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| <h1><img src="../../../../boost.png" align="middle" />Overview of Matrix and Vector Operations</h1> |
| <div class="toc" id="toc"></div> |
| |
| <dl> |
| <dt>Contents:</dt> |
| <dd><a href="#blas">Basic Linear Algebra</a></dd> |
| <dd><a href="#advanced">Advanced Functions</a></dd> |
| <dd><a href="#sub">Submatrices, Subvectors</a></dd> |
| <dd><a href="#speed">Speed Improvements</a></dd> |
| </dl> |
| |
| <h2>Definitions</h2> |
| |
| <table style="" summary="notation"> |
| <tr><td><code>A, B, C</code></td> |
| <td> are matrices</td></tr> |
| <tr><td><code>u, v, w</code></td> |
| <td>are vectors</td></tr> |
| <tr><td><code>i, j, k</code></td> |
| <td>are integer values</td></tr> |
| <tr><td><code>t, t1, t2</code></td> |
| <td>are scalar values</td></tr> |
| <tr><td><code>r, r1, r2</code></td> |
| <td>are <a href="range.htm">ranges</a>, e.g. <code>range(0, 3)</code></td></tr> |
| <tr><td><code>s, s1, s2</code></td> |
| <td>are <a href="range.htm#slice">slices</a>, e.g. <code>slice(0, 1, 3)</code></td></tr> |
| </table> |
| |
| <h2><a name="blas">Basic Linear Algebra</a></h2> |
| |
| <h3>standard operations: addition, subtraction, multiplication by a |
| scalar</h3> |
| |
| <pre><code> |
| C = A + B; C = A - B; C = -A; |
| w = u + v; w = u - v; w = -u; |
| C = t * A; C = A * t; C = A / t; |
| w = t * u; w = u * t; w = u / t; |
| </code></pre> |
| |
| <h3>computed assignments</h3> |
| |
| <pre><code> |
| C += A; C -= A; |
| w += u; w -= u; |
| C *= t; C /= t; |
| w *= t; w /= t; |
| </code></pre> |
| |
| <h3>inner, outer and other products</h3> |
| |
| <pre><code> |
| t = inner_prod(u, v); |
| C = outer_prod(u, v); |
| w = prod(A, u); w = prod(u, A); w = prec_prod(A, u); w = prec_prod(u, A); |
| C = prod(A, B); C = prec_prod(A, B); |
| w = element_prod(u, v); w = element_div(u, v); |
| C = element_prod(A, B); C = element_div(A, B); |
| </code></pre> |
| |
| <h3>transformations</h3> |
| |
| <pre><code> |
| w = conj(u); w = real(u); w = imag(u); |
| C = trans(A); C = conj(A); C = herm(A); C = real(A); C = imag(A); |
| </code></pre> |
| |
| <h2><a name="advanced">Advanced functions</a></h2> |
| |
| <h3>norms</h3> |
| |
| <pre><code> |
| t = norm_inf(v); i = index_norm_inf(v); |
| t = norm_1(v); t = norm_2(v); |
| t = norm_inf(A); i = index_norm_inf(A); |
| t = norm_1(A); t = norm_frobenius(A); |
| </code></pre> |
| |
| <h3>products</h3> |
| |
| <pre><code> |
| axpy_prod(A, u, w, true); // w = A * u |
| axpy_prod(A, u, w, false); // w += A * u |
| axpy_prod(u, A, w, true); // w = trans(A) * u |
| axpy_prod(u, A, w, false); // w += trans(A) * u |
| axpy_prod(A, B, C, true); // C = A * B |
| axpy_prod(A, B, C, false); // C += A * B |
| </code></pre> |
| <p><em>Note:</em> The last argument (<code>bool init</code>) of |
| <code>axpy_prod</code> is optional. Currently it defaults to |
| <code>true</code>, but this may change in the future. Setting the |
| <code>init</code> to <code>true</code> is equivalent to calling |
| <code>w.clear()</code> before <code>axpy_prod</code>. |
| There are some specialisation for products of compressed matrices that give a |
| large speed up compared to <code>prod</code>.</p> |
| <pre><code> |
| w = block_prod<matrix_type, 64> (A, u); // w = A * u |
| w = block_prod<matrix_type, 64> (u, A); // w = trans(A) * u |
| C = block_prod<matrix_type, 64> (A, B); // C = A * B |
| </code></pre> |
| <p><em>Note:</em> The blocksize can be any integer. However, the |
| actual speed depends very significantly on the combination of blocksize, |
| CPU and compiler. The function <code>block_prod</code> is designed |
| for large dense matrices.</p> |
