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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<meta name="GENERATOR" content="Microsoft FrontPage Express 2.0">
<title>uBLAS Overview</title>
</head>
<body bgcolor="#ffffff">
<h1><img src="c++boost.gif" alt="c++boost.gif" align="Center">
uBLAS Overview</h1>
<h2>Rationale</h2>
<p><cite>It would be nice if every kind of numeric software could be written
in C++ without loss of efficiency, but unless something can be found that
achieves this without compromising the C++ type system it may be preferable
to rely on Fortran, assembler or architecture-specific extensions (Bjarne
Stroustrup).</cite></p>
<p>This C++ library is directed towards scientific computing on the level
of basic linear algebra constructions with matrices and vectors and their
corresponding abstract operations. The primary design goals were:</p>
<ul type="Disc">
<li>mathematical notation</li>
<li>efficiency</li>
<li>functionality</li>
<li>compatibility</li>
</ul>
<p>Another intention was to evaluate, if the abstraction penalty resulting
from the use of such matrix and vector classes is acceptable.</p>
<h2>Resources</h2>
<p>The development of this library was guided by a couple of similar efforts:</p>
<ul type="Disc">
<li><a href="http://www.netlib.org/blas/index.html">BLAS</a>
by Jack Dongarra et al.</li>
<li><a href="http://www.oonumerics.org/blitz/">Blitz++</a>
by Todd Veldhuizen</li>
<li><a href="http://www.acl.lanl.gov/pooma/">POOMA</a>
by Scott Haney et al.</li>
<li><a href="http://www.lsc.nd.edu/research/mtl/">MTL</a>
by Jeremy Siek et al.</li>
</ul>
<p>BLAS seems to be the most widely used library for basic linear algebra
constructions, so it could be called a de-facto standard. Its interface is
procedural, the individual functions are somewhat abstracted from simple linear
algebra operations. Due to the fact that is has been implemented using Fortran
and its optimizations, it also seems to be one of the fastest libraries available.
As we decided to design and implement our library in an object-oriented way,
the technical approaches are distinct. However anyone should be able to express
BLAS abstractions in terms of our library operators and to compare the efficiency
of the implementations.</p>
<p>Blitz++ is an impressive library implemented in C++. Its main design seems
to be oriented towards multidimensional arrays and their associated operators
including tensors. The author of Blitz++ states, that the library achieves
performance on par or better than corresponding Fortran code due to his implementation
technique using expression templates and template metaprograms. However
we see some reasons, to develop an own design and implementation approach.
We do not know whether anybody tries to implement traditional linear algebra
and other numerical algorithms using Blitz++. We also presume that even today
Blitz++ needs the most advanced C++ compiler technology due to its implementation
idioms. On the other hand, Blitz++ convinced us, that the use of expression
templates is mandatory to reduce the abstraction penalty to an acceptable
limit.</p>
<p>POOMA's design goals seem to parallel Blitz++'s in many parts . It extends
Blitz++'s concepts with classes from the domains of partial differential equations
and theoretical physics. The implementation supports even parallel architectures.</p>
<p>MTL is another approach supporting basic linear algebra operations in C++.
Its design mainly seems to be influenced by BLAS and the C++ Standard Template
Library. We share the insight that a linear algebra library has to provide
functionality comparable to BLAS. On the other hand we think, that the concepts
of the C++ standard library have not yet been proven to support numerical
computations as needed. As another difference MTL currently does not seem
to use expression templates. This may result in one of two consequences:
a possible loss of expressiveness or a possible loss of performance.</p>
<h2>Concepts</h2>
<h3>Mathematical Notation</h3>
<p>The usage of mathematical notation may ease the development of scientific
algorithms. So a C++ library implementing basic linear algebra concepts carefully
should overload selected C++ operators on matrix and vector classes.</p>
<p>We decided to use operator overloading for the following primitives:</p>
<table border="1">
<tbody>
<tr>
<th align="Left">Description</th>
<th align="Left">Operator</th>
</tr>
<tr>
<td>Indexing of vectors and matrices</td>
<td><code>vector::operator(size_t i);<br>
matrix::operator(size_t i, size_t j);</code></td>
</tr>
<tr>
<td>Assignment of vectors and matrices</td>
<td><code>vector::operator = (const vector_expression &amp;);<br>
vector::operator += (const vector_expression &amp;);<br>
vector::operator -= (const vector_expression &amp;);<br>
vector::operator *= (const scalar_expression &amp;);<br>
matrix::operator = (const matrix_expression &amp;);<br>
matrix::operator += (const matrix_expression &amp;);<br>
matrix::operator -= (const matrix_expression &amp;);<br>
matrix::operator *= (const scalar_expression &amp;);</code></td>
</tr>
<tr>
<td>Unary operations on vectors and matrices</td>
<td><code>vector_expression operator - (const vector_expression
&amp;);<br>
matrix_expression operator - (const matrix_expression &amp;);</code></td>
</tr>
<tr>
<td>Binary operations on vectors and matrices</td>
<td><code>vector_expression operator + (const vector_expression
&amp;, const vector_expression &amp;);<br>
vector_expression operator - (const vector_expression &amp;,
const vector_expression &amp;);<br>
matrix_expression operator + (const matrix_expression &amp;,
const matrix_expression &amp;);<br>
matrix_expression operator - (const matrix_expression &amp;,
const matrix_expression &amp;);</code></td>
</tr>
<tr>
<td>Multiplication of vectors and matrices with a scalar</td>
<td><code>vector_expression operator * (const scalar_expression
&amp;, const vector_expression &amp;);<br>
vector_expression operator * (const vector_expression &amp;,
