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https://github.com/sfilippone/psblas3.git
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Docs updates.
This commit is contained in:
sfilippone
@@ -26,7 +26,7 @@ defined in the library as follows:
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\item[psb\_mpk\_] Kind parameter for 4-bytes integer data, as is
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always used by MPI;
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\item[psb\_epk\_] Kind parameter for 8-bytes integer data, as is
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always used by the \fortinline|sizeof| methods;
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always returned by the \fortinline|sizeof| methods;
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\item[psb\_ipk\_] Kind parameter for ``local'' integer indices and data;
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with default build options this is a 4 bytes integer;
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\item[psb\_lpk\_] Kind parameter for ``global'' integer indices and data;
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@@ -47,9 +47,9 @@ developer's documentation.
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\subsection{Descriptor data structure}
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\label{sec:desc}
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All the general matrix informations and elements to be
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exchanged among processes are stored within a data structure of the
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type \hypertarget{descdata}{{\tt psb\_desc\_type}}.
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All the general matrix information and the identification of elements
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to be exchanged among processes are stored within a data structure of
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the type \hypertarget{descdata}{{\tt psb\_desc\_type}}.
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Every structure of this type is associated with a discretization
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pattern and enables data communications and other operations that are
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necessary for implementing the various algorithms of interest to us.
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+8
-8
@@ -2,11 +2,11 @@
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\section{Error handling\label{sec:errors}}
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The PSBLAS library error handling policy has been completely rewritten
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in version 2.0. The idea behind the design of this new error handling
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strategy is to keep error messages on a stack allowing the user to
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trace back up to the point where the first error message has been
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generated. Every routine in the PSBLAS-2.0 library has, as last
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The PSBLAS library error handling policy has been defined at the time
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version 2.0 was written. The idea behind the design of error
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handling strategy is to keep error messages on a stack allowing the
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user to trace back up to the point where the first error message has
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been generated. Every routine in the PSBLAS library has, as last
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non-optional argument, an integer \verb|info| variable; whenever,
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inside the routine, an error is detected, this variable is set to a
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value corresponding to a specific error code. Then this error code is
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@@ -21,7 +21,7 @@ levels of nested calls until the level where the user decides to abort
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the program execution.
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Figure~\ref{fig:routerr} shows the layout of a generic \verb|psb_foo|
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routine with respect to the PSBLAS-2.0 error handling policy. It is
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routine with respect to the PSBLAS error handling policy. It is
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possible to see how, whenever an error condition is detected, the
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\verb|info| variable is set to the corresponding error code which is,
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then, pushed on top of the stack by means of the
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@@ -110,7 +110,7 @@ end subroutine psb_foo
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Figure~\ref{fig:errormsg} reports a sample error message generated by
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the PSBLAS-2.0 library. This error has been generated by the fact that
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the PSBLAS library. This error has been generated by the fact that
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the user has chosen the invalid ``FOO'' storage format to represent
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the sparse matrix. From this error message it is possible to see that
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the error has been detected inside the \verb|psb_cest| subroutine
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@@ -161,7 +161,7 @@ Aborting...