| <h3>rank-k updates</h3> |
| <pre><code> |
| opb_prod(A, B, C, true); // C = A * B |
| opb_prod(A, B, C, false); // C += A * B |
| </code></pre> |
| <p><em>Note:</em> The last argument (<code>bool init</code>) of |
| <code>opb_prod</code> is optional. Currently it defaults to |
| <code>true</code>, but this may change in the future. This function |
| may give a speedup if <code>A</code> has less columns than rows, |
| because the product is computed as a sum of outer products.</p> |
| |
| <h2><a name="sub">Submatrices, Subvectors</a></h2> |
| <p>Accessing submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p> |
| <pre><code> |
| w = project(u, r); // the subvector of u specifed by the index range r |
| w = project(u, s); // the subvector of u specifed by the index slice s |
| C = project(A, r1, r2); // the submatrix of A specified by the two index ranges r1 and r2 |
| C = project(A, s1, s2); // the submatrix of A specified by the two index slices s1 and s2 |
| w = row(A, i); w = column(A, j); // a row or column of matrix as a vector |
| </code></pre> |
| <p>Assigning to submatrices and subvectors via <b>proxies</b> using <code>project</code> functions:</p> |
| <pre><code> |
| project(u, r) = w; // assign the subvector of u specifed by the index range r |
| project(u, s) = w; // assign the subvector of u specifed by the index slice s |
| project(A, r1, r2) = C; // assign the submatrix of A specified by the two index ranges r1 and r2 |
| project(A, s1, s2) = C; // assign the submatrix of A specified by the two index slices s1 and s2 |
| row(A, i) = w; column(A, j) = w; // a row or column of matrix as a vector |
| </code></pre> |
| <p><em>Note:</em> A range <code>r = range(start, stop)</code> |
| contains all indices <code>i</code> with <code>start <= i < |
| stop</code>. A slice is something more general. The slice |
| <code>s = slice(start, stride, size)</code> contains the indices |
| <code>start, start+stride, ..., start+(size-1)*stride</code>. The |
| stride can be 0 or negative! If <code>start >= stop</code> for a range |
| or <code>size == 0</code> for a slice then it contains no elements.</p> |
| <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> |
| <pre><code> |
| w = subrange(u, 0, 2); // the 2 element subvector of u |
| w = subslice(u, 0, 1, 2); // the 2 element subvector of u |
| C = subrange(A, 0,2, 0,3); // the 2x3 element submatrix of A |
| C = subslice(A, 0,1,2, 0,1,3); // the 2x3 element submatrix of A |
| subrange(u, 0, 2) = w; // assign the 2 element subvector of u |
| subslice(u, 0, 1, 2) = w; // assign the 2 element subvector of u |
| subrange(A, 0,2, 0,3) = C; // assign the 2x3 element submatrix of A |
| subrange(A, 0,1,2, 0,1,3) = C; // assigne the 2x3 element submatrix of A |
| </code></pre> |
| <p>There are to more ways to access some matrix elements as a |
| vector:</p> |
| <pre><code>matrix_vector_range<matrix_type> (A, r1, r2); |
| matrix_vector_slice<matrix_type> (A, s1, s2); |
| </code></pre> |
| <p><em>Note:</em> These matrix proxies take a sequence of elements |
| of a matrix and allow you to access these as a vector. In |
| particular <code>matrix_vector_slice</code> can do this in a very |
| general way. <code>matrix_vector_range</code> is less useful as the |
| elements must lie along a diagonal.</p> |
| <p><em>Example:</em> To access the first two elements of a sub |
| column of a matrix we access the row with a slice with stride 1 and |
| the column with a slice with stride 0 thus:<br /> |
| <code>matrix_vector_slice<matrix_type> (A, slice(0,1,2), |
| slice(0,0,2)); |
| </code></p> |
| |
| <h2><a name="speed">Speed improvements</a></h2> |