const scalar_expression &amp;);<br>
matrix_expression operator * (const scalar_expression &amp;,
const matrix_expression &amp;);<br>
matrix_expression operator * (const matrix_expression &amp;,
const scalar_expression &amp;);</code></td>
</tr>
</tbody>
</table>
<p>We decided to use no operator overloading for the following other primitives:</p>
<table border="1">
<tbody>
<tr>
<th align="Left">Description</th>
<th align="Left">Function</th>
</tr>
<tr>
<td>Left multiplication of vectors with a matrix</td>
<td><code>vector_expression prod&lt;</code><code><em>vector_type</em></code><code>
&gt; (const matrix_expression &amp;, const vector_expression
&amp;);<br>
vector_expression prod (const matrix_expression &amp;,
const vector_expression &amp;);</code></td>
</tr>
<tr>
<td>Right multiplication of vectors with a matrix</td>
<td><code>vector_expression prod&lt;</code><code><em>vector_type</em></code><code>
&gt; (const vector_expression &amp;, const matrix_expression
&amp;);<br>
vector_expression prod (const vector_expression &amp;,
const matrix_expression &amp;);<br>
</code></td>
</tr>
<tr>
<td>Multiplication of matrices</td>
<td><code>matrix_expression prod&lt;</code><code><em>matrix_type</em></code><code>
&gt; (const matrix_expression &amp;, const matrix_expression
&amp;);<br>
matrix_expression prod (const matrix_expression &amp;,
const matrix_expression &amp;);</code></td>
</tr>
<tr>
<td>Inner product of vectors</td>
<td><code>scalar_expression inner_prod (const vector_expression
&amp;, const vector_expression &amp;);</code></td>
</tr>
<tr>
<td>Outer product of vectors</td>
<td><code>matrix_expression outer_prod (const vector_expression
&amp;, const vector_expression &amp;);</code></td>
</tr>
<tr>
<td>Transpose of a matrix</td>
<td><code>matrix_expression trans (const matrix_expression
&amp;);</code></td>
</tr>
</tbody>
</table>
<h3>Efficiency</h3>
<p>To achieve the goal of efficiency for numerical computing, one has to overcome
two difficulties in formulating abstractions with C++, namely temporaries
and virtual function calls. Expression templates solve these problems, but
tend to slow down compilation times.</p>
<h4>Eliminating Temporaries</h4>
<p>Abstract formulas on vectors and matrices normally compose a couple of
unary and binary operations. The conventional way of evaluating such a formula
is first to evaluate every leaf operation of a composition into a temporary
and next to evaluate the composite resulting in another temporary. This method
is expensive in terms of time especially for small and space especially for
large vectors and matrices. The approach to solve this problem is to use lazy
evaluation as known from modern functional programming languages. The principle
of this approach is to evaluate a complex expression element wise and to
assign it directly to the target.</p>
<p>Two interesting and dangerous facts result.</p>
<p>First one may get serious side effects using element wise evaluation on
vectors or matrices. Consider the matrix vector product <em>x = A x</em>.
Evaluation of <em>A</em><sub><em>1</em></sub><em>x</em> and assignment to
<em>x</em><sub><em>1</em></sub> changes the right hand side, so that the evaluation
of <em>A</em><sub><em>2</em></sub><em>x </em>returns a wrong result. Our
solution for this problem is to evaluate the right hand side of an assignment
into a temporary and then to assign this temporary to the left hand side.
To allow further optimizations, we provide a corresponding member function
for every assignment operator. By using this member function a programmer
can confirm, that the left and right hand sides of an assignment are independent,
so that element wise evaluation and direct assignment to the target is safe.</p>
<p>Next one can get worse computational complexity under certain cirumstances.
Consider the chained matrix vector product <em>A (B x)</em>. Conventional
evaluation of <em>A (B x)</em> is quadratic. Deferred evaluation of <em>B
x</em><sub><em>i</em></sub> is linear. As every element <em>B x</em><sub><em>
i</em></sub> is needed linearly depending of the size, a completely deferred
evaluation of the chained matrix vector product <em>A (B x)</em> is cubic.
In such cases one needs to reintroduce temporaries in the expression.</p>
<h4>Eliminating Virtual Function Calls</h4>
<p>Lazy expression evaluation normally leads to the definition of a class
hierarchy of terms. This results in the usage of dynamic polymorphism to access
single elements of vectors and matrices, which is also known to be expensive
in terms of time. A solution was found a couple of years ago independently
by David Vandervoorde and Todd Veldhuizen and is commonly called expression
templates. Expression templates contain lazy evaluation and replace dynamic
polymorphism with static, i.e. compile time polymorphism. Expression templates
heavily depend on the famous Barton-Nackman trick, also coined 'curiously
defined recursive templates' by Jim Coplien.</p>
<p>Expression templates form the base of our implementation.</p>
<h4>Compilation times</h4>
<p>It is also a well known fact, that expression templates challenge currently
available compilers. We were able to significantly reduce the amount of needed
expression templates using the Barton-Nackman trick consequently.</p>
<p>We also decided to support a dual conventional implementation (i.e. not
using expression templates) with extensive bounds and type checking of vector
and matrix operations to support the development cycle. Switching from debug
mode to release mode is controlled by the <code>NDEBUG</code> preprocessor
symbol of <code>&lt;cassert&gt;</code>.</p>
<h3>Functionality</h3>
<p>Every C++ library supporting linear algebra will be measured against the
long-standing Fortran package BLAS. We now describe how BLAS calls may be
mapped onto our classes. </p>
<h4>Blas Level 1</h4>
<table border="1">
<tbody>
<tr>
<th align="Left">BLAS Call</th>
<th align="Left">Mapped Library Expression</th>