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\fbox{\TheSbox}
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\end{center}
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\fi
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\caption{\label{fig:errormsg}A sample PSBLAS-3.0 error
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\caption{\label{fig:errormsg}A sample PSBLAS error
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message. Process 0 detected an error condition inside the {\textrm
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psb\_cest} subroutine}
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\end{listing}
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+22
-58
@@ -1,10 +1,9 @@
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%%Creator: fig2dev Version 3.2 Patchlevel 5d
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%%CreationDate: Thu Dec 15 14:55:15 2011
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%%For: sfilippo@donald (Salvatore Filippone)
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%%CreationDate: 2025-12-23 13:28:52
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%%Magnification: 0.5000
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%%EndComments
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%%BeginProlog
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%% here ends figure;
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pagefooter
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showpage
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%%Trailer
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+13
-12
@@ -1,15 +1,16 @@
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#FIG 3.2 Produced by xfig version 3.2.5b
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#FIG 3.2 Produced by xfig version 3.2.9
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#encoding: UTF-8
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Landscape
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Center
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Metric
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Letter
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Letter
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50.00
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Single
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-2
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1200 2
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6 540 5940 2430 6660
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4 0 0 50 -1 0 25 0.0000 4 375 2250 540 6210 Serial Sparse\001
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4 0 0 50 -1 0 25 0.0000 4 285 1080 1080 6660 BLAS\001
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4 0 0 50 -1 0 25 0.0000 4 376 2163 540 6210 Serial Sparse\001
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4 0 0 50 -1 0 25 0.0000 4 288 1032 1080 6660 BLAS\001
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-6
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2 4 0 1 0 7 50 -1 -1 0.000 0 0 15 0 0 5
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4500 1575 1800 1575 1800 225 4500 225 4500 1575
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@@ -31,11 +32,11 @@ Single
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945 4995 5490 4995
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2 4 0 1 0 7 50 -1 -1 0.000 0 0 15 0 0 5
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6120 7020 3420 7020 3420 5625 6120 5625 6120 7020
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4 0 0 50 -1 0 25 0.0000 4 375 2010 2295 990 Application\001
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4 0 0 50 -1 0 25 0.0000 4 285 1665 2250 3735 PSBLAS \001
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4 0 0 50 -1 0 25 0.0000 4 285 1515 4050 2565 Interface\001
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4 0 0 50 -1 0 25 0.0000 4 285 885 4140 4860 Inner\001
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4 0 0 50 -1 0 25 0.0000 4 285 1515 4140 5310 Interface\001
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4 0 0 50 -1 0 25 0.0000 4 375 2910 3420 6120 Message Passing\001
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4 0 0 50 -1 0 25 0.0000 4 285 750 4275 6660 MPI\001
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4 0 0 50 -1 0 25 0.0000 4 285 2190 4050 2160 Fortran 2003\001
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4 0 0 50 -1 0 25 0.0000 4 376 1956 2295 990 Application\001
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4 0 0 50 -1 0 25 0.0000 4 288 1524 2250 3735 PSBLAS \001
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4 0 0 50 -1 0 25 0.0000 4 280 873 4140 4860 Inner\001
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4 0 0 50 -1 0 25 0.0000 4 288 1466 4140 5310 Interface\001
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4 0 0 50 -1 0 25 0.0000 4 376 2818 3420 6120 Message Passing\001
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4 0 0 50 -1 0 25 0.0000 4 276 730 4275 6660 MPI\001
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4 0 0 50 -1 0 25 0.0000 4 288 2137 4050 2160 Fortran 2008\001
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+44
-45
@@ -21,12 +21,13 @@ The software architecture allows us to offer support for many
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alternatives in the implementation, including usage of
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heterogeneous platforms, and computations performed on GPUs throuh
|
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CUDA.
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There is support for GPU computations through OpenACC, but it is at
|
||||
this time a highly experimental version; we plan to also look at using
|
||||
accelerators through OpenMP as support from compilers improves.
|
||||
There is also support for GPU computations through OpenACC, but it is
|
||||
at this time a highly experimental version; we plan to also look at
|
||||
using accelerators through OpenMP as support from compilers improves.
|
||||
|
||||
The project is lead by Salvatore Filippone; a number of people have been contributing to this package over the
|
||||
years; contributors in roughly reverse chronological order:
|
||||
The project is lead by Salvatore Filippone; a number of people have
|
||||
been contributing to this package over the years; contributors in
|
||||
roughly reverse chronological order:
|
||||
\begin{obeylines}
|
||||
Luca Pepè Sciarria
|
||||
Theophane Loloum
|
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@@ -81,18 +82,19 @@ works discussing advanced programming in Fortran~2008
|
||||
include~\cite{DesPat:11,RouXiaXu:11}; sufficient support for
|
||||
Fortran~2008 is now available from many compilers, including recent
|
||||
versions of the GNU Fortran compiler from the Free Software
|
||||
Foundation, and the FLANG compiler from the LLVM project.
