85 | | <a name="l00036"></a>00036 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a832d1a18bbf33ee7681adf01d5d08e21" title="Inplace symmetric multiplication by a SQUARE matrix , i.e. .">mult_sym</a> ( <span class="keyword">const</span> mat &C , <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &U ) <span class="keyword">const</span>; |
86 | | <a name="l00037"></a>00037 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a7a4f54f1ea8b802f318de77a78f6ddb0" title="Inplace symmetric multiplication by a SQUARE transpose of matrix , i.e. .">mult_sym_t</a> ( <span class="keyword">const</span> mat &C ); |
87 | | <a name="l00038"></a>00038 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a7a4f54f1ea8b802f318de77a78f6ddb0" title="Inplace symmetric multiplication by a SQUARE transpose of matrix , i.e. .">mult_sym_t</a> ( <span class="keyword">const</span> mat &C, <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &U ) <span class="keyword">const</span>; |
88 | | <a name="l00039"></a>00039 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a949ccd174ed19f9cfe36366cbd5c56a4" title="Logarithm of a determinant.">logdet</a>() <span class="keyword">const</span>; |
89 | | <a name="l00040"></a>00040 vec <a class="code" href="classbdm_1_1chmat.html#aec8d3a1cbf3b50183f246a858663c92d" title="Multiplies square root of by vector .">sqrt_mult</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
90 | | <a name="l00041"></a>00041 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a6e0aee9691909602d341fe5ebd1f1ca8" title="Evaluates quadratic form ;.">qform</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
91 | | <a name="l00042"></a>00042 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a3ac8d00a14a22a21760a8374fe5c5757" title="Evaluates quadratic form ;.">invqform</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
92 | | <a name="l00043"></a>00043 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#abc753cb54aa8946ce41d95e82fc148b5" title="Clearing matrix so that it corresponds to zeros.">clear</a>(); |
93 | | <a name="l00045"></a><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7">00045</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2, <span class="keywordtype">double</span> w = 1.0 ) { |
94 | | <a name="l00046"></a>00046 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> == A2.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, <span class="stringliteral">"Matrices of unequal dimension"</span> ); |
95 | | <a name="l00047"></a>00047 mat pre = concat_vertical ( <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>, sqrt ( w ) * A2.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ); |
96 | | <a name="l00048"></a>00048 mat post = zeros ( pre.rows(), pre.cols() ); |
97 | | <a name="l00049"></a>00049 <span class="keywordflow">if</span> ( !qr ( pre, post ) ) { |
98 | | <a name="l00050"></a>00050 <a class="code" href="bdmerror_8h.html#a59c5a63b3878ebcd6cc952fc2b47191c" title="Display a warning message.">bdm_warning</a> ( <span class="stringliteral">"Unstable QR in chmat add"</span> ); |
99 | | <a name="l00051"></a>00051 } |
100 | | <a name="l00052"></a>00052 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = post ( 0, <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> - 1, 0, <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> - 1 ); |
101 | | <a name="l00053"></a>00053 }; |
102 | | <a name="l00055"></a><a class="code" href="classbdm_1_1chmat.html#acbf3389db96dff41fb2e9532d59b13c0">00055</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#acbf3389db96dff41fb2e9532d59b13c0" title="Inversion in the same form, i.e. cholesky.">inv</a> ( <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &Inv )<span class="keyword"> const </span>{ |
103 | | <a name="l00056"></a>00056 ( Inv.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ) = itpp::inv ( <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ).T(); |
104 | | <a name="l00057"></a>00057 }; <span class="comment">//Fixme: can be more efficient</span> |
105 | | <a name="l00058"></a>00058 ; |
106 | | <a name="l00059"></a>00059 <span class="comment">// void inv ( mat &Inv );</span> |
107 | | <a name="l00060"></a>00060 |