<th align="Left">Mathematical Description</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td><code>_asum</code></td>
<td><code>norm_1 (x)</code> </td>
<td><em>sum |x</em><sub><em>i</em></sub><em>|</em></td>
<td>Computes the sum norm of a vector.</td>
</tr>
<tr>
<td><code>_nrm2</code></td>
<td><code>norm_2 (x)</code></td>
<td><em>sqrt (sum |x</em><sub><em>i</em></sub>|<sup><em>2</em></sup><em>
)</em></td>
<td>Computes the euclidean norm of a vector.</td>
</tr>
<tr>
<td><code>i_amax</code></td>
<td><code>norm_inf (x)<br>
norm_inf_index (x)</code></td>
<td><em>max |x</em><sub><em>i</em></sub><em>|</em></td>
<td>Computes the maximum norm of a vector.<br>
BLAS computes the index of the first element having this
value.</td>
</tr>
<tr>
<td><code>_dot<br>
_dotu<br>
_dotc</code></td>
<td><code>inner_prod (x, y)</code>or<code><br>
inner_prod (conj (x), y)</code></td>
<td><em>x</em><sup><em>T</em></sup><em> y</em> or<br>
<em>x</em><sup><em>H</em></sup><em> y</em></td>
<td>Computes the inner product of two vectors. <br>
BLAS implements certain loop unrollment.</td>
</tr>
<tr>
<td><code>dsdot<br>
sdsdot</code></td>
<td><code>a + prec_inner_prod (x, y)</code></td>
<td><em>a + x</em><sup><em>T</em></sup><em> y</em></td>
<td>Computes the inner product in double precision. </td>
</tr>
<tr>
<td><code>_copy</code></td>
<td><code>x = y <br>
y.assign (x)</code></td>
<td><em>x &lt;- y</em></td>
<td>Copies one vector to another. <br>
BLAS implements certain loop unrollment.</td>
</tr>
<tr>
<td><code>_swap</code></td>
<td><code>swap (x, y)</code></td>
<td><em>x &lt;-&gt; y</em></td>
<td>Swaps two vectors. <br>
BLAS implements certain loop unrollment.</td>
</tr>
<tr>
<td><code>_scal<br>
csscal<br>
zdscal</code></td>
<td><code>x *= a</code></td>
<td><em>x &lt;- a x</em></td>
<td>Scales a vector. <br>
BLAS implements certain loop unrollment.</td>
</tr>
<tr>
<td><code>_axpy</code></td>
<td><code>y += a * x</code></td>
<td><em>y &lt;- a x + y</em></td>
<td>Adds a scaled vector. <br>
BLAS implements certain loop unrollment.</td>
</tr>
<tr>
<td><code>_rot<br>
_rotm<br>
csrot<br>
zdrot</code></td>
<td><code>t.assign (a * x + b * y), <br>
y.assign (- b * x + a * y),<br>
x.assign (t)</code></td>
<td><em>(x, y) &lt;- (a x + b y, -b x + a y)</em></td>
<td>Applies a plane rotation.</td>
</tr>
<tr>
<td><code>_rotg<br>
_rotmg</code></td>
<td>&nbsp;</td>
<td><em>(a, b) &lt;- <br>
&nbsp; (? a / sqrt (a</em><sup><em>2</em></sup> + <em>b</em><sup><em>
2</em></sup><em>), <br>
&nbsp; &nbsp; ? b / sqrt (a</em><sup><em>2</em></sup> + <em>b</em><sup><em>
2</em></sup><em>)) </em>or<em><br>
(1, 0) &lt;- (0, 0)</em></td>
<td>Constructs a plane rotation.</td>
</tr>
</tbody>
</table>
<h4>Blas Level 2</h4>
<table border="1">
<tbody>
<tr>
<th align="Left">BLAS Call</th>
<th align="Left">Mapped Library Expression</th>
<th align="Left">Mathematical Description</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td><code>_t_mv</code></td>
<td><code>x = prod (A, x)</code> or<code><br>
x = prod (trans (A), x)</code> or<code><br>
x = prod (herm (A), x)</code></td>
<td><em>x &lt;- A x </em>or<em><br>
x &lt;- A</em><sup><em>T</em></sup><em> x </em>or<em><br>
x &lt;- A</em><sup><em>H</em></sup><em> x</em></td>
<td>Computes the product of a matrix with a vector.</td>
</tr>
<tr>
<td><code>_t_sv</code></td>
<td><code>y = solve (A, x, tag) </code>or<br>
<code>inplace_solve (A, x, tag) </code>or<br>
<code>y = solve (trans (A), x, tag) </code>or<code><br>
inplace_solve (trans (A), x, tag) </code>or<font face="Courier New"><code><br>
</code></font><code>y = solve (herm (A), x, tag)</code>or<code><br>
inplace_solve (herm (A), x, tag)</code></td>
<td><em>y &lt;- A</em><sup><em>-1</em></sup><em> x </em>or<em><br>
x &lt;- A</em><sup><em>-1</em></sup><em> x </em>or<em><br>
y &lt;- A</em><sup><em>T</em></sup><sup><sup><em>-1</em></sup></sup><em>
x </em>or<em><br>
x &lt;- A</em><sup><em>T</em></sup><sup><sup><em>-1</em></sup></sup><em>
x </em>or<em><br>
y &lt;- A</em><sup><em>H</em></sup><sup><sup><em>-1</em></sup></sup><em>
x</em> or<em><br>
x &lt;- A</em><sup><em>H</em></sup><sup><sup><em>-1</em></sup></sup><em>
x</em></td>
<td>Solves a system of linear equations with triangular
form, i.e. <em>A </em>is triangular.</td>
</tr>
<tr>
<td><code>_g_mv<br>
_s_mv<br>
_h_mv</code></td>
<td><code>y = a * prod (A, x) + b * y </code>or<code><br>
y = a * prod (trans (A), x) + b * y </code>or<code><br>
y = a * prod (herm (A), x) + b * y</code></td>
<td><em>y &lt;- a A x + b y </em>or<em><br>
y &lt;- a A</em><sup><em>T</em></sup><em> x + b y<br>
y &lt;- a A</em><sup><em>H</em></sup><em> x + b y</em></td>
<td>Adds the scaled product of a matrix with a vector.</td>
</tr>
<tr>
<td><code>_g_r<br>
_g_ru<br>
_g_rc</code></td>
<td><code>A += a * outer_prod (x, y)</code> or<code><br>
A += a * outer_prod (x, conj (y))</code></td>
<td><em>A &lt;- a x y</em><sup><em>T</em></sup><em> + A </em>or<em><br>
A &lt;- a x y</em><sup><em>H</em></sup><em> + A</em></td>
<td>Performs a rank <em>1</em> update.</td>
</tr>
<tr>
<td><code>_s_r<br>
_h_r</code></td>
<td><code>A += a * outer_prod (x, x)</code> or<code><br>
A += a * outer_prod (x, conj (x))</code></td>
<td><em>A &lt;- a x x</em><sup><em>T</em></sup><em> + A</em>
or<em><br>
A &lt;- a x x</em><sup><em>H</em></sup><em> + A</em></td>
<td>Performs a symmetric or hermitian rank <em>1</em> update.</td>
</tr>
<tr>
<td><code>_s_r2<br>
_h_r2</code></td>
<td><code>A += a * outer_prod (x, y) +<br>
&nbsp;a * outer_prod (y, x)) </code>or<code><br>
A += a * outer_prod (x, conj (y)) +<br>
&nbsp;conj (a) * outer_prod (y, conj (x)))</code></td>
<td><em>A &lt;- a x y</em><sup><em>T</em></sup><em> + a y
x</em><sup><em>T</em></sup><em> + A</em> or<em><br>
A &lt;- a x y</em><sup><em>H</em></sup><em> + a</em><sup><em>-</em></sup><em>
y x</em><sup><em>H</em></sup><em> + A</em> </td>
<td>Performs a symmetric or hermitian rank <em>2</em> update.</td>
</tr>
</tbody>
</table>
<h4>Blas Level 3</h4>
<table border="1">
<tbody>
<tr>
<th align="Left">BLAS Call</th>
<th align="Left">Mapped Library Expression</th>
