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dune-istl
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39d16545
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Commit
39d16545
authored
8 years ago
by
Christoph Grüninger
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[doc] Fix typos and duplicate words
parent
4bbc8f79
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39d16545
...
...
@@ -83,8 +83,8 @@ there are many libraries available on the internet for doing sparse matrix/vecto
computations. A comprehensive overview is given in
\cite
{
LALinks
}
.
The
widely availabl
y
Basic Linear Algebra Subprograms (BLAS) standard has
been extended to cover a
l
s sparse matrices
\cite
{
BLASTForum
}
. BLAS
widely availabl
e
Basic Linear Algebra Subprograms (BLAS) standard has
been extended to cover as sparse matrices
\cite
{
BLASTForum
}
. BLAS
divides the available functions into level 1 (vector operations),
level 2 (vector/matrix operations) and level 3 (matrix/matrix
operations). BLAS for sparse matrices contains only level 1 and 2
...
...
@@ -94,7 +94,7 @@ only a FORTRAN and C interface. As a consequence, the interface is
``coarse grained'', meaning that ``small'' functions such as access to
individual matrix elements is relatively slow.
Generic programming techniqes in C++ offer the possibility to combine
Generic programming techniq
u
es in C++ offer the possibility to combine
flexibility and reuse (``efficiency of the programmer'') with fast
execution (``efficieny of the program'') as has been demonstrated with
the Standard Template Library (STL),
\cite
{
Stroustrup
}
or the Blitz++
...
...
@@ -174,7 +174,7 @@ In mathemetics vectors are elements of a vector space. A vector space
$
V
(
\K
)
$
, defined over a field
$
\K
$
, is a set of elements with two
operations: (i) vector space addition
$
+
: V
\times
V
\to
V
$
and (ii) scalar
multiplication
$
*
:
\K\times
V
\to
V
$
. These operations obey certain formal
rules, see your favo
u
rite textbook on linear algebra,
rules, see your favorite textbook on linear algebra,
e.~g.
\cite
{
LaBook
}
. In addition a
vector space may be normed, i.~e.~there is a function (obeying certain
rules)
$
\|
.
\|
: V
\to
\R
$
which measures distance in the vector
...
...
@@ -187,7 +187,7 @@ field, such as $\K=\R$ or $\K=\C$ and take a tensor product:
V =
\K
^
n =
\underbrace
{
\K\times\K\times\ldots\times\K
}_{
\text
{$
n
$
times
}}
.
\end{equation*}
$
n
\in\N
$
is called the dimension of the vector space. There are also
infinite-dimensional vector spaceswhich are, however, not of interest
infinite-dimensional vector spaces
which are, however, not of interest
in the context here. The idea of tensor products can be generalized.
If we have vector spaces
$
V
_
1
(
\K
)
,
\ldots
,V
_
n
(
\K
)
$
we can construct a
new vector space by setting
...
...
@@ -628,8 +628,8 @@ The base class
linear maps. The
template parameter
\lstinline
!X! is the type of the domain and
\lstinline
!Y! is the type of the range of the operator. A linear
operator provides the methods
\lstinline
!apply(const X
&
x, Y
&
y)! and
apply
\lstinline
!applyscaledadd(const X
&
x, Y
&
y)! performing the
operator provides the methods
\lstinline
!apply(const X
&
x, Y
&
y)! and
\lstinline
!applyscaledadd(const X
&
x, Y
&
y)! performing the
operations
$
y
=
A
(
x
)
$
and
$
y
=
y
+
\alpha
A
(
x
)
$
, respectively.
The subclass
\lstinline
!template<class M, class X, class Y> AssembledLinearOperator!
...
...
@@ -726,7 +726,7 @@ providing them with the vector implementation used.
\hline
\textbf
{
class
}&
\textbf
{
implements
}
\\\hline\hline
\lstinline
!LoopSolver!
&
only apply precoditioner multiple time
\\
\lstinline
!GradientSolver!
&
preconditioned radient method
\\
\lstinline
!GradientSolver!
&
preconditioned
g
radient method
\\
\lstinline
!CGSolver!
&
preconditioned conjugate gradient method
\\
\lstinline
!BiCGStab!
&
preconditioned biconjugate gradient stabilized method
\\\hline
\end{tabular}
...
...
@@ -812,10 +812,10 @@ operation ($x = y + \alpha z$) in Tables
\hline
\end{tabular}
}}
\end{table}
The code was comiled with the GNU C++
The code was com
p
iled with the GNU C++
compiler version 4.0 with -O3 optimization. In the tables
$
N
$
is the
number of
unknown blocks (equals the number of unknows for the scalar cases in
unknown blocks (equals the number of unknow
n
s for the scalar cases in
Tables
\ref
{
tab:perf
_
sp
}
,
\ref
{
tab:perf
_
daxpy
}
,
\ref
{
tab:perf
_
gs
}
).
The performance for the scalarproduct,
see Table
\ref
{
tab:perf
_
sp
}
,
...
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