108 | | <a name="l00062"></a><a class="code" href="classbdm_1_1chmat.html#aa888e866a163bffe94b51e58af65901b">00062</a> <span class="keyword">virtual</span> <a class="code" href="classbdm_1_1chmat.html#aa888e866a163bffe94b51e58af65901b" title="Destructor for future use;.">~chmat</a>() {}; |
109 | | <a name="l00064"></a>00064 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> ( ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> (), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( ) {}; |
110 | | <a name="l00066"></a><a class="code" href="classbdm_1_1chmat.html#a947c52c64022ef364d8ef17c94a65ef1">00066</a> <a class="code" href="classbdm_1_1chmat.html#a947c52c64022ef364d8ef17c94a65ef1" title="Default constructor.">chmat</a> ( <span class="keyword">const</span> <span class="keywordtype">int</span> dim0 ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( dim0 ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( dim0, dim0 ) {}; |
111 | | <a name="l00068"></a><a class="code" href="classbdm_1_1chmat.html#aae4dd65ff29d7d951b18903b100bd39d">00068</a> <a class="code" href="classbdm_1_1chmat.html#aae4dd65ff29d7d951b18903b100bd39d" title="Default constructor.">chmat</a> ( <span class="keyword">const</span> vec &v ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( v.length() ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( diag ( sqrt ( v ) ) ) {}; |
112 | | <a name="l00070"></a><a class="code" href="classbdm_1_1chmat.html#a57513fca80f91194faf90297f2ab2b24">00070</a> <a class="code" href="classbdm_1_1chmat.html#a57513fca80f91194faf90297f2ab2b24" title="Copy constructor.">chmat</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &Ch0 ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> ) { |
113 | | <a name="l00071"></a>00071 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = Ch0.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
114 | | <a name="l00072"></a>00072 } |
115 | | <a name="l00073"></a>00073 |
116 | | <a name="l00075"></a><a class="code" href="classbdm_1_1chmat.html#a796ee52e1343bc53d278f17c55bb1cc7">00075</a> <a class="code" href="classbdm_1_1chmat.html#a796ee52e1343bc53d278f17c55bb1cc7" title="Default constructor (m3k:cholform).">chmat</a> ( <span class="keyword">const</span> mat &M ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( M.<a class="code" href="classbdm_1_1sqmat.html#a73e639221343dcce76c3305524d67590" title="Reimplementing common functions of mat: rows().">rows</a>() ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( M.<a class="code" href="classbdm_1_1sqmat.html#a73e639221343dcce76c3305524d67590" title="Reimplementing common functions of mat: rows().">rows</a>(), M.<a class="code" href="classbdm_1_1sqmat.html#a445ef762cf5d2d6cfdf53c72942363cf" title="Reimplementing common functions of mat: cols().">cols</a>() ) { |
117 | | <a name="l00076"></a>00076 mat Q; |
118 | | <a name="l00077"></a>00077 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( M.rows() == M.cols(), <span class="stringliteral">"chmat:: input matrix must be square!"</span> ); |
119 | | <a name="l00078"></a>00078 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = chol ( M ); |
120 | | <a name="l00079"></a>00079 } |
121 | | <a name="l00080"></a>00080 |
122 | | <a name="l00085"></a><a class="code" href="classbdm_1_1chmat.html#a3c4e675e7efd91f78faa3dcf23aec153">00085</a> <a class="code" href="classbdm_1_1chmat.html#a3c4e675e7efd91f78faa3dcf23aec153">chmat</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &M, <span class="keyword">const</span> ivec &perm ) { |
123 | | <a name="l00086"></a>00086 <a class="code" href="bdmerror_8h.html#a7c43f3a72afe68ab0c85663a1bb3521a" title="Unconditionally throw std::runtime_error.">bdm_error</a> ( <span class="stringliteral">"not implemented"</span> ); |
124 | | <a name="l00087"></a>00087 } |
125 | | <a name="l00088"></a>00088 |
126 | | <a name="l00090"></a><a class="code" href="classbdm_1_1chmat.html#a17daa8c5c5914bd3194cb3053c5793a5">00090</a> mat & <a class="code" href="classbdm_1_1chmat.html#a17daa8c5c5914bd3194cb3053c5793a5" title="Access function.">_Ch</a>() { |
127 | | <a name="l00091"></a>00091 <span class="keywordflow">return</span> <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