<th align="Left">Mathematical Description</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td><code>_t_mm</code></td>
<td><code>B = a * prod (A, B) </code>or<br>
<code>B = a * prod (trans (A), B) </code>or<br>
<code>B = a * prod (A, trans (B)) </code>or<br>
<code>B = a * prod (trans (A), trans (B)) </code>or<br>
<code>B = a * prod (herm (A), B) </code>or<br>
<code>B = a * prod (A, herm (B)) </code>or<br>
<code>B = a * prod (herm (A), trans (B)) </code>or<br>
<code>B = a * prod (trans (A), herm (B)) </code>or<br>
<code>B = a * prod (herm (A), herm (B))</code></td>
<td><em>B &lt;- a op (A) op (B) </em>with<br>
&nbsp; <em>op (X) = X </em>or<br>
&nbsp; <em>op (X) = X</em><sup><em>T</em></sup><em> </em>or<br>
&nbsp; <em>op (X) = X</em><sup><em>H</em></sup></td>
<td>Computes the scaled product of two matrices.</td>
</tr>
<tr>
<td><code>_t_sm</code></td>
<td><code>C = solve (A, B, tag) </code>or<br>
<code>inplace_solve (A, B, tag) </code>or<br>
<code>C = solve (trans (A), B, tag) </code>or<code><br>
inplace_solve (trans (A), B, tag) </code>or<code><br>
C = solve (herm (A), B, tag)</code> or<code><br>
inplace_solve (herm (A), B, tag)</code></td>
<td><em>C &lt;- A</em><sup><em>-1</em></sup><em> B </em>or<em><br>
B &lt;- A</em><sup><em>-1</em></sup><em> B </em>or<em><br>
C &lt;- A</em><sup><em>T</em></sup><sup><sup><em>-1</em></sup></sup><em>
B </em>or<em><br>
B &lt;- A</em><sup><em>-1</em></sup><em> B </em>or<em><br>
C &lt;- A</em><sup><em>H</em></sup><sup><sup><em>-1</em></sup></sup><em>
B</em> or<em><br>
B &lt;- A</em><sup><em>H</em></sup><sup><sup><em>-1</em></sup></sup><em>
B</em></td>
<td>Solves a system of linear equations with triangular
form, i.e. <em>A </em>is triangular.</td>
</tr>
<tr>
<td><code>_g_mm<br>
_s_mm<br>
_h_mm</code></td>
<td><code>C = a * prod (A, B) + b * C </code>or<br>
<code>C = a * prod (trans (A), B) + b * C </code>or<br>
<code>C = a * prod (A, trans (B)) + b * C </code>or<br>
<code>C = a * prod (trans (A), trans (B)) + b * C </code>or<br>
<code>C = a * prod (herm (A), B) + b * C </code>or<br>
<code>C = a * prod (A, herm (B)) + b * C </code>or<br>
<code>C = a * prod (herm (A), trans (B)) + b * C </code>or<br>
<code>C = a * prod (trans (A), herm (B)) + b * C </code>or<br>
<code>C = a * prod (herm (A), herm (B)) + b * C</code></td>
<td><em>C &lt;- a op (A) op (B) + b C </em>with<br>
&nbsp; <em>op (X) = X </em>or<br>
&nbsp; <em>op (X) = X</em><sup><em>T</em></sup><em> </em>or<br>
&nbsp; <em>op (X) = X</em><sup><em>H</em></sup></td>
<td>Adds the scaled product of two matrices.</td>
</tr>
<tr>
<td><code>_s_rk<br>
_h_rk</code></td>
<td><code>B = a * prod (A, trans (A)) + b * B </code>or<br>
<code>B = a * prod (trans (A), A) + b * B </code>or<br>
<code>B = a * prod (A, herm (A)) + b * B </code>or<br>
<code>B = a * prod (herm (A), A) + b * B</code></td>
<td><em>B &lt;- a A A</em><sup><em>T</em></sup><em> + b B
</em>or<em><br>
B &lt;- a A</em><sup><em>T</em></sup><em> A + b B </em>or<br>
<em>B &lt;- a A A</em><sup><em>H</em></sup><em> + b B </em>or<em><br>
B &lt;- a A</em><sup><em>H</em></sup><em> A + b B </em></td>
<td>Performs a symmetric or hermitian rank <em>k</em> update.</td>
</tr>
<tr>
<td><code>_s_r2k<br>
_h_r2k</code></td>
<td><code>C = a * prod (A, trans (B)) +<br>
&nbsp;a * prod (B, trans (A)) + b * C </code>or<br>
<code>C = a * prod (trans (A), B) +<br>
&nbsp;a * prod (trans (B), A) + b * C </code>or<br>
<code>C = a * prod (A, herm (B)) +<br>
&nbsp;conj (a) * prod (B, herm (A)) + b * C </code>or<br>
<code>C = a * prod (herm (A), B) +<br>
&nbsp;conj (a) * prod (herm (B), A) + b * C</code></td>
<td><em>C &lt;- a A B</em><sup><em>T</em></sup><em> + a B
A</em><sup><em>T</em></sup><em> + b C </em>or<em><br>
C &lt;- a A</em><sup><em>T</em></sup><em> B + a B</em><sup><em>
T</em></sup><em> A + b C </em>or<em><br>
C &lt;- a A B</em><sup><em>H</em></sup><em> + a</em><sup><em>-</em></sup><em>
B A</em><sup><em>H</em></sup><em> + b C</em> or<em><br>
C &lt;- a A</em><sup><em>H</em></sup><em> B + a</em><sup><em>-</em></sup><em>
B</em><sup><em>H</em></sup><em> A + b C</em></td>
<td>Performs a symmetric or hermitian rank <em>2 k</em>
update.</td>
</tr>
</tbody>
</table>
<h4>Storage Layouts</h4>
<p>The library supports conventional dense, packed and basic sparse vector
and matrix storage layouts. The description of the most common constructions
of vectors and matrices comes next.</p>
<table border="1">
<tbody>
<tr>
<th align="Left">Construction</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td><code>vector&lt;T,<br>
&nbsp;std::vector&lt;T&gt; &gt; <br>
&nbsp;&nbsp;v (size)</code></td>
<td>Constructs a dense vector, storage is provided by a
standard vector.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>vector&lt;T,<br>
&nbsp;unbounded_array&lt;T&gt; &gt; <br>
&nbsp;&nbsp;v (size)</code></td>
<td>Constructs a dense vector, storage is provided by a
heap-based array.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>vector&lt;T,<br>
&nbsp;bounded_array&lt;T, N&gt; &gt; <br>
&nbsp;&nbsp;v (size)</code></td>
<td>Constructs a dense vector, storage is provided by a
stack-based array.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>unit_vector&lt;T&gt; <br>
&nbsp;&nbsp;v (size, index)</code></td>
<td>Constructs the <code>index</code>-th canonical unit
vector.</td>
</tr>
<tr>
<td><code>zero_vector&lt;T&gt; <br>
&nbsp;&nbsp;v (size)</code></td>
<td>Constructs a zero vector.</td>
</tr>
<tr>
<td><code>sparse_vector&lt;T,<br>
&nbsp;std::map&lt;std::size_t, T&gt; &gt; <br>
&nbsp;&nbsp;v (size, non_zeros)</code></td>
<td>Constructs a sparse vector, storage is provided by a
standard map.</td>
</tr>
<tr>
<td><code>sparse_vector&lt;T,<br>
&nbsp;map_array&lt;std::size_t, T&gt; &gt; <br>
&nbsp;&nbsp;v (size, non_zeros)</code></td>
<td>Constructs a sparse vector, storage is provided by a
map array.</td>
</tr>
<tr>
<td><code>compressed_vector&lt;T&gt; <br>
&nbsp;&nbsp;v (size, non_zeros)</code></td>
<td>Constructs a compressed vector.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>coordinate_vector&lt;T&gt; <br>