|
||||
Foundation, the FLANG compiler from the LLVM project, and the Intel
|
||||
OneAPI compiler.
|
||||
The README file contains a list of compilers against which we have
|
||||
successfully tested the current release.
|
||||
|
||||
|
||||
Previous approaches have been based on mixing Fortran~95, with its
|
||||
support for object-based design, with other languages; these have
|
||||
been advocated by a number of authors,
|
||||
e.g.~\cite{machiels}. Moreover, the Fortran~95 facilities for dynamic
|
||||
memory management and interface overloading greatly enhance the
|
||||
usability of the PSBLAS
|
||||
subroutines. In this way, the library can take care of runtime memory
|
||||
requirements that are quite difficult or even impossible to predict at
|
||||
implementation or compilation time.
|
||||
e.g.~\cite{machiels}. The Fortran~95 facilities for dynamic
|
||||
memory management and interface overloading ensure that the library
|
||||
can take care of runtime memory requirements that are quite difficult
|
||||
or even impossible to predict at implementation or compilation time.
|
||||
|
||||
The presentation of the
|
||||
PSBLAS library follows the general structure of the proposal for
|
||||
@@ -101,13 +103,13 @@ proposal for BLAS on dense matrices~\cite{BLAS1,BLAS2,BLAS3}.
|
||||
|
||||
The applicability of sparse iterative solvers to many different areas
|
||||
causes some terminology problems because the same concept may be
|
||||
denoted through different names depending on the application area. The
|
||||
PSBLAS features presented in this document will be discussed referring
|
||||
to a finite difference discretization of a Partial Differential
|
||||
Equation (PDE). However, the scope of the library is wider than
|
||||
that: for example, it can be applied to finite element discretizations
|
||||
of PDEs, and even to different classes of problems such as nonlinear
|
||||
optimization, for example in optimal control problems.
|
||||
denoted by different names depending on the application area. The
|
||||
PSBLAS features presented in this document will be discussed taking as
|
||||
a reference a finite difference discretization of a Partial
|
||||
Differential Equation (PDE). However, the scope of the library is
|
||||
wider than that: it can be applied to finite element and other
|
||||
discretizations of PDEs, and even to different classes of problems
|
||||
such as nonlinear optimization, for example in optimal control problems.
|
||||
|
||||
The design of a solver for sparse linear systems is driven by many
|
||||
conflicting objectives, such as limiting occupation of storage
|
||||
@@ -145,7 +147,7 @@ application layer.
|
||||
The serial parts of the computation on each process are executed through
|
||||
calls to the serial sparse BLAS subroutines.
|
||||
In a similar way, the inter-process message exchanges are encapsulated
|
||||
in an applicaiton layer that has been strongly inspired by the Basic
|
||||
in an application layer that has been strongly inspired by the Basic
|
||||
Linear Algebra Communication Subroutines (BLACS) library~\cite{BLACS}.
|
||||
Usually there is no need to deal directly with MPI; however, in some
|
||||
cases, MPI routines are used directly to improve efficiency. For
|
||||
@@ -184,10 +186,9 @@ the variable associated to each mesh point is assigned to a process
|
||||
that will own the corresponding row in the coefficient matrix and
|
||||
will carry out all related computations. This allocation strategy
|
||||
is equivalent to a partition of the discretization mesh into {\em
|
||||
sub-domains}.
|
||||
Our library supports any distribution that keeps together
|
||||
the coefficients of each matrix row; there are no other constraints on
|
||||
the variable assignment.