128 | | <a name="l00092"></a>00092 } |
129 | | <a name="l00094"></a><a class="code" href="classbdm_1_1chmat.html#a3e9e137c1a6f46d13c2b6790cfddaa92">00094</a> <span class="keyword">const</span> mat & <a class="code" href="classbdm_1_1chmat.html#a3e9e137c1a6f46d13c2b6790cfddaa92" title="Access function.">_Ch</a>()<span class="keyword"> const </span>{ |
130 | | <a name="l00095"></a>00095 <span class="keywordflow">return</span> <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
131 | | <a name="l00096"></a>00096 } |
132 | | <a name="l00098"></a><a class="code" href="classbdm_1_1chmat.html#ab192df5b54b173ba9bc6922ee9983bc9">00098</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#ab192df5b54b173ba9bc6922ee9983bc9" title="Access functions.">setD</a> ( <span class="keyword">const</span> vec &nD ) { |
133 | | <a name="l00099"></a>00099 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = diag ( sqrt ( nD ) ); |
134 | | <a name="l00100"></a>00100 } |
135 | | <a name="l00102"></a><a class="code" href="classbdm_1_1chmat.html#ab8fd18754b0e5e0463f818cd16c710c8">00102</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#ab8fd18754b0e5e0463f818cd16c710c8" title="Access functions.">setCh</a> ( <span class="keyword">const</span> vec &chQ ) { |
136 | | <a name="l00103"></a>00103 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( chQ.length() == <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> * <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, <span class="stringliteral">"wrong length"</span> ); |
137 | | <a name="l00104"></a>00104 copy_vector ( dim*dim, chQ._data(), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>._data() ); |
138 | | <a name="l00105"></a>00105 } |
139 | | <a name="l00106"></a>00106 |
140 | | <a name="l00108"></a><a class="code" href="classbdm_1_1chmat.html#a18f4b97bde650086fe5e3d9461a13514">00108</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a18f4b97bde650086fe5e3d9461a13514" title="Access functions.">setD</a> ( <span class="keyword">const</span> vec &nD, <span class="keywordtype">int</span> i ) { |
141 | | <a name="l00109"></a>00109 <span class="keywordflow">for</span> ( <span class="keywordtype">int</span> j = i; j < nD.length(); j++ ) { |
142 | | <a name="l00110"></a>00110 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( j, j ) = sqrt ( nD ( j - i ) ); <span class="comment">//Fixme can be more general</span> |
143 | | <a name="l00111"></a>00111 } |
144 | | <a name="l00112"></a>00112 } |
145 | | <a name="l00113"></a>00113 |
146 | | <a name="l00115"></a>00115 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9" title="Operators.">operator += </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2 ); |
147 | | <a name="l00116"></a>00116 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba" title="mapping of negative add operation to operators">operator -= </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2 ); |
148 | | <a name="l00117"></a>00117 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& operator * ( <span class="keyword">const</span> <span class="keywordtype">double</span> &d ) { |
149 | | <a name="l00118"></a>00118 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>*sqrt ( d ); |
150 | | <a name="l00119"></a>00119 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
151 | | <a name="l00120"></a>00120 }; |
152 | | <a name="l00121"></a>00121 chmat& operator = ( <span class="keyword">const</span> chmat &A2 ) { |
153 | | <a name="l00122"></a>00122 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = A2.Ch; |
154 | | <a name="l00123"></a>00123 <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> = A2.dim; |
155 | | <a name="l00124"></a>00124 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
156 | | <a name="l00125"></a>00125 } |
157 | | <a name="l00126"></a>00126 chmat& operator *= ( <span class="keywordtype">double</span> x ) { |
158 | | <a name="l00127"></a>00127 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> *= sqrt ( x ); |
159 | | <a name="l00128"></a>00128 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
160 | | <a name="l00129"></a>00129 }; |
161 | | <a name="l00130"></a>00130 }; |