&nbsp;&nbsp;v (size, non_zeros)</code></td>
<td>Constructs a coordinate vector.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>vector_range&lt;V&gt; <br>
&nbsp;&nbsp;vr (v, range)</code></td>
<td>Constructs a sub vector of a dense, packed or sparse
vector.</td>
</tr>
<tr>
<td><code>vector_slice&lt;V&gt; <br>
&nbsp;&nbsp;vs (v, slice)</code></td>
<td>Constructs a sub vector of a dense, packed or sparse
vector.<br>
This class usually can be used to emulate BLAS vector slices.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;row_major,<br>
&nbsp;std::vector&lt;T&gt; &gt; <br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is row major,
storage is provided by a standard vector.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;column_major,<br>
&nbsp;std::vector&lt;T&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is column major,
storage is provided by a standard vector.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;row_major, <br>
&nbsp;unbounded_array&lt;T&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is row major,
storage is provided by a heap-based array.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;column_major,<br>
&nbsp;unbounded_array&lt;T&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is column major,
storage is provided by a heap-based array.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;row_major,<br>
&nbsp;bounded_array&lt;T, N1 * N2&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is row major,
storage is provided by a stack-based array.</td>
</tr>
<tr>
<td><code>matrix&lt;T,<br>
&nbsp;column_major,<br>
&nbsp;bounded_array&lt;T, N1 * N2&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a dense matrix, orientation is column major,
storage is provided by a stack-based array.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>identity_matrix&lt;T&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs an identity matrix.</td>
</tr>
<tr>
<td><code>zero_matrix&lt;T&gt;<br>
&nbsp;&nbsp;m (size1, size2)</code></td>
<td>Constructs a zero matrix.</td>
</tr>
<tr>
<td><code>triangular_matrix&lt;T,<br>
&nbsp;row_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed triangular matrix, orientation is
row major.</td>
</tr>
<tr>
<td><code>triangular_matrix&lt;T,<br>
&nbsp;column_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed triangular matrix, orientation is
column major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>banded_matrix&lt;T,<br>
&nbsp;row_major, A&gt;<br>
&nbsp;&nbsp;m (size1, size2, lower, upper)</code></td>
<td>Constructs a packed banded matrix, orientation is row
major.</td>
</tr>
<tr>
<td><code>banded_matrix&lt;T,<br>
&nbsp;column_major, A&gt;<br>
&nbsp;&nbsp;m (size1, size2, lower, upper)</code></td>
<td>Constructs a packed banded matrix, orientation is column
major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>symmetric_matrix&lt;T,<br>
&nbsp;row_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed symmetric matrix, orientation is
row major.</td>
</tr>
<tr>
<td><code>symmetric_matrix&lt;T,<br>
&nbsp;column_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed symmetric matrix, orientation is
column major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>hermitian_matrix&lt;T,<br>
&nbsp;row_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed hermitian matrix, orientation is
row major.</td>
</tr>
<tr>
<td><code>hermitian_matrix&lt;T,<br>
&nbsp;column_major, F, A&gt;<br>
&nbsp;&nbsp;m (size)</code></td>
<td>Constructs a packed hermitian matrix, orientation is
column major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>sparse_matrix&lt;T, <br>
&nbsp;row_major,<br>
&nbsp;std::map&lt;std::size_t, T&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a sparse matrix, orientation is row major,
storage is provided by a standard map.</td>
</tr>
<tr>
<td><code>sparse_matrix&lt;T, <br>
&nbsp;column_major,<br>
&nbsp;std::map&lt;std::size_t, T&gt; &gt;<br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a sparse matrix, orientation is column major,
storage is provided by a standard map.</td>
</tr>
<tr>
<td><code>sparse_matrix&lt;T, <br>
&nbsp;row_major,<br>
&nbsp;map_array&lt;std::size_t, T&gt; &gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a sparse matrix, orientation is row major,
storage is provided by a map array.</td>
</tr>
<tr>
<td><code>sparse_matrix&lt;T, <br>
&nbsp;column_major,<br>
&nbsp;map_array&lt;std::size_t, T&gt; &gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a sparse matrix, orientation is column major,
storage is provided by a map array.</td>
</tr>
<tr>
<td><code>compressed_matrix&lt;T, <br>
&nbsp;row_major&gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a compressed matrix, orientation is row
major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>compressed_matrix&lt;T, <br>
&nbsp;column_major&gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a compressed matrix, orientation is column
major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>coordinate_matrix&lt;T, <br>
&nbsp;row_major&gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a coordinate matrix, orientation is row
major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>coordinate_matrix&lt;T, <br>
&nbsp;column_major&gt; <br>
&nbsp;&nbsp;m (size1, size2, non_zeros)</code></td>
<td>Constructs a coordinate matrix, orientation is column
major.<br>
The storage layout usually is BLAS compliant.</td>
</tr>
<tr>
<td><code>matrix_row&lt;M&gt; <br>
&nbsp;&nbsp;mr (m, i)</code></td>
<td>Constructs a sub vector of a dense, packed or sparse
matrix, containing the i-th row.</td>
</tr>
<tr>
<td><code>matrix_column&lt;M&gt; <br>
&nbsp;&nbsp;mc (m, j)</code></td>
<td>Constructs a sub vector of a dense, packed or sparse
matrix, containing the j-th column.</td>
</tr>
<tr>
<td><code>matrix_range&lt;M&gt; <br>
&nbsp;&nbsp;mr (m, range1, range2)</code></td>
<td>Constructs a sub matrix of a dense, packed or sparse
matrix.<br>
This class usually can be used to emulate BLAS leading
dimensions.</td>
</tr>
<tr>
