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14<title>uBLAS operations overview</title>
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17<h1><img src="../../../../boost.png" align="middle" />Overview of Matrix and Vector Operations</h1>
18<div class="toc" id="toc"></div>
19
20<dl>
21<dt>Contents:</dt>
22<dd><a href="#blas">Basic Linear Algebra</a></dd>
23<dd><a href="#advanced">Advanced Functions</a></dd>
24<dd><a href="#sub">Submatrices, Subvectors</a></dd>
25<dd><a href="#speed">Speed Improvements</a></dd>
26</dl>
27
28<h2>Definitions</h2>
29
30<table style="" summary="notation">
31<tr><td><code>A, B, C</code></td>
32<td> are matrices</td></tr>
33<tr><td><code>u, v, w</code></td>
34<td>are vectors</td></tr>
35<tr><td><code>i, j, k</code></td>
36<td>are integer values</td></tr>
37<tr><td><code>t, t1, t2</code></td>
38<td>are scalar values</td></tr>
39<tr><td><code>r, r1, r2</code></td>
40<td>are <a href="range.html">ranges</a>, e.g. <code>range(0, 3)</code></td></tr>
41<tr><td><code>s, s1, s2</code></td>
42<td>are <a href="range.html#slice">slices</a>, e.g. <code>slice(0, 1, 3)</code></td></tr>
43</table>
44
45<h2><a name="blas">Basic Linear Algebra</a></h2>
46
47<h3>standard operations: addition, subtraction, multiplication by a
48scalar</h3>
49
50<pre><code>
51C = A + B; C = A - B; C = -A;
52w = u + v; w = u - v; w = -u;
53C = t * A; C = A * t; C = A / t;
54w = t * u; w = u * t; w = u / t;
55</code></pre>
56
57<h3>computed assignments</h3>
58
59<pre><code>
60C += A; C -= A;
61w += u; w -= u;
62C *= t; C /= t;
63w *= t; w /= t;
64</code></pre>
65
66<h3>inner, outer and other products</h3>
67
68<pre><code>
69t = inner_prod(u, v);
70C = outer_prod(u, v);
71w = prod(A, u); w = prod(u, A); w = prec_prod(A, u); w = prec_prod(u, A);
72C = prod(A, B); C = prec_prod(A, B);
73w = element_prod(u, v); w = element_div(u, v);
74C = element_prod(A, B); C = element_div(A, B);
75</code></pre>
76
77<h3>transformations</h3>
78
79<pre><code>
80w = conj(u); w = real(u); w = imag(u);
81C = trans(A); C = conj(A); C = herm(A); C = real(A); C = imag(A);
82</code></pre>
83
84<h2><a name="advanced">Advanced functions</a></h2>
85
86<h3>norms</h3>
87
88<pre><code>
89t = norm_inf(v); i = index_norm_inf(v);
90t = norm_1(v); t = norm_2(v);
91t = norm_inf(A); i = index_norm_inf(A);
92t = norm_1(A); t = norm_frobenius(A);
93</code></pre>
94
95<h3>products</h3>
96
97<pre><code>
98axpy_prod(A, u, w, true); // w = A * u
99axpy_prod(A, u, w, false); // w += A * u
100axpy_prod(u, A, w, true); // w = trans(A) * u
101axpy_prod(u, A, w, false); // w += trans(A) * u
102axpy_prod(A, B, C, true); // C = A * B
103axpy_prod(A, B, C, false); // C += A * B
104</code></pre>
105<p><em>Note:</em> The last argument (<code>bool init</code>) of
106<code>axpy_prod</code> is optional. Currently it defaults to
107<code>true</code>, but this may change in the future. Setting the
108<code>init</code> to <code>true</code> is equivalent to calling
109<code>w.clear()</code> before <code>axpy_prod</code>.