162 | | <a name="l00131"></a>00131 |
163 | | <a name="l00132"></a>00132 |
164 | | <a name="l00135"></a><a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9">00135</a> <span class="keyword">inline</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9" title="Operators.">chmat::operator += </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> & A2 ) { |
165 | | <a name="l00136"></a>00136 this-><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( A2 ); |
166 | | <a name="l00137"></a>00137 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
167 | | <a name="l00138"></a>00138 } |
168 | | <a name="l00140"></a><a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba">00140</a> <span class="keyword">inline</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba" title="mapping of negative add operation to operators">chmat::operator -= </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> & A2 ) { |
169 | | <a name="l00141"></a>00141 this-><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( A2, -1.0 ); |
170 | | <a name="l00142"></a>00142 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
171 | | <a name="l00143"></a>00143 } |
172 | | <a name="l00144"></a>00144 |
173 | | <a name="l00145"></a>00145 } |
174 | | <a name="l00146"></a>00146 |
175 | | <a name="l00147"></a>00147 <span class="preprocessor">#endif // CHMAT_H</span> |
| 85 | <a name="l00037"></a>00037 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a832d1a18bbf33ee7681adf01d5d08e21" title="Inplace symmetric multiplication by a SQUARE matrix , i.e. .">mult_sym</a> ( <span class="keyword">const</span> mat &C , <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &U ) <span class="keyword">const</span>; |
| 86 | <a name="l00038"></a>00038 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a7a4f54f1ea8b802f318de77a78f6ddb0" title="Inplace symmetric multiplication by a SQUARE transpose of matrix , i.e. .">mult_sym_t</a> ( <span class="keyword">const</span> mat &C ); |
| 87 | <a name="l00040"></a>00040 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a7a4f54f1ea8b802f318de77a78f6ddb0" title="Inplace symmetric multiplication by a SQUARE transpose of matrix , i.e. .">mult_sym_t</a> ( <span class="keyword">const</span> mat &C, <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &U ) <span class="keyword">const</span>; |
| 88 | <a name="l00041"></a>00041 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a949ccd174ed19f9cfe36366cbd5c56a4" title="Logarithm of a determinant.">logdet</a>() <span class="keyword">const</span>; |
| 89 | <a name="l00042"></a>00042 vec <a class="code" href="classbdm_1_1chmat.html#aec8d3a1cbf3b50183f246a858663c92d" title="Multiplies square root of by vector .">sqrt_mult</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
| 90 | <a name="l00043"></a>00043 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a6e0aee9691909602d341fe5ebd1f1ca8" title="Evaluates quadratic form ;.">qform</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
| 91 | <a name="l00044"></a>00044 <span class="keywordtype">double</span> <a class="code" href="classbdm_1_1chmat.html#a3ac8d00a14a22a21760a8374fe5c5757" title="Evaluates quadratic form ;.">invqform</a> ( <span class="keyword">const</span> vec &v ) <span class="keyword">const</span>; |
| 92 | <a name="l00045"></a>00045 <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#abc753cb54aa8946ce41d95e82fc148b5" title="Clearing matrix so that it corresponds to zeros.">clear</a>(); |
| 93 | <a name="l00047"></a><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7">00047</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2, <span class="keywordtype">double</span> w = 1.0 ) { |
| 94 | <a name="l00048"></a>00048 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> == A2.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, <span class="stringliteral">"Matrices of unequal dimension"</span> ); |
| 95 | <a name="l00049"></a>00049 mat pre = concat_vertical ( <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>, sqrt ( w ) * A2.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ); |
| 96 | <a name="l00050"></a>00050 mat post = zeros ( pre.rows(), pre.cols() ); |