<td><code>matrix_slice&lt;M&gt; <br>
&nbsp;&nbsp;ms (m, slice1, slice2)</code></td>
<td>Constructs a sub matrix of a dense, packed or sparse
matrix.</td>
</tr>
<tr>
<td><code>triangular_adaptor&lt;M, F&gt;<br>
&nbsp;&nbsp;ta (m)</code></td>
<td>Constructs a triangular view of a dense, packed or
sparse matrix.<br>
This class usually can be used to generate corresponding
BLAS matrix types.</td>
</tr>
<tr>
<td><code>banded_adaptor&lt;M&gt;<br>
&nbsp;&nbsp;ba (m, lower, upper)</code></td>
<td>Constructs a banded view of a dense, packed or sparse
matrix.<br>
This class usually can be used to generate corresponding
BLAS matrix types.</td>
</tr>
<tr>
<td><code>symmetric_adaptor&lt;M&gt;<br>
&nbsp;&nbsp;sa (m)</code></td>
<td>Constructs a symmetric view of a dense, packed or sparse
matrix.<br>
This class usually can be used to generate corresponding
BLAS matrix types.</td>
</tr>
<tr>
<td><code>hermitian_adaptor&lt;M&gt;<br>
&nbsp;&nbsp;ha (m)</code></td>
<td>Constructs a hermitian view of a dense, packed or sparse
matrix.<br>
This class usually can be used to generate corresponding
BLAS matrix types.</td>
</tr>
</tbody>
</table>
<h3>Compatibility</h3>
<p>For compatibility reasons we provide array like indexing for vectors and
matrices, although this could be expensive for matrices due to the needed
temporary proxy objects.</p>
<p>To support the most widely used C++ compilers our design and implementation
do not depend on partial template specialization essentially.</p>
<p>The library presumes standard compliant allocation through <code>operator
new </code>and <code>operator delete</code>. So programs which are intended
to run under MSVC 6.0 should set a correct new handler throwing a <code>std::bad_alloc</code>
exception via <code>_set_new_handler</code> to detect out of memory conditions.</p>
<p>To get the most performance out of the box with MSVC 6.0, you should change
the preprocessor definition of <code>BOOST_UBLAS_INLINE </code>to <code>__forceinline
</code>in the header file config.hpp. But we suspect this optimization to
be fragile.</p>
<h2>Reference</h2>
<ul>
<li><a href="expression.htm">Expression Concepts</a>
<ul type="Circle">
<li><a href="expression.htm#scalar_expression">Scalar
Expression</a>
</li>
<li><a href="expression.htm#vector_expression">Vector
Expression</a>
</li>
<li><a href="expression.htm#matrix_expression">Matrix
Expression</a>
</li>
</ul>
</li>
<li><a href="container.htm">Container Concepts</a>
<ul type="Circle">
<li><a href="container.htm#vector">Vector</a>
</li>
<li><a href="container.htm#matrix">Matrix</a>
</li>
</ul>
</li>
<li><a href="iterator.htm">Iterator Concepts</a>
<ul type="Circle">
<li><a href="iterator.htm#indexed_bidirectional_iterator">Indexed
Bidirectional Iterator</a>
</li>
<li><a href="iterator.htm#indexed_random_access_iterator">Indexed
Random Access Iterator</a>
</li>
<li><a href="iterator.htm#indexed_bidirectional_cr_iterator">
Indexed Bidirectional Column/Row Iterator</a>
</li>
<li><a href="iterator.htm#indexed_random_access_cr_iterator">
Indexed Random Access Column/Row Iterator</a>
</li>
</ul>
</li>
<li><a href="storage.htm">Storage</a>
<ul type="Circle">
<li><a href="storage.htm#unbounded_array">Unbounded
Array</a>
</li>
<li><a href="storage.htm#bounded_array">Bounded Array</a>
</li>
<li><a href="storage.htm#range">Range</a>
</li>
<li><a href="storage.htm#slice">Slice</a>
</li>
</ul>
</li>
<li><a href="storage_sparse.htm">Sparse Storage</a>
<ul type="Circle">
<li><a href="storage_sparse.htm#map_array">Map Array</a>
</li>
</ul>
</li>
<li><a href="vector.htm">Vector</a>
<ul type="Circle">
<li><a href="vector.htm#vector">Vector</a>
</li>
<li><a href="vector.htm#unit_vector">Unit Vector</a>
</li>
<li><a href="vector.htm#zero_vector">Zero Vector</a>
</li>
</ul>
</li>
<li><a href="vector_sparse.htm">Sparse Vector</a>
<ul type="Circle">
<li><a href="vector_sparse.htm#sparse_vector">Sparse
Vector</a>
</li>
<li><a href="vector_sparse.htm#compressed_vector">Compressed
Vector</a>
</li>
<li><a href="vector_sparse.htm#coordinate_vector">Coordinate
Vector</a>
</li>
</ul>
</li>
<li><a href="vector_proxy.htm">Vector Proxies</a>
<ul type="Circle">
<li><a href="vector_proxy.htm#vector_range">Vector
Range</a>
</li>
<li><a href="vector_proxy.htm#vector_slice">Vector
Slice</a>
</li>
</ul>
</li>
<li><a href="vector_expression.htm">Vector Expressions</a>
<ul type="Circle">
<li><a href="vector_expression.htm#vector_expression">Vector
Expression</a>
</li>
<li><a href="vector_expression.htm#vector_references">Vector
References</a>
</li>
<li><a href="vector_expression.htm#vector_operations">Vector
Operations</a>
</li>
<li><a href="vector_expression.htm#vector_reductions">Vector
Reductions</a>
</li>
</ul>
</li>
<li><a href="matrix.htm">Matrix</a>
<ul type="Circle">
<li><a href="matrix.htm#matrix">Matrix</a>
</li>
<li><a href="matrix.htm#identity_matrix">Identity
Matrix</a>
</li>
<li><a href="matrix.htm#zero_matrix">Zero Matrix</a>
</li>
</ul>
</li>
<li><a href="triangular.htm">Triangular Matrix</a>
<ul type="Circle">
<li><a href="triangular.htm#triangular_matrix">Triangular
Matrix</a>
</li>
<li><a href="triangular.htm#triangular_adaptor">Triangular
Adaptor</a>
</li>
</ul>
</li>
<li><a href="symmetric.htm">Symmetric Matrix</a>
<ul type="Circle">
<li><a href="symmetric.htm#symmetric_matrix">Symmetric
Matrix</a>
</li>
<li><a href="symmetric.htm#symmetric_adaptor">Symmetric
Adaptor</a>
</li>
</ul>
</li>
<li><a href="hermitian.htm">Hermitian Matrix</a>
<ul type="Circle">
<li><a href="hermitian.htm#hermitian_matrix">Hermitian
Matrix</a>
</li>
<li><a href="hermitian.htm#hermitian_adaptor">Hermitian
Adaptor</a>
</li>
</ul>
</li>
<li><a href="banded.htm">Banded Matrix</a>
<ul type="Circle">
<li><a href="banded.htm#banded_matrix">Banded Matrix</a>
</li>
<li><a href="banded.htm#banded_adaptor">Banded
Adaptor</a>
</li>
</ul>
</li>
<li><a href="matrix_sparse.htm">Sparse Matrix</a>
<ul type="Circle">
<li><a href="matrix_sparse.htm#sparse_matrix">Sparse
Matrix</a>
</li>
<li><a href="matrix_sparse.htm#compressed_matrix">Compressed
