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<link href="ublas.css" type="text/css" />
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<title>uBLAS operations overview</title>
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<body>
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<h1><img src="../../../../boost.png" align="middle" />
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Overview of Matrix and Vector Operations</h1>
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<dl>
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<dt>Contents:</dt>
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<dd><a href="#blas">Basic Linear Algebra</a></dd>
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<dd><a href="#advanced">Advanced Functions</a></dd>
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<dd><a href="#sub">Submatrices, Subvectors</a></dd>
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<dd><a href="#speed">Speed Improvements</a></dd>
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</dl>
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<h3>Definitions:</h3>
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<table style="" summary="notation">
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<tr><td><code>A, B, C</code></td>
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<td> are matrices</td></tr>
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<tr><td><code>u, v, w</code></td>
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<td>are vectors</td></tr>
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<tr><td><code>i, j, k</code></td>
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<td>are integer values</td></tr>
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<tr><td><code>t, t1, t2</code></td>
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<td>are scalar values</td></tr>
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<tr><td><code>r, r1, r2</code></td>
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<td>are <a href="range.htm">ranges</a>, e.g. <code>range(0, 3)</code></td></tr>
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<tr><td><code>s, s1, s2</code></td>
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<td>are <a href="range.htm#slice">slices</a>, e.g. <code>slice(0, 1, 3)</code></td></tr>
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</table>
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<h2><a name="blas">Basic Linear Algebra</a></h2>
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<h3>standard operations: addition, subtraction, multiplication by a
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scalar</h3>
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<pre><code>
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C = A + B; C = A - B; C = -A;
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w = u + v; w = u - v; w = -u;
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C = t * A; C = A * t; C = A / t;
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w = t * u; w = u * t; w = u / t;
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</code></pre>
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<h3>computed assignements</h3>
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<pre><code>
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C += A; C -= A;
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w += u; w -= u;
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C *= t; C /= t;
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w *= t; w /= t;
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</code></pre>
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<h3>inner, outer and other products</h3>
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<pre><code>
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t = inner_prod(u, v);
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C = outer_prod(u, v);
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w = prod(A, u); w = prod(u, A); w = prec_prod(A, u); w = prec_prod(u, A);
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C = prod(A, B); C = prec_prod(A, B);
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w = element_prod(u, v); w = element_div(u, v);
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C = element_prod(A, B); C = element_div(A, B);
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</code></pre>
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<h3>transformations</h3>
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<pre><code>
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w = conj(u); w = real(u); w = imag(u);
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C = trans(A); C = conj(A); C = herm(A); C = real(A); C = imag(A);
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</code></pre>
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<h2><a name="advanced">Advanced functions</a></h2>
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<h3>norms</h3>
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<pre><code>
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t = norm_inf(v); i = index_norm_inf(v);
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t = norm_1(v); t = norm_2(v);
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t = norm_inf(A); i = index_norm_inf(A);
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t = norm_1(A); t = norm_frobenius(A);
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</code></pre>
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<h3>products</h3>
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<pre><code>
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axpy_prod(A, u, w, true); // w = A * u
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axpy_prod(A, u, w, false); // w += A * u
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axpy_prod(u, A, w, true); // w = trans(A) * u
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axpy_prod(u, A, w, false); // w += trans(A) * u
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axpy_prod(A, B, C, true); // C = A * B
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axpy_prod(A, B, C, false); // C += A * B
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</code></pre>
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<p><em>Note:</em> The last argument (<code>bool init</code>) of
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<code>axpy_prod</code> is optional. Currently it defaults to
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<code>true</code>, but this may change in the future. Set the
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<code>init</code> to <code>true</code> is equivalent to call
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<code>w.clear()</code> before <code>axpy_prod</code>. Up to now
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there are some specialisation for compressed matrices that give a
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large speed up compared to <code>prod</code>.</p>
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<pre><code>
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w = block_prod<matrix_type, 64> (A, u); // w = A * u
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w = block_prod<matrix_type, 64> (u, A); // w = trans(A) * u
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C = block_prod<matrix_type, 64> (A, B); // w = A * B
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</code></pre>
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<p><em>Note:</em> The blocksize can be any integer. However, the
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total speed depends very strong on the combination of blocksize,
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CPU and compiler. The function <code>block_prod</code> is designed
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for large dense matrices.</p>
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<h3>rank-k updates</h3>
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<pre><code>
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opb_prod(A, B, C, true); // C = A * B