| 97 | <a name="l00051"></a>00051 <span class="keywordflow">if</span> ( !qr ( pre, post ) ) { |
| 98 | <a name="l00052"></a>00052 <a class="code" href="bdmerror_8h.html#a59c5a63b3878ebcd6cc952fc2b47191c" title="Display a warning message.">bdm_warning</a> ( <span class="stringliteral">"Unstable QR in chmat add"</span> ); |
| 99 | <a name="l00053"></a>00053 } |
| 100 | <a name="l00054"></a>00054 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = post ( 0, <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> - 1, 0, <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> - 1 ); |
| 101 | <a name="l00055"></a>00055 }; |
| 102 | <a name="l00057"></a><a class="code" href="classbdm_1_1chmat.html#acbf3389db96dff41fb2e9532d59b13c0">00057</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#acbf3389db96dff41fb2e9532d59b13c0" title="Inversion in the same form, i.e. cholesky.">inv</a> ( <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &Inv )<span class="keyword"> const </span>{ |
| 103 | <a name="l00058"></a>00058 ( Inv.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ) = itpp::inv ( <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ).T(); |
| 104 | <a name="l00059"></a>00059 }; <span class="comment">//Fixme: can be more efficient</span> |
| 105 | <a name="l00060"></a>00060 ; |
| 106 | <a name="l00061"></a>00061 <span class="comment">// void inv ( mat &Inv );</span> |
| 107 | <a name="l00062"></a>00062 |
| 108 | <a name="l00064"></a><a class="code" href="classbdm_1_1chmat.html#aa888e866a163bffe94b51e58af65901b">00064</a> <span class="keyword">virtual</span> <a class="code" href="classbdm_1_1chmat.html#aa888e866a163bffe94b51e58af65901b" title="Destructor for future use;.">~chmat</a>() {}; |
| 109 | <a name="l00066"></a>00066 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> ( ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> (), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( ) {}; |
| 110 | <a name="l00068"></a><a class="code" href="classbdm_1_1chmat.html#a947c52c64022ef364d8ef17c94a65ef1">00068</a> <a class="code" href="classbdm_1_1chmat.html#a947c52c64022ef364d8ef17c94a65ef1" title="Default constructor.">chmat</a> ( <span class="keyword">const</span> <span class="keywordtype">int</span> dim0 ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( dim0 ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( dim0, dim0 ) {}; |
| 111 | <a name="l00070"></a><a class="code" href="classbdm_1_1chmat.html#aae4dd65ff29d7d951b18903b100bd39d">00070</a> <a class="code" href="classbdm_1_1chmat.html#aae4dd65ff29d7d951b18903b100bd39d" title="Default constructor.">chmat</a> ( <span class="keyword">const</span> vec &v ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( v.length() ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( diag ( sqrt ( v ) ) ) {}; |
| 112 | <a name="l00072"></a><a class="code" href="classbdm_1_1chmat.html#a57513fca80f91194faf90297f2ab2b24">00072</a> <a class="code" href="classbdm_1_1chmat.html#a57513fca80f91194faf90297f2ab2b24" title="Copy constructor.">chmat</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &Ch0 ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, Ch0.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> ) { |
| 113 | <a name="l00073"></a>00073 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = Ch0.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
| 114 | <a name="l00074"></a>00074 } |
| 115 | <a name="l00075"></a>00075 |
| 116 | <a name="l00077"></a><a class="code" href="classbdm_1_1chmat.html#a796ee52e1343bc53d278f17c55bb1cc7">00077</a> <a class="code" href="classbdm_1_1chmat.html#a796ee52e1343bc53d278f17c55bb1cc7" title="Default constructor (m3k:cholform).">chmat</a> ( <span class="keyword">const</span> mat &M ) : <a class="code" href="classbdm_1_1sqmat.html" title="Abstract class for representation of double symmetric matrices in square-root form...">sqmat</a> ( M.<a class="code" href="classbdm_1_1sqmat.html#a73e639221343dcce76c3305524d67590" title="Reimplementing common functions of mat: rows().">rows</a>() ), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( M.<a class="code" href="classbdm_1_1sqmat.html#a73e639221343dcce76c3305524d67590" title="Reimplementing common functions of mat: rows().">rows</a>(), M.<a class="code" href="classbdm_1_1sqmat.html#a445ef762cf5d2d6cfdf53c72942363cf" title="Reimplementing common functions of mat: cols().">cols</a>() ) { |