Matrix</a>
</li>
<li><a href="matrix_sparse.htm#coordinate_matrix">Coordinate
Matrix</a>
</li>
</ul>
</li>
<li><a href="matrix_proxy.htm">Matrix Proxies</a>
<ul type="Circle">
<li><a href="matrix_proxy.htm#matrix_row">Matrix Row</a>
</li>
<li><a href="matrix_proxy.htm#matrix_column">Matrix
Column</a>
</li>
<li><a href="matrix_proxy.htm#vector_range">Vector
Range</a>
</li>
<li><a href="matrix_proxy.htm#vector_slice">Vector
Slice</a>
</li>
<li><a href="matrix_proxy.htm#matrix_range">Matrix
Range</a>
</li>
<li><a href="matrix_proxy.htm#matrix_slice">Matrix
Slice</a>
</li>
</ul>
</li>
<li><a href="matrix_expression.htm">Matrix Expressions</a>
<ul type="Circle">
<li><a href="matrix_expression.htm#matrix_expression">Matrix
Expression</a>
</li>
<li><a href="matrix_expression.htm#matrix_references">Matrix
References</a>
</li>
<li><a href="matrix_expression.htm#matrix_operations">Matrix
Operations</a>
</li>
<li><a href="matrix_expression.htm#matrix_vector_operations">
Matrix Vector Operations</a>
</li>
<li><a href="matrix_expression.htm#matrix_matrix_operations">
Matrix Matrix Operations</a>
</li>
</ul>
</li>
</ul>
<h2>Benchmark Results</h2>
<p>The following tables contain results of one of our benchmarks. This benchmark
compares a native C implementation ('C array') and some library based implementations.
The safe variants based on the library assume aliasing, the fast variants
do not use temporaries and are functionally equivalent to the native C implementation.
Besides the generic vector and matrix classes the benchmark utilizes special
classes <code>c_vector</code> and <code>c_matrix</code>, which are intended
to avoid every overhead through genericity.</p>
<p>The benchmark program was compiled with MSVC 6.0 and run on an Intel Pentium
II with 333 Mhz. First we comment the results for double vectors and matrices
of dimension 3 and 3 x 3, respectively.</p>
<table border="1">
<tbody>
<tr>
<th align="Left">Operation</th>
<th align="Left">Implementation</th>
<th align="Left">Elapsed [s]</th>
<th align="Left">MFLOP/s</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td rowspan="4">inner_prod</td>
<td>C array</td>
<td align="Right">0.1 </td>
<td align="Right">47.6837 </td>
<td rowspan="4">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_vector</td>
<td align="Right">0.06 </td>
<td align="Right">79.4729 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt;</td>
<td align="Right">0.11 </td>
<td align="Right">43.3488 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt;</td>
<td align="Right">0.11 </td>
<td align="Right">43.3488 </td>
</tr>
<tr>
<td rowspan="7">vector + vector</td>
<td>C array</td>
<td align="Right">0.05 </td>
<td align="Right">114.441 </td>
<td rowspan="7">Abstraction penalty: factor 2</td>
</tr>
<tr>
<td>c_vector safe</td>
<td align="Right">0.22 </td>
<td align="Right">26.0093 </td>
</tr>
<tr>
<td>c_vector fast</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt; safe</td>
<td align="Right">1.05 </td>
<td align="Right">5.44957 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt; fast</td>
<td align="Right">0.16 </td>
<td align="Right">35.7628 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt; safe</td>
<td align="Right">1.16 </td>
<td align="Right">4.9328 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt; fast</td>
<td align="Right">0.16 </td>
<td align="Right">35.7628 </td>
</tr>
<tr>
<td rowspan="7">outer_prod</td>
<td>C array</td>
<td align="Right">0.06 </td>
<td align="Right">85.8307 </td>
<td rowspan="7">Abstraction penalty: factor 2</td>
</tr>
<tr>
<td>c_matrix, c_vector safe</td>
<td align="Right">0.22 </td>
<td align="Right">23.4084 </td>
</tr>
<tr>
<td>c_matrix, c_vector fast</td>
<td align="Right">0.11 </td>
<td align="Right">46.8167 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
safe</td>
<td align="Right">0.38 </td>
<td align="Right">13.5522 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
fast</td>
<td align="Right">0.16 </td>
<td align="Right">32.1865 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
safe</td>
<td align="Right">0.5 </td>
<td align="Right">10.2997 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
fast</td>
<td align="Right">0.11 </td>
<td align="Right">46.8167 </td>
</tr>
<tr>
<td rowspan="7">prod (matrix, vector)</td>
<td>C array</td>
<td align="Right">0.06 </td>
<td align="Right">71.5256 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix, c_vector safe</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>c_matrix, c_vector fast</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
safe</td>
<td align="Right">0.33 </td>
<td align="Right">13.0046 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
fast</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
safe</td>
<td align="Right">0.38 </td>
<td align="Right">11.2935 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
fast</td>
<td align="Right">0.05 </td>
<td align="Right">85.8307 </td>
</tr>
<tr>
<td rowspan="7">matrix + matrix</td>
<td>C array</td>
<td align="Right">0.11 </td>
<td align="Right">46.8167 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix safe</td>
<td align="Right">0.17 </td>
<td align="Right">30.2932 </td>
</tr>
<tr>
<td>c_matrix fast</td>
<td align="Right">0.11 </td>
<td align="Right">46.8167 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; safe</td>
<td align="Right">0.44 </td>
<td align="Right">11.7042 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; fast</td>
<td align="Right">0.16 </td>
<td align="Right">32.1865 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; safe</td>
<td align="Right">0.6 </td>
<td align="Right">8.58307 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; fast</td>
<td align="Right">0.17 </td>
<td align="Right">30.2932 </td>
</tr>
<tr>
<td rowspan="7">prod (matrix, matrix)</td>
<td>C array</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix safe</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>c_matrix fast</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; safe</td>
