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Bill Somerville
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head>
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<meta name="generator" content=
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"HTML Tidy for Linux/x86 (vers 1st March 2004), see www.w3.org" />
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<meta name="GENERATOR" content="Quanta Plus" />
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<meta http-equiv="Content-Type" content=
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"text/html; charset=us-ascii" />
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<link rel="stylesheet" href="../../../../boost.css" type="text/css"/>
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<link rel="stylesheet" href="ublas.css" type="text/css" />
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<script type="text/javascript" src="js/jquery-1.3.2.min.js" async="async" ></script>
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<script type="text/javascript" src="js/jquery.toc-gw.js" async="async" ></script>
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<title>uBLAS operations overview</title>
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</head>
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<body>
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<h1><img src="../../../../boost.png" align="middle" />Overview of Matrix and Vector Operations</h1>
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<div class="toc" id="toc"></div>
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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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<h2>Definitions</h2>
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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.html">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.html#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 assignments</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. Setting the
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<code>init</code> to <code>true</code> is equivalent to calling
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<code>w.clear()</code> before <code>axpy_prod</code>.
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There are some specialisation for products of 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); // C = 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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actual speed depends very significantly 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>Accessing submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p>
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<pre><code>
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w = project(u, r); // the subvector of u specifed by the index range r
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w = project(u, s); // the subvector of u specifed by the index slice s
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C = project(A, r1, r2); // the submatrix of A specified by the two index ranges r1 and r2
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C = project(A, s1, s2); // the 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>Assigning to submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p>
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<pre><code>
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project(u, r) = w; // assign the subvector of u specifed by the index range r
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project(u, s) = w; // assign the subvector of u specifed by the index slice s
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project(A, r1, r2) = C; // assign the submatrix of A specified by the two index ranges r1 and r2
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project(A, s1, s2) = C; // assign the submatrix of A specified by the two index slices s1 and s2
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row(A, i) = w; column(A, j) = w; // a row or column of matrix as a vector
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</code></pre>
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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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<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>
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<pre><code>
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w = subrange(u, 0, 2); // the 2 element subvector of u
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w = subslice(u, 0, 1, 2); // the 2 element subvector of u
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C = subrange(A, 0,2, 0,3); // the 2x3 element submatrix of A
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C = subslice(A, 0,1,2, 0,1,3); // the 2x3 element submatrix of A
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subrange(u, 0, 2) = w; // assign the 2 element subvector of u
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subslice(u, 0, 1, 2) = w; // assign the 2 element subvector of u
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subrange(A, 0,2, 0,3) = C; // assign the 2x3 element submatrix of A
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subrange(A, 0,1,2, 0,1,3) = C; // assigne the 2x3 element submatrix of A
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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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<h3><a name='noalias'>Matrix / Vector assignment</a></h3>
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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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<h3>Controlling the complexity of nested products</h3>
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<p>What is the complexity (the number of add and multiply operations) required to compute the following?
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</p>
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<pre>
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R = prod(A, prod(B,C));
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</pre>
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<p>Firstly the complexity depends on matrix size. Also since prod is transitive (not commutative)
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the bracket order affects the complexity.
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</p>
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<p>uBLAS evaluates expressions without matrix or vector temporaries and honours
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the bracketing structure. However avoiding temporaries for nested product unnecessarly increases the complexity.
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Conversly by explictly using temporary matrices the complexity of a nested product can be reduced.
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</p>
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<p>uBLAS provides 3 alternative syntaxes for this purpose:
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</p>
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<pre>
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temp_type T = prod(B,C); R = prod(A,T); // Preferable if T is preallocated
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</pre>
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<pre>
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prod(A, temp_type(prod(B,C));
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</pre>
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<pre>
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prod(A, prod<temp_type>(B,C));
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</pre>
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<p>The 'temp_type' is important. Given A,B,C are all of the same type. Say
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matrix<float>, the choice is easy. However if the value_type is mixed (int with float or double)
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or the matrix type is mixed (sparse with symmetric) the best solution is not so obvious. It is up to you! It
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depends on numerical properties of A and the result of the prod(B,C).
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</p>
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<hr />
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<p>Copyright (©) 2000-2007 Joerg Walter, Mathias Koch, Gunter
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Winkler, Michael Stevens<br />
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Use, modification and distribution are subject to the
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Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt
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or copy at <a href="http://www.boost.org/LICENSE_1_0.txt">
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http://www.boost.org/LICENSE_1_0.txt
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</a>).
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</p>
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<script type="text/javascript">
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(function($) {
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$('#toc').toc();
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})(jQuery);
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</script>
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</body>
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</html>
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