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opb_prod(A, B, C, false); // C += A * B
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</code></pre>
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<p><em>Note:</em> The last argument (<code>bool init</code>) of
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<code>opb_prod</code> is optional. Currently it defaults to
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<code>true</code>, but this may change in the future. This function
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may give a speedup if <code>A</code> has less columns than rows,
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because the product is computed as a sum of outer products.</p>
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<h2><a name="sub">Submatrices, Subvectors</a></h2>
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<p><em>Note:</em> A range <code>r = range(start, stop)</code>
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contains all indices <code>i</code> with <code>start <= i <
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stop</code>. A slice is something more general. The slice
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<code>s = slice(start, stride, size)</code> contains the indices
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<code>start, start+stride, ..., start+(size-1)*stride</code>. The
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stride can be 0 or negative! If <code>start >= stop</code> for a range
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or <code>size == 0</code> for a slice then it contains no elements.</p>
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<pre><code>
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w = project(u, r); // a subvector of u specifed by the index range r
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w = project(u, s); // a subvector of u specifed by the index slice s
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C = project(A, r1, r2); // a submatrix of A specified by the two index ranges r1 and r2
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C = project(A, s1, s2); // a submatrix of A specified by the two index slices s1 and s2
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w = row(A, i); w = column(A, j); // a row or column of matrix as a vector
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</code></pre>
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<p>There are to more ways to access some matrix elements as a
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vector:</p>
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<pre><code>matrix_vector_range<matrix_type> (A, r1, r2);
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matrix_vector_slice<matrix_type> (A, s1, s2);
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</code></pre>
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<p><em>Note:</em> These matrix proxies take a sequence of elements
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of a matrix and allow you to access these as a vector. In
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particular <code>matrix_vector_slice</code> can do this in a very
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general way. <code>matrix_vector_range</code> is less useful as the
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elements must lie along a diagonal.</p>
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<p><em>Example:</em> To access the first two elements of a sub
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column of a matrix we access the row with a slice with stride 1 and
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the column with a slice with stride 0 thus:<br />
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<code>matrix_vector_slice<matrix_type> (A, slice(0,1,2),
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slice(0,0,2));
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</code></p>
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<h2><a name="speed">Speed improvements</a></h2>
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<a name='noalias'>
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<h3>Matrix / Vector assignment</h3></a>
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<p>If you know for sure that the left hand expression and the right
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hand expression have no common storage, then assignment has
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no <em>aliasing</em>. A more efficient assignment can be specified
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in this case:</p>
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<pre><code>noalias(C) = prod(A, B);
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</code></pre>
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<p>This avoids the creation of a temporary matrix that is required in a normal assignment.
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'noalias' assignment requires that the left and right hand side be size conformant.</p>
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<h3>Sparse element access</h3>
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<p>The matrix element access function <code>A(i1,i2)</code> or the equivalent vector
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element access functions (<code>v(i) or v[i]</code>) usually create 'sparse element proxies'
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when applied to a sparse matrix or vector. These <em>proxies</em> allow access to elements
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without having to worry about nasty C++ issues where references are invalidated.</p>
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<p>These 'sparse element proxies' can be implemented more efficiently when applied to <code>const</code>
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objects.
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Sadly in C++ there is no way to distinguish between an element access on the left and right hand side of
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an assignment. Most often elements on the right hand side will not be changed and therefore it would
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be better to use the <code>const</code> proxies. We can do this by making the matrix or vector
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<code>const</code> before accessing it's elements. For example:</p>
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<pre><code>value = const_cast<const VEC&>(v)[i]; // VEC is the type of V
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</code></pre>
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<p>If more then one element needs to be accessed <code>const_iterator</code>'s should be used
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in preference to <code>iterator</code>'s for the same reason. For the more daring 'sparse element proxies'
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can be completely turned off in uBLAS by defining the configuration macro <code>BOOST_UBLAS_NO_ELEMENT_PROXIES</code>.
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</p>
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<hr />
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<p>Copyright (©) 2000-2004 Joerg Walter, Mathias Koch, Gunter
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Winkler, Michael Stevens<br />
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Permission to copy, use, modify, sell and distribute this document
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is granted provided this copyright notice appears in all copies.
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This document is provided ``as is'' without express or implied
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warranty, and with no claim as to its suitability for any
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purpose.</p>
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</body>
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</html>
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