| <h3><a name='noalias'>Matrix / Vector assignment</a></h3> |
| <p>If you know for sure that the left hand expression and the right |
| hand expression have no common storage, then assignment has |
| no <em>aliasing</em>. A more efficient assignment can be specified |
| in this case:</p> |
| <pre><code>noalias(C) = prod(A, B); |
| </code></pre> |
| <p>This avoids the creation of a temporary matrix that is required in a normal assignment. |
| 'noalias' assignment requires that the left and right hand side be size conformant.</p> |
| |
| <h3>Sparse element access</h3> |
| <p>The matrix element access function <code>A(i1,i2)</code> or the equivalent vector |
| element access functions (<code>v(i) or v[i]</code>) usually create 'sparse element proxies' |
| when applied to a sparse matrix or vector. These <em>proxies</em> allow access to elements |
| without having to worry about nasty C++ issues where references are invalidated.</p> |
| <p>These 'sparse element proxies' can be implemented more efficiently when applied to <code>const</code> |
| objects. |
| Sadly in C++ there is no way to distinguish between an element access on the left and right hand side of |
| an assignment. Most often elements on the right hand side will not be changed and therefore it would |
| be better to use the <code>const</code> proxies. We can do this by making the matrix or vector |
| <code>const</code> before accessing it's elements. For example:</p> |
| <pre><code>value = const_cast<const VEC>(v)[i]; // VEC is the type of V |
| </code></pre> |
| <p>If more then one element needs to be accessed <code>const_iterator</code>'s should be used |
| in preference to <code>iterator</code>'s for the same reason. For the more daring 'sparse element proxies' |
| can be completely turned off in uBLAS by defining the configuration macro <code>BOOST_UBLAS_NO_ELEMENT_PROXIES</code>. |
| </p> |
| |
| |
| <h3>Controlling the complexity of nested products</h3> |
| |
| <p>What is the complexity (the number of add and multiply operations) required to compute the following? |
| </p> |
| <pre> |
| R = prod(A, prod(B,C)); |
| </pre> |
| <p>Firstly the complexity depends on matrix size. Also since prod is transitive (not commutative) |
| the bracket order affects the complexity. |
| </p> |
| <p>uBLAS evaluates expressions without matrix or vector temporaries and honours |
| the bracketing structure. However avoiding temporaries for nested product unnecessarly increases the complexity. |
| Conversly by explictly using temporary matrices the complexity of a nested product can be reduced. |
| </p> |
| <p>uBLAS provides 3 alternative syntaxes for this purpose: |
| </p> |
| <pre> |
| temp_type T = prod(B,C); R = prod(A,T); // Preferable if T is preallocated |
| </pre> |
| <pre> |
| prod(A, temp_type(prod(B,C)); |
| </pre> |
| <pre> |
| prod(A, prod<temp_type>(B,C)); |
| </pre> |
| <p>The 'temp_type' is important. Given A,B,C are all of the same type. Say |
| matrix<float>, the choice is easy. However if the value_type is mixed (int with float or double) |
| or the matrix type is mixed (sparse with symmetric) the best solution is not so obvious. It is up to you! It |
| depends on numerical properties of A and the result of the prod(B,C). |
| </p> |
| |
| <hr /> |
| <p>Copyright (©) 2000-2007 Joerg Walter, Mathias Koch, Gunter |
| Winkler, Michael Stevens<br /> |
| Use, modification and distribution are subject to the |
| Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt |
| or copy at <a href="http://www.boost.org/LICENSE_1_0.txt"> |
| http://www.boost.org/LICENSE_1_0.txt |
| </a>). |
| </p> |
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