|
||||
sub-domains}; our library supports any distribution that keeps
|
||||
together the coefficients of each matrix row; there are no other
|
||||
constraints on the variable assignment.
|
||||
This choice is consistent with simple data distributions
|
||||
%commonly used in ScaLAPACK
|
||||
such as \verb|CYCLIC(N)| and \verb|BLOCK|,
|
||||
@@ -201,8 +202,8 @@ matrices, that is, the entries of a vector follow the same distribution
|
||||
of the matrix rows.
|
||||
|
||||
We assume that the sparse matrix is built in parallel, where each
|
||||
process generates its own portion. We never require that the entire
|
||||
matrix be available on a single node. However, it is possible
|
||||
process generates its own portion: we never \emph{require} that the
|
||||
entire matrix be available on a single node. However, it is possible
|
||||
to hold the entire matrix in one process and distribute it
|
||||
explicitly\footnote{In our prototype implementation we provide
|
||||
sample scatter/gather routines.}, even though the resulting memory
|
||||
@@ -227,33 +228,31 @@ assigned to the parallel processes,
|
||||
we classify the points of a given sub-domain as following.
|
||||
\begin{description}
|
||||
\item[Internal.] An internal point of
|
||||
a given domain {\em depends} only on points of the
|
||||
same domain.
|
||||
If all points of a domain are assigned to one
|
||||
process, then a computational step (e.g., a
|
||||
matrix-vector product) of the
|
||||
equations associated with the internal points requires no data
|
||||
items from other domains and no communications.
|
||||
a given sub-domain {\em depends} only on points of the
|
||||
same sub-domain.
|
||||
If all points of a sub-domain are assigned to one
|
||||
process, then a computational step (e.g., a matrix-vector product) of
|
||||
the equations associated with the internal points requires no data
|
||||
items from other sub-domains and no communications.
|
||||
|
||||
\item[Boundary.] A point of
|
||||
a given domain is a boundary point if it {\em depends} on points
|
||||
belonging to other domains.
|
||||
\item[Boundary.] A point of a given sub-domain is a boundary point if
|
||||
it {\em depends} on points belonging to other sub-domains.
|
||||
|
||||
\item[Halo.] A halo point for a given domain is a point belonging to
|
||||
another domain such that there is a boundary point which {\em depends\/}
|
||||
\item[Halo.] A halo point for a given sub-domain is a point belonging to
|
||||
another sub-domain such that there is a boundary point which {\em depends\/}
|
||||
on it. Whenever performing a computational step, such as a
|
||||
matrix-vector product, the values associated with halo points are
|
||||
requested from other domains. A boundary point of a given
|
||||
domain is usually a halo point for some other domain\footnote{This is
|
||||
requested from other sub-domains. A boundary point of a given
|
||||
sub-domain is usually a halo point for some other sub-domain\footnote{This is
|
||||
the normal situation when the pattern of the sparse matrix is
|
||||
symmetric, which is equivalent to say that the interaction between
|
||||
two variables is reciprocal. If the matrix pattern is non-symmetric
|
||||
we may have one-way interactions, and these could cause a situation
|
||||
in which a boundary point is not a halo point for its neighbour.}; therefore
|
||||
the cardinality of the boundary points set determines the amount of data
|
||||
sent to other domains.
|
||||
sent to other sub-domains.
|
||||
\item[Overlap.] An overlap point is a boundary point assigned to
|
||||
multiple domains. Any operation that involves an overlap point
|
||||
multiple sub-domains. Any operation that involves an overlap point
|
||||
has to be replicated for each assignment.
|
||||
\end{description}
|
||||
Overlap points do not usually exist in the basic data
|
||||
@@ -378,7 +377,7 @@ $1\dots n_{\hbox{row}_i}$, each element of which corresponds to a certain
|
||||
element of $1\dots n$. The user does not set explicitly this mapping;
|
||||
when the application needs to indicate to which element of the index
|
||||
space a certain item is related, such as the row and column index of a
|
||||
matrix coefficient, it does so in the ``global'' numbering, and the
|
||||
matrix coefficient, it usually does so in the ``global'' numbering, and the
|
||||
library will translate into the appropriate ``local'' numbering.