110There are some specialisation for products of compressed matrices that give a
111large speed up compared to <code>prod</code>.</p>
112<pre><code>
113w = block_prod&lt;matrix_type, 64&gt; (A, u); // w = A * u
114w = block_prod&lt;matrix_type, 64&gt; (u, A); // w = trans(A) * u
115C = block_prod&lt;matrix_type, 64&gt; (A, B); // C = A * B
116</code></pre>
117<p><em>Note:</em> The blocksize can be any integer. However, the
118actual speed depends very significantly on the combination of blocksize,
119CPU and compiler. The function <code>block_prod</code> is designed
120for large dense matrices.</p>
121<h3>rank-k updates</h3>
122<pre><code>
123opb_prod(A, B, C, true); // C = A * B
124opb_prod(A, B, C, false); // C += A * B
125</code></pre>
126<p><em>Note:</em> The last argument (<code>bool init</code>) of
127<code>opb_prod</code> is optional. Currently it defaults to
128<code>true</code>, but this may change in the future. This function
129may give a speedup if <code>A</code> has less columns than rows,
130because the product is computed as a sum of outer products.</p>
131
132<h2><a name="sub">Submatrices, Subvectors</a></h2>
133<p>Accessing submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p>
134<pre><code>
135w = project(u, r); // the subvector of u specifed by the index range r
136w = project(u, s); // the subvector of u specifed by the index slice s
137C = project(A, r1, r2); // the submatrix of A specified by the two index ranges r1 and r2
138C = project(A, s1, s2); // the submatrix of A specified by the two index slices s1 and s2
139w = row(A, i); w = column(A, j); // a row or column of matrix as a vector
140</code></pre>
141<p>Assigning to submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p>
142<pre><code>
143project(u, r) = w; // assign the subvector of u specifed by the index range r
144project(u, s) = w; // assign the subvector of u specifed by the index slice s
145project(A, r1, r2) = C; // assign the submatrix of A specified by the two index ranges r1 and r2
146project(A, s1, s2) = C; // assign the submatrix of A specified by the two index slices s1 and s2
147row(A, i) = w; column(A, j) = w; // a row or column of matrix as a vector
148</code></pre>
149<p><em>Note:</em> A range <code>r = range(start, stop)</code>
150contains all indices <code>i</code> with <code>start &lt;= i &lt;
151stop</code>. A slice is something more general. The slice
152<code>s = slice(start, stride, size)</code> contains the indices
153<code>start, start+stride, ..., start+(size-1)*stride</code>. The
154stride can be 0 or negative! If <code>start >= stop</code> for a range
155or <code>size == 0</code> for a slice then it contains no elements.</p>
156<p>Sub-ranges and sub-slices of vectors and matrices can be created directly with the <code>subrange</code> and <code>sublice</code> functions:</p>
157<pre><code>
158w = subrange(u, 0, 2); // the 2 element subvector of u
159w = subslice(u, 0, 1, 2); // the 2 element subvector of u
160C = subrange(A, 0,2, 0,3); // the 2x3 element submatrix of A
161C = subslice(A, 0,1,2, 0,1,3); // the 2x3 element submatrix of A
162subrange(u, 0, 2) = w; // assign the 2 element subvector of u
163subslice(u, 0, 1, 2) = w; // assign the 2 element subvector of u
164subrange(A, 0,2, 0,3) = C; // assign the 2x3 element submatrix of A
165subrange(A, 0,1,2, 0,1,3) = C; // assigne the 2x3 element submatrix of A
166</code></pre>
167<p>There are to more ways to access some matrix elements as a
168vector:</p>
169<pre><code>matrix_vector_range&lt;matrix_type&gt; (A, r1, r2);
170matrix_vector_slice&lt;matrix_type&gt; (A, s1, s2);
171</code></pre>
172<p><em>Note:</em> These matrix proxies take a sequence of elements
173of a matrix and allow you to access these as a vector. In
174particular <code>matrix_vector_slice</code> can do this in a very
175general way. <code>matrix_vector_range</code> is less useful as the
176elements must lie along a diagonal.</p>
177<p><em>Example:</em> To access the first two elements of a sub
178column of a matrix we access the row with a slice with stride 1 and
179the column with a slice with stride 0 thus:<br />
180<code>matrix_vector_slice&lt;matrix_type&gt; (A, slice(0,1,2),
181slice(0,0,2));
182</code></p>
183
184<h2><a name="speed">Speed improvements</a></h2>
185<h3><a name='noalias'>Matrix / Vector assignment</a></h3>
186<p>If you know for sure that the left hand expression and the right
187hand expression have no common storage, then assignment has
188no <em>aliasing</em>. A more efficient assignment can be specified
189in this case:</p>
190<pre><code>noalias(C) = prod(A, B);
191</code></pre>
192<p>This avoids the creation of a temporary matrix that is required in a normal assignment.
