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<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"
width="277" height="86"> 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">
<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>
</table>
<p>We decided to use no operator overloading for the following
other primitives:</p>
<table border="1">
<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>
</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">
<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>
</table>
<h4>Blas Level 2</h4>
<table border="1">
<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>
</table>
<h4>Blas Level 3</h4>
<table border="1">
<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>
</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">
<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>
</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_reductions">Matrix
Reductions</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">
<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>
</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">
<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>
</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 (<28>) 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: 8/3/2002</p>
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