|
||||
|
||||
For a given index space $1\dots n$ there are many possible associated
|
||||
@@ -390,7 +389,7 @@ with a call to \verb|psb_cdasb| and a sparse matrix with a call to
|
||||
\verb|psb_spasb|. After \verb|psb_cdasb| each process $i$ will have
|
||||
defined a set of ``halo'' (or ``ghost'') indices
|
||||
$n_{\hbox{row}_i}+1\dots n_{\hbox{col}_i}$, denoting elements of the index
|
||||
space that are \emph{not} assigned to process $i$; however the
|
||||
space that are \emph{not} assigned to process $i$; the
|
||||
variables associated with them are needed to complete computations
|
||||
associated with the sparse matrix $A$, and thus they have to be
|
||||
fetched from (neighbouring) processes. The descriptor of the index
|
||||
@@ -521,7 +520,7 @@ computation. This is certainly true for the data allocation and
|
||||
assembly routines, for all the computational routines and for some of
|
||||
the tools routines.
|
||||
|
||||
However there are many cases where no synchronization, and indeed no
|
||||
However there are cases where no synchronization, and indeed no
|
||||
communication among processes, is implied; for instance, all the routines in
|
||||
sec.~\ref{sec:datastruct} are only acting on the local data structures,
|
||||
and thus may be called independently. The most important case is that
|
||||
|
||||
@@ -156,8 +156,8 @@ An integer value; 0 means no error has been detected.
|
||||
Richardson Iteration Driver Routine}
|
||||
|
||||
This subroutine is a driver implementig a Richardson iteration
|
||||
\[ x_{k+1} = M^-1 (b-Ax_k) +x_k,\]
|
||||
with the preconditioner operator $M$ defined in the previous section.
|
||||
\[ x_{k+1} = M^{-1} (b-Ax_k) +x_k,\]
|
||||
with the preconditioner operator $M$ defined in section~\ref{sec:precs}.
|
||||
|
||||
The stopping criterion can take the following values:
|
||||
\begin{description}
|
||||
|
||||
+5
-4
@@ -94,12 +94,12 @@ Specified as: an integer variable.
|
||||
Scope: {\bf local}.\\
|
||||
Type: {\bf required}.\\
|
||||
Intent: {\bf out}.\\
|
||||
Specified as: an integer value. $-1 \le iam \le np-1$\
|
||||
Returned as: an integer value. $-1 \le iam \le np-1$\
|
||||
\item[np] Number of processes in the PSBLAS virtual parallel machine.\\
|
||||
Scope: {\bf global}.\\
|
||||
Type: {\bf required}.\\
|
||||
Intent: {\bf out}.\\
|
||||
Specified as: an integer variable. \
|
||||
Returned as: an integer variable. \
|
||||
\end{description}
|
||||
|
||||
|
||||
@@ -153,8 +153,9 @@ Specified as: a logical variable, default value: true.
|
||||
same program, or to enter and exit multiple times into the parallel
|
||||
environment, this routine may be called to
|
||||
selectively close the contexts with \verb|close=.false.|, while on
|
||||
the last call it should be called with \verb|close=.true.| to
|
||||
shutdown in a clean way the entire parallel environment.
|
||||
the last instance it should close in a clean way the entire
|
||||
parallel environment with \verb|close=.true.|
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
|
||||
@@ -1246,7 +1246,7 @@ An integer value; 0 means no error has been detected.