193'noalias' assignment requires that the left and right hand side be size conformant.</p>
194
195<h3>Sparse element access</h3>
196<p>The matrix element access function <code>A(i1,i2)</code> or the equivalent vector
197element access functions (<code>v(i) or v[i]</code>) usually create 'sparse element proxies'
198when applied to a sparse matrix or vector. These <em>proxies</em> allow access to elements
199without having to worry about nasty C++ issues where references are invalidated.</p>
200<p>These 'sparse element proxies' can be implemented more efficiently when applied to <code>const</code>
201objects.
202Sadly in C++ there is no way to distinguish between an element access on the left and right hand side of
203an assignment. Most often elements on the right hand side will not be changed and therefore it would
204be better to use the <code>const</code> proxies. We can do this by making the matrix or vector
205<code>const</code> before accessing it's elements. For example:</p>
206<pre><code>value = const_cast&lt;const VEC&gt;(v)[i]; // VEC is the type of V
207</code></pre>
208<p>If more then one element needs to be accessed <code>const_iterator</code>'s should be used
209in preference to <code>iterator</code>'s for the same reason. For the more daring 'sparse element proxies'
210can be completely turned off in uBLAS by defining the configuration macro <code>BOOST_UBLAS_NO_ELEMENT_PROXIES</code>.
211</p>
212
213
214<h3>Controlling the complexity of nested products</h3>
215
216<p>What is the complexity (the number of add and multiply operations) required to compute the following?
217</p>
218<pre>
219 R = prod(A, prod(B,C));
220</pre>
221<p>Firstly the complexity depends on matrix size. Also since prod is transitive (not commutative)
222the bracket order affects the complexity.
223</p>
224<p>uBLAS evaluates expressions without matrix or vector temporaries and honours
225the bracketing structure. However avoiding temporaries for nested product unnecessarly increases the complexity.
226Conversly by explictly using temporary matrices the complexity of a nested product can be reduced.
227</p>
228<p>uBLAS provides 3 alternative syntaxes for this purpose:
229</p>
230<pre>
231 temp_type T = prod(B,C); R = prod(A,T); // Preferable if T is preallocated
232</pre>
233<pre>
234 prod(A, temp_type(prod(B,C));
235</pre>
236<pre>
237 prod(A, prod&lt;temp_type&gt;(B,C));
238</pre>
239<p>The 'temp_type' is important. Given A,B,C are all of the same type. Say
240matrix&lt;float&gt;, the choice is easy. However if the value_type is mixed (int with float or double)
241or the matrix type is mixed (sparse with symmetric) the best solution is not so obvious. It is up to you! It
242depends on numerical properties of A and the result of the prod(B,C).
243</p>
244
245<hr />
246<p>Copyright (&copy;) 2000-2007 Joerg Walter, Mathias Koch, Gunter
247Winkler, Michael Stevens<br />
248 Use, modification and distribution are subject to the
249 Boost Software License, Version 1.0.
250 (See accompanying file LICENSE_1_0.txt
251 or copy at <a href="http://www.boost.org/LICENSE_1_0.txt">
252 http://www.boost.org/LICENSE_1_0.txt
253 </a>).
254</p>
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