<td align="Right">0.22 </td>
<td align="Right">19.507 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; fast</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; safe</td>
<td align="Right">0.27 </td>
<td align="Right">15.8946 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; fast</td>
<td align="Right">0.11 </td>
<td align="Right">39.0139 </td>
</tr>
</tbody>
</table>
<p>We notice a twofold performance loss for small vectors and matrices: first
the general abstraction penalty for using classes, and then a small loss when
using the generic vector and matrix classes. The difference w.r.t. alias assumptions
is also significant.</p>
<p>Next we comment the results for double vectors and matrices of dimension
100 and 100 x 100, respectively.</p>
<table border="1">
<tbody>
<tr>
<th align="Left">Operation</th>
<th align="Left">Implementation</th>
<th align="Left">Elapsed [s]</th>
<th align="Left">MFLOP/s</th>
<th align="Left">Comment</th>
</tr>
<tr>
<td rowspan="4">inner_prod</td>
<td>C array</td>
<td align="Right">0.05 </td>
<td align="Right">113.869 </td>
<td rowspan="4">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_vector</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt;</td>
<td align="Right">0.05 </td>
<td align="Right">113.869 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt;</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
</tr>
<tr>
<td rowspan="7">vector + vector</td>
<td>C array</td>
<td align="Right">0.05 </td>
<td align="Right">114.441 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_vector safe</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td>c_vector fast</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt; safe</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td>vector&lt;unbounded_array&gt; fast</td>
<td align="Right">0.06 </td>
<td align="Right">95.3674 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt; safe</td>
<td align="Right">0.17 </td>
<td align="Right">33.6591 </td>
</tr>
<tr>
<td>vector&lt;std::vector&gt; fast</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td rowspan="7">outer_prod</td>
<td>C array</td>
<td align="Right">0.05 </td>
<td align="Right">114.441 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix, c_vector safe</td>
<td align="Right">0.28 </td>
<td align="Right">20.4359 </td>
</tr>
<tr>
<td>c_matrix, c_vector fast</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
safe</td>
<td align="Right">0.27 </td>
<td align="Right">21.1928 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
fast</td>
<td align="Right">0.06 </td>
<td align="Right">95.3674 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
safe</td>
<td align="Right">0.28 </td>
<td align="Right">20.4359 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
fast</td>
<td align="Right">0.11 </td>
<td align="Right">52.0186 </td>
</tr>
<tr>
<td rowspan="7">prod (matrix, vector)</td>
<td>C array</td>
<td align="Right">0.11 </td>
<td align="Right">51.7585 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix, c_vector safe</td>
<td align="Right">0.11 </td>
<td align="Right">51.7585 </td>
</tr>
<tr>
<td>c_matrix, c_vector fast</td>
<td align="Right">0.05 </td>
<td align="Right">113.869 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
safe</td>
<td align="Right">0.11 </td>
<td align="Right">51.7585 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt;, vector&lt;unbounded_array&gt;
fast</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
safe</td>
<td align="Right">0.1 </td>
<td align="Right">56.9344 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt;, vector&lt;std::vector&gt;
fast</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
</tr>
<tr>
<td rowspan="7">matrix + matrix</td>
<td>C array</td>
<td align="Right">0.22 </td>
<td align="Right">26.0093 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix safe</td>
<td align="Right">0.49 </td>
<td align="Right">11.6776 </td>
</tr>
<tr>
<td>c_matrix fast</td>
<td align="Right">0.22 </td>
<td align="Right">26.0093 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; safe</td>
<td align="Right">0.39 </td>
<td align="Right">14.6719 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; fast</td>
<td align="Right">0.22 </td>
<td align="Right">26.0093 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; safe</td>
<td align="Right">0.44 </td>
<td align="Right">13.0046 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; fast</td>
<td align="Right">0.27 </td>
<td align="Right">21.1928 </td>
</tr>
<tr>
<td rowspan="7">prod (matrix, matrix)</td>
<td>C array</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
<td rowspan="7">No significant abstraction penalty</td>
</tr>
<tr>
<td>c_matrix safe</td>
<td align="Right">0.06 </td>
<td align="Right">94.8906 </td>
</tr>
<tr>
<td>c_matrix fast</td>
<td align="Right">0.05 </td>
<td align="Right">113.869 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; safe</td>
<td align="Right">0.11 </td>
<td align="Right">51.7585 </td>
</tr>
<tr>
<td>matrix&lt;unbounded_array&gt; fast</td>
<td align="Right">0.17 </td>
<td align="Right">33.4908 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; safe</td>
<td align="Right">0.11 </td>
<td align="Right">51.7585 </td>
</tr>
<tr>
<td>matrix&lt;std::vector&gt; fast</td>
<td align="Right">0.16 </td>
<td align="Right">35.584 </td>
</tr>
</tbody>
</table>
<p>For larger vectors and matrices the general abstraction penalty for using
classes seems to decrease, the small loss when using generic vector and matrix
classes seems to remain. The difference w.r.t. alias assumptions remains visible,
too.</p>
<hr>
<p>Copyright (&copy;) 2000-2002 Joerg Walter, Mathias Koch <br>
Permission to copy, use, modify, sell and distribute this document is granted
provided this copyright notice appears in all copies. This document is provided
``as is'' without express or implied warranty, and with no claim as to its
suitability for any purpose.</p>
<p>Last revised: 1/15/2003</p>
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