| 117 | <a name="l00078"></a>00078 mat Q; |
| 118 | <a name="l00079"></a>00079 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( M.rows() == M.cols(), <span class="stringliteral">"chmat:: input matrix must be square!"</span> ); |
| 119 | <a name="l00080"></a>00080 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = chol ( M ); |
| 120 | <a name="l00081"></a>00081 } |
| 121 | <a name="l00082"></a>00082 |
| 122 | <a name="l00087"></a><a class="code" href="classbdm_1_1chmat.html#a3c4e675e7efd91f78faa3dcf23aec153">00087</a> <a class="code" href="classbdm_1_1chmat.html#a3c4e675e7efd91f78faa3dcf23aec153">chmat</a> ( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &M, <span class="keyword">const</span> ivec &perm ) { |
| 123 | <a name="l00088"></a>00088 <a class="code" href="bdmerror_8h.html#a7c43f3a72afe68ab0c85663a1bb3521a" title="Unconditionally throw std::runtime_error.">bdm_error</a> ( <span class="stringliteral">"not implemented"</span> ); |
| 124 | <a name="l00089"></a>00089 } |
| 125 | <a name="l00090"></a>00090 |
| 126 | <a name="l00092"></a><a class="code" href="classbdm_1_1chmat.html#a17daa8c5c5914bd3194cb3053c5793a5">00092</a> mat & <a class="code" href="classbdm_1_1chmat.html#a17daa8c5c5914bd3194cb3053c5793a5" title="Access function.">_Ch</a>() { |
| 127 | <a name="l00093"></a>00093 <span class="keywordflow">return</span> <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
| 128 | <a name="l00094"></a>00094 } |
| 129 | <a name="l00096"></a><a class="code" href="classbdm_1_1chmat.html#a3e9e137c1a6f46d13c2b6790cfddaa92">00096</a> <span class="keyword">const</span> mat & <a class="code" href="classbdm_1_1chmat.html#a3e9e137c1a6f46d13c2b6790cfddaa92" title="Access function.">_Ch</a>()<span class="keyword"> const </span>{ |
| 130 | <a name="l00097"></a>00097 <span class="keywordflow">return</span> <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
| 131 | <a name="l00098"></a>00098 } |
| 132 | <a name="l00100"></a><a class="code" href="classbdm_1_1chmat.html#ab192df5b54b173ba9bc6922ee9983bc9">00100</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#ab192df5b54b173ba9bc6922ee9983bc9" title="Access functions.">setD</a> ( <span class="keyword">const</span> vec &nD ) { |
| 133 | <a name="l00101"></a>00101 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = diag ( sqrt ( nD ) ); |
| 134 | <a name="l00102"></a>00102 } |
| 135 | <a name="l00104"></a><a class="code" href="classbdm_1_1chmat.html#ab8fd18754b0e5e0463f818cd16c710c8">00104</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#ab8fd18754b0e5e0463f818cd16c710c8" title="Access functions.">setCh</a> ( <span class="keyword">const</span> vec &chQ ) { |
| 136 | <a name="l00105"></a>00105 <a class="code" href="bdmerror_8h.html#a89a0f906b242b79c5d3d342291b2cab4" title="Throw std::runtime_exception if t is not true and NDEBUG is not defined.">bdm_assert_debug</a> ( chQ.length() == <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> * <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>, <span class="stringliteral">"wrong length"</span> ); |
| 137 | <a name="l00106"></a>00106 copy_vector ( dim*dim, chQ._data(), <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>._data() ); |
| 138 | <a name="l00107"></a>00107 } |
| 139 | <a name="l00108"></a>00108 |
| 140 | <a name="l00110"></a><a class="code" href="classbdm_1_1chmat.html#a18f4b97bde650086fe5e3d9461a13514">00110</a> <span class="keywordtype">void</span> <a class="code" href="classbdm_1_1chmat.html#a18f4b97bde650086fe5e3d9461a13514" title="Access functions.">setD</a> ( <span class="keyword">const</span> vec &nD, <span class="keywordtype">int</span> i ) { |
| 141 | <a name="l00111"></a>00111 <span class="keywordflow">for</span> ( <span class="keywordtype">int</span> j = i; j < nD.length(); j++ ) { |
| 142 | <a name="l00112"></a>00112 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> ( j, j ) = sqrt ( nD ( j - i ) ); <span class="comment">//Fixme can be more general</span> |