|
||||
|
||||
This function computes the entrywise product between two vectors $x$ and
|
||||
$y$
|
||||
\[dot \leftarrow x(i) y(i).\]
|
||||
\[y(i) \leftarrow x(i) y(i).\]
|
||||
|
||||
\fortinline|psb_gemlt(x, y, desc_a, info)|
|
||||
|
||||
@@ -1313,7 +1313,7 @@ $y$
|
||||
|
||||
This function computes the entrywise division between two vectors $x$ and
|
||||
$y$
|
||||
\[/ \leftarrow x(i)/y(i).\]
|
||||
\[y(i) \leftarrow x(i)/y(i).\]
|
||||
|
||||
\fortinline|psb_gediv(x, y, desc_a, info, [flag)|
|
||||
|
||||
@@ -1385,7 +1385,7 @@ $y$
|
||||
|
||||
This function computes the entrywise inverse of a vector $x$ and puts it into
|
||||
$y$
|
||||
\[/ \leftarrow 1/x(i).\]
|
||||
\[y(i) \leftarrow 1/x(i).\]
|
||||
|
||||
\fortinline|psb_geinv(x, y, desc_a, info, [flag)|
|
||||
|
||||
|
||||
+18
-16
@@ -598,7 +598,7 @@ An integer value; 0 means no error has been detected.
|
||||
state.
|
||||
\item The descriptor may be in either the build or assembled state.
|
||||
\item Providing a good estimate for the number of nonzeroes $nnz$ in
|
||||
the assembled matrix may substantially improve performance in the
|
||||
the assembled matrix may improve performance in the
|
||||
matrix build phase, as it will reduce or eliminate the need for
|
||||
(potentially multiple) data reallocations;
|
||||
\item Using \verb|psb_matbld_remote_| is likely to cause a runtime overhead at
|
||||
@@ -722,7 +722,7 @@ An integer value; 0 means no error has been detected.
|
||||
entirety to a single call to this routine: the buildup of a row
|
||||
may be split into as many calls as desired (even in the CSR format);
|
||||
\item Coefficients from different rows may also be mixed up freely
|
||||
in a single call, according to the application needs;
|
||||
in a single call (in COO format), according to the application needs;
|
||||
\item Coefficients from matrix rows not owned by the calling
|
||||
process are treated according to the value of \verb|bldmode|
|
||||
specified at allocation time; if
|
||||
@@ -1246,7 +1246,7 @@ An integer value; 0 means no error has been detected.
|
||||
%% psb_glob_to_loc %%
|
||||
%
|
||||
\clearpage\subsection{psb\_glob\_to\_loc --- Global to local indices
|
||||
convertion}
|
||||
conversion}
|
||||
%\addcontentsline{toc}{subsection}{psb\_glob\_to\_loc}
|
||||
|
||||
\begin{verbatim}
|
||||
@@ -1261,7 +1261,7 @@ call psb_glob_to_loc(x, desc_a, info, iact,owned)
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in, inout}.\\
|
||||
Specified as: a rank one integer array.\\
|
||||
Specified as: a rank one integer array of global indices, i.e. \verb|psb_lpk_|.\\
|
||||
\item[desc\_a] the communication descriptor.\\
|
||||
Scope:{\bf local}.\\
|
||||
Type:{\bf required}.\\
|
||||
@@ -1292,7 +1292,7 @@ Intent: {\bf inout}.\\
|
||||
Specified as: a rank one integer array.
|
||||
\item[y] If $y$ is present,
|
||||
then $y$ is overwritten with the translated integer indices, and $x$
|
||||
is left unchanged.
|
||||
is left unchanged; since $y$ contains local indices it should use \verb|psb_ipk_|.
