cbf01dabd475d1">cusp::multiply</a>(A, v1, w);
<a name="l00057"></a>00057
<a name="l00058"></a>00058 <span class="keywordflow">if</span>(j >= 1)
<a name="l00059"></a>00059 {
<a name="l00060"></a>00060 H_(j - 1, j) = beta;
<a name="l00061"></a>00061 <a class="code" href="group__blas.html#ga583040af87e11cd57a5101497f6dd66a">cusp::blas::axpy</a>(v0, w, -beta);
<a name="l00062"></a>00062 }
<a name="l00063"></a>00063
<a name="l00064"></a>00064 alpha = <a class="code" href="group__blas.html#ga44ee4c52ab982060c48876da74db7bde">cusp::blas::dot</a>(w, v1);
<a name="l00065"></a>00065 H_(j,j) = alpha;
<a name="l00066"></a>00066
<a name="l00067"></a>00067 <a class="code" href="group__blas.html#ga583040af87e11cd57a5101497f6dd66a">cusp::blas::axpy</a>(v1, w, -alpha);
<a name="l00068"></a>00068
<a name="l00069"></a>00069 beta = <a class="code" href="group__blas.html#gaf7ada70d0229eb01d4bd81f0f6f66809">cusp::blas::nrm2</a>(w);
<a name="l00070"></a>00070 H_(j + 1, j) = beta;
<a name="l00071"></a>00071
<a name="l00072"></a>00072 <span class="keywordflow">if</span>(beta < 1e-10) <span class="keywordflow">break</span>;
<a name="l00073"></a>00073
<a name="l00074"></a>00074 <a class="code" href="group__blas.html#ga9ce2f79064a8d258884a33e98c7d79bd">cusp::blas::scal</a>(w, ValueType(1) / beta);
<a name="l00075"></a>00075
<a name="l00076"></a>00076 <span class="comment">// [v0 v1 w] - > [v1 w v0]</span>
<a name="l00077"></a>00077 v0.swap(v1);
<a name="l00078"></a>00078 v1.swap(w);
<a name="l00079"></a>00079 }
<a name="l00080"></a>00080
<a name="l00081"></a>00081 H.resize(j,j);
<a name="l00082"></a>00082 <span class="keywordflow">for</span>(<span class="keywordtype">size_t</span> row = 0; row < j; row++)
<a name="l00083"></a>00083 <span class="keywordflow">for</span>(<span class="keywordtype">size_t</span> col = 0; col < j; col++)
<a name="l00084"></a>00084 H(row,col) = H_(row,col);
<a name="l00085"></a>00085 }
<a name="l00086"></a>00086
<a name="l00087"></a>00087 <span class="keyword">template</span> <<span class="keyword">typename</span> Matrix, <span class="keyword">typename</span> Array2d>
<a name="l00088"></a>00088 <span class="keywordtype">void</span> arnoldi(<span class="keyword">const</span> Matrix& A, Array2d& H, <span class="keywordtype">size_t</span> k = 10)
<a name="l00089"></a>00089 {
<a name="l00090"></a>00090 <span class="keyword">typedef</span> <span class="keyword">typename</span> Matrix::value_type ValueType;
<a name="l00091"></a>00091 <span class="keyword">typedef</span> <span class="keyword">typename</span> Matrix::memory_space MemorySpace;
<a name="l00092"></a>00092
<a name="l00093"></a>00093 <span class="keywordtype">size_t</span> N = A.num_rows;
<a name="l00094"></a>00094
<a name="l00095"></a>00095 <span class="keywordtype">size_t</span> maxiter = std::min(N, k);
<a name="l00096"></a>00096
<a name="l00097"></a>00097 Array2d H_(maxiter + 1, maxiter, 0);
<a name="l00098"></a>00098
<a name="l00099"></a>00099 <span class="comment">// allocate workspace of k + 1 vectors</span>
<a name="l00100"></a>00100 std::vector< cusp::array1d<ValueType,MemorySpace> > V(maxiter + 1);
<a name="l00101"></a>00101 <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> i = 0; i < maxiter + 1; i++)
<a name="l00102"></a>00102 V[i].resize(N);
<a name="l00103"></a>00103
<a name="l00104"></a>00104 <span class="comment">// initialize starting vector to random values in [0,1)</span>
<a name="l00105"></a>00105 <a class="code" href="group__algorithms.html#ga74fcfc1486aa8b2311631f60c0bda938">cusp::copy</a>(cusp::detail::random_reals<ValueType>(N), V[0]);
<a name="l00106"></a>00106
<a name="l00107"></a>00107 <span class="comment">// normalize v0</span>
<a name="l00108"></a>00108 <a class="code" href="group__blas.html#ga9ce2f79064a8d258884a33e98c7d79bd">cusp::blas::scal</a>(V[0], ValueType(1) / cusp::blas::nrm2(V[0]));
<a name="l00109"></a>00109
<a name="l00110"></a>00110 <span class="keywordtype">size_t</span> j;
<a name="l00111"></a>00111
<a name="l00112"></a>00112 <span class="keywordflow">for</span>(j = 0; j < maxiter; j++)
<a name="l00113"></a>00113 {
<a name="l00114"></a>00114 <a class="code" href="group__algorithms.html#ga9ebf00a9f894c28ef6cbf01dabd475d1">cusp::multiply</a>(A, V[j], V[j + 1]);
<a name="l00115"></a>00115
<a name="l00116"></a>00116 <span class="keywordflow">for</span>(<span class="keywordtype">size_t</span> i = 0; i <= j; i++)
<a name="l00117"></a>00117 {
<a name="l00118"></a>00118 H_(i,j) = <a class="code" href="group__blas.html#ga44ee4c52ab982060c48876da74db7bde">cusp::blas::dot</a>(V[i], V[j + 1]);
<a name="l00119"></a>00119
<a name="l00120"></a>00120 <a class="code" href="group__blas.html#ga583040af87e11cd57a5101497f6dd66a">cusp::blas::axpy</a>(V[i], V[j + 1], -H_(i,j));
<a name="l00121"></a>00121 }
<a name="l00122"></a>00122
<a name="l00123"></a>00123 H_(j+1,j) = <a class="code" href="group__blas.html#gaf7ada70d0229eb01d4bd81f0f6f66809">cusp::blas::nrm2</a>(V[j + 1]);
<a name="l00124"></a>00124
<a name="l00125"></a>00125 <span class="keywordflow">if</span>(H_(j+1,j) < 1e-10) <span class="keywordflow">break</span>;
<a name="l00126"></a>00126
<a name="l00127"></a>00127 <a class="code" href="group__blas.html#ga9ce2f79064a8d258884a33e98c7d79bd">cusp::blas::scal</a>(V[j + 1], ValueType(1) / H_(j+1,j));
<a name="l00128"></a>00128 }
<a name="l00129"></a>00129
<a name="l00130"></a>00130 H.resize(j,j);
<a name="l00131"></a>00131 <span class="keywordflow">for</span>( <span class="keywordtype">size_t</span> row = 0; row < j; row++ )
<a name="l00132"></a>00132 <span class="keywordflow">for</span>( <span class="keywordtype">size_t</span> col = 0; col < j; col++ )
<a name="l00133"></a>00133 H(row,col) = H_(row,col);
<a name="l00134"></a>00134 }
<a name="l00135"></a>00135
<a name="l00136"></a>00136 } <span class="comment">// end namespace krylov</span>
<a name="l00137"></a>00137 } <span class="comment">// end namespace cusp</span>
<a name="l00138"></a>00138
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