| 143 | <a name="l00113"></a>00113 } |
| 144 | <a name="l00114"></a>00114 } |
| 145 | <a name="l00115"></a>00115 |
| 146 | <a name="l00117"></a>00117 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9" title="Operator.">operator += </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2 ); |
| 147 | <a name="l00119"></a>00119 <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba" title="Operator.">operator -= </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2 ); |
| 148 | <a name="l00121"></a><a class="code" href="classbdm_1_1chmat.html#a62ce85eb2461b51821c0d8507dfb9e95">00121</a> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a62ce85eb2461b51821c0d8507dfb9e95" title="Operator.">operator * </a>( <span class="keyword">const</span> <span class="keywordtype">double</span> &d ) { |
| 149 | <a name="l00122"></a>00122 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>*sqrt ( d ); |
| 150 | <a name="l00123"></a>00123 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
| 151 | <a name="l00124"></a>00124 }; |
| 152 | <a name="l00126"></a><a class="code" href="classbdm_1_1chmat.html#a8727d34435b7aa241c47a7b2fc0bed16">00126</a> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a8727d34435b7aa241c47a7b2fc0bed16" title="Operator.">operator = </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> &A2 ) { |
| 153 | <a name="l00127"></a>00127 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> = A2.<a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a>; |
| 154 | <a name="l00128"></a>00128 <a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a> = A2.<a class="code" href="classbdm_1_1sqmat.html#a05111b5744a494880e62cb0f3d42293f" title="dimension of the square matrix">dim</a>; |
| 155 | <a name="l00129"></a>00129 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
| 156 | <a name="l00130"></a>00130 } |
| 157 | <a name="l00132"></a><a class="code" href="classbdm_1_1chmat.html#ab05aac3f788d953a78c96295a6ad3d3b">00132</a> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#ab05aac3f788d953a78c96295a6ad3d3b" title="Operator.">operator *= </a>( <span class="keywordtype">double</span> x ) { |
| 158 | <a name="l00133"></a>00133 <a class="code" href="classbdm_1_1chmat.html#a3e4f39b1895a0a870f8db98329635223" title="Upper triangle of the cholesky matrix.">Ch</a> *= sqrt ( x ); |
| 159 | <a name="l00134"></a>00134 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
| 160 | <a name="l00135"></a>00135 }; |
| 161 | <a name="l00136"></a>00136 }; |
| 162 | <a name="l00137"></a>00137 |
| 163 | <a name="l00138"></a>00138 |
| 164 | <a name="l00141"></a><a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9">00141</a> <span class="keyword">inline</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#a029a0997afc656fde0c73bbf7724aca9" title="Operator.">chmat::operator += </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> & A2 ) { |
| 165 | <a name="l00142"></a>00142 this-><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( A2 ); |
| 166 | <a name="l00143"></a>00143 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
| 167 | <a name="l00144"></a>00144 } |
| 168 | <a name="l00146"></a><a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba">00146</a> <span class="keyword">inline</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a>& <a class="code" href="classbdm_1_1chmat.html#af1d0180fb795d38b8c352df2686ec5ba" title="Operator.">chmat::operator -= </a>( <span class="keyword">const</span> <a class="code" href="classbdm_1_1chmat.html" title="Symmetric matrix stored in square root decomposition using upper cholesky.">chmat</a> & A2 ) { |
| 169 | <a name="l00147"></a>00147 this-><a class="code" href="classbdm_1_1chmat.html#a6b3d5b772f791fc7c5df544a418278c7" title="add another chmat A2 with weight w.">add</a> ( A2, -1.0 ); |
| 170 | <a name="l00148"></a>00148 <span class="keywordflow">return</span> *<span class="keyword">this</span>; |
| 171 | <a name="l00149"></a>00149 } |
| 172 | <a name="l00150"></a>00150 |
| 173 | <a name="l00151"></a>00151 } |
| 174 | <a name="l00152"></a>00152 |
| 175 | <a name="l00153"></a>00153 <span class="preprocessor">#endif // CHMAT_H</span> |