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf optional}\\
|
||||
Intent: {\bf out}.\\
|
||||
@@ -1325,7 +1325,8 @@ call psb_loc_to_glob(x, desc_a, info, iact)
|
||||
\begin{description}
|
||||
\item[Type:] Asynchronous.
|
||||
\item[\bf On Entry]
|
||||
\item[x] An integer vector of indices to be converted.\\
|
||||
\item[x] An integer vector of indices to be converted; if $y$ is present,
|
||||
they are local indices, i.e. \verb|psb_ipk_| \\
|
||||
Scope: {\bf local} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf in, inout}.\\
|
||||
@@ -1346,14 +1347,14 @@ Specified as: a character variable \verb|I|gnore, \verb|W|arning or
|
||||
\begin{description}
|
||||
\item[\bf On Return]
|
||||
\item[x] If $y$ is not present,
|
||||
then $x$ is overwritten with the translated integer indices.
|
||||
then $x$ is overwritten with the translated integer global indices, i.e. \verb|psb_lpk_|
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf required}\\
|
||||
Intent: {\bf inout}.\\
|
||||
Specified as: a rank one integer array.
|
||||
\item[y] If $y$ is not present,
|
||||
then $y$ is overwritten with the translated integer indices, and $x$
|
||||
is left unchanged.
|
||||
\item[y] If $y$ not present,
|
||||
then $y$ is overwritten with the translated global
|
||||
indices i.e. \verb|psb_lpk_|, and $x$ is left unchanged.
|
||||
Scope: {\bf global} \\
|
||||
Type: {\bf optional}\\
|
||||
Intent: {\bf out}.\\
|
||||
@@ -1773,7 +1774,7 @@ Specified as: a preconditioner data structure \precdata.
|
||||
\item[Function value] The memory occupation of the object specified in
|
||||
the calling sequence, in bytes.\\
|
||||
Scope: {\bf local} \\
|
||||
Returned as: an \verb|integer(psb_long_int_k_)| number.
|
||||
Returned as: an \verb|integer(psb_lpk_)| number.
|
||||
\end{description}
|
||||
|
||||
|
||||
@@ -1873,9 +1874,10 @@ the (sorted) value of $x$ in the original sequence.
|
||||
$O(n^2)$; of the other three, in the average case quicksort will be the
|
||||
fastest and merge-sort the slowest. However note that:
|
||||
\begin{enumerate}
|
||||
\item The the best case running time for insertion sort is $\Omega(n)$ while the average
|
||||
and worst case are $O(n^2)$; however for very short input sequences this is
|
||||
likely to be the fastest method;
|
||||
\item The best case running time for insertion sort is $\Omega(n)$
|
||||
while the average and worst case are $O(n^2)$; moreover, for
|
||||
very short input sequences this is likely to be the fastest
|
||||
method;
|
||||
\item The worst case running time for quicksort is $O(n^2)$; the algorithm
|
||||
implemented here follows the well-known median-of-three heuristics,
|
||||
but the worst case may still apply;
|
||||
@@ -1885,8 +1887,8 @@ the (sorted) value of $x$ in the original sequence.
|
||||
subsequences that may be already in the desired ordering prior to
|
||||
the subroutine call; this situation is relatively common when
|
||||
dealing with groups of indices of sparse matrix entries, thus
|
||||
merge-sort is the preferred choice when a sorting is needed
|
||||
by other routines in the library.
|
||||
merge-sort is the preferred choice when a sorting routine is needed
|
||||
for preprocessing matrix data.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
@@ -136,7 +136,7 @@
|
||||
by Salvatore Filippone\\
|
||||
Alfredo Buttari \\
|
||||
Fabio Durastante}\\
|
||||
June 9th, 2025
|
||||
December 23rd, 2025
|
||||
\end{minipage}}
|
||||
}
|
||||
%\addtolength{\textwidth}{\centeroffset}
|
||||
|
||||
@@ -106,7 +106,7 @@ Fabio Durastante } \\
|
||||
%\today
|
||||
Software version: 3.9.0\\
|
||||
%\today
|
||||
June 9th, 2025
|
||||
December 23rd, 2025
|
||||
\cleardoublepage
|
||||
\begingroup
|
||||
\renewcommand*{\thepage}{toc}
|
||||
|
||||
Reference in New Issue
Block a user