Mathematics of Scientific Computing, 3rd Edition
David Kincaid & Ward Cheney
Brooks/Cole Publ. Co., 2002, ISBN 0-534-38905-8
Appendix: An Overview of Mathematical Software
A tremendous amount of mathematical software is available worldwide,
and more is being developed each day.
To find the most up-to-date information,
a browser to the World Wide Web (www) on the Internet should be used.
On the Internet, one can execute a search for available mathematical software in
a particular application area of interest.
It is helpful to classify mathematical software into three categories:
(a) public domain,
(b) freely accessible (some usage restrictions apply), and
(c) proprietary (license agreement required).
Public domain has a specific legal meaning, implying that any use is permitted,
including modification, resale, and so on. From the Internet, one can download
either public domain or freely accessible software, and in many cases, one can
obtain free demonstration copies of commercial software packages. Some of the
software on the Internet has usage restrictions imposed by the authors, such
as copyright, that allow unrestricted use for research and educational purposes
only. (An example of this is code from the ACM Collected Algorithms (p. ),
which is subject to the ACM Software Copyright and License Agreement.) On the
other hand, proprietary software must be purchased or leased from the developing
company or from a computer store that sells software. Notice that whether money
changes hands before the software can be used is not addressed in this breakdown
of software into categories. For example, consider these seemingly contradictory
(a) paying for public domain software (Netlib on CD-ROM),
(b) paying for accessible software-you can freely download some
software, but you had better send in a check if your use it for non-educational
(c) software that is given away for free but is proprietary in the
sense that a license agreement must be signed before you can get it.
In the following, we give a quick overview of some available
mathematical software with pointers to the www-addresses where additional
information and some of the software can be found.
Also mentioned are several systems for helping in the search
for mathematical software that solves specific problems.
This is not a complete and comprehensive listing since
developments are proceeding at such a rapid rate that any listing is soon
out of date!
Since mathematical software is written in a variety of different programming
languages and for a wide range of different computer architectures, it is difficult
to know how to organize the discussion. We are guided by an excellent overview
of mathematical special functions by Lozier and Olver . They organize
software into the categories of Software Packages, Intermediate Libraries,
Comprehensive Libraries, and Interactive Systems. First, Software
Packages contain one or more subroutines for solving a particular problem
in a subfield of numerical mathematics. Second, Intermediate Libraries
are usually collections of subprograms for use on small computers or PCs. Some
of these libraries contain a collection of useful mathematical functions in
one or more computer languages. They may have been written by the manufacturer
of the computer equipment or the developer of the compiler. Also, some intermediate
libraries are available for PCs that are subsets of general-purpose mathematical
libraries. Third, Comprehensive Libraries contain subroutines that have
been assimilated into high-quality software containing many unifying features
such as uniform documentation, uniform style of usage, and error-handling conditions.
Finally, Interactive Systems are fully interactive computer environments
with a powerful set of keyboard commands so that the user can avoid the compile-link-execute
cycle. Usually there is an evolutionary process that mathematical software goes
through from the introduction of the original idea in a journal or report to
the general acceptance by the scientific community of the algorithm. Eventually,
the algorithm is incorporation into a large mathematical library or a sophisticated
interactive computer system.
We start by mentioning some searching systems for finding
available mathematical software on the Internet
and then giving general information related to mathematical software
research and developments.
Finally, we discuss a number of comprehensive libraries,
interactive systems, and
some examples of intermediate libraries or mathematical software packages.
- Guide to Available Mathematical Software (GAMS)
- is a large online and cross-indexed virtual repository of mathematical
and statistical software for use in computational science and engineering.
It was developed as a National Institute of Standards and Technology (NIST)
project for providing scientists and engineers with improved access to reusable
computer software. The user is guided through a problem decision tree to search
for appropriate software for the particular problem to be solved or for the
package/module name. Rather than providing a physical repository, the guide
provides transparent access to multiple repositories maintained by NIST and
others. One can obtain abstracts, documentation, and source code from the
URL site. http://gams.nist.gov
A Java interface to GAMS with additional search features is available at the
URL site. http://gams.nist.gov/HotGAMS
Some other sources of information about mathematical software can be found
which includes directories, journals, repositories, freely available packages,
software vendors, and educational software vendors.
- is a repository for mathematical software, documents (papers, reports,
etc.), databases (address lists-e-mail and mail, conferences, performance
data, etc.), and other useful mathematical information. There is a keyword
searching capability for obtaining mathematical software as well as for searching
back issues of the weekly NA-Digest online newsletter. This is a system developed
to serve the community of numerical analysts and other researchers and is
available at http://www.netlib.org
Also, the NA-Net system is available at http://netlib.org/na-net/
and serves the scientific computing community by providing an Email database
for its members, a White Pages directory service, and the NA-Digest, which
is a weekly collections of articles on topics of interest to them. Some other
related efforts are the following.
- HPC-Netlib is a high performance branch of Netlib available at
It provides information about high performance mathematical software,
whether research or commercial, as well as giving a roadmap to issues
such as software selection and performance.
- Matrix Market is accessible at http://math.nist.gov/MatrixMarket/
and provides a convenient access to a repository of test data for use
in comparative studies of numerical linear algebra algorithms.
- PTLib is a source of information about high quality software
and tools for parallel systems. It can be found at http://www.nhse.org/ptlib/
and is part of the National High Performance Computing and Communication
- are announced and information on them are available on the Internet. For
example, the Netlib Conference Database is accessible at http://www.netlib.org/confdb/Conferences.html
and it contains information about upcoming conferences, lectures, and other
meetings relevant to the fields of mathematics and computer science. Also,
the Atlas Mathematical Conferences Abstracts provides conference announcements
and conference abstracts for mathematicians worldwide at http://at.yorku.ca/amca.
Many conference announcements are available directly from scientific societies
and associations such as those listed below.
- Graduate Programs
- involving numerical analysis and scientific computing exist throughout
the world such as the Computational and Applied Mathematics (CAM) program
at The University of Texas at Austin with details available at http://www.ticam.utexas.edu/cam
Many other graduate programs are available on the World Wide Web.
- have been established by various research interest groups. An example of
some of these are:
Individuals have homepages such as the authors of this textbook:
Many professional societies and associations have homepages such as those
- are published for the dissemination of recent research developments and
associated algorithms and mathematical software. The bibliography to this
book contains a listing of many of the primary numerical analysis journals.
For some of these journals, tables of contents, and even full-text articles
are available to readers over the Internet. Some of the organizations that
publish research journals and disseminate mathematical software are:
For example, the ACM Transactions on Mathematical Software (TOMS) publishes
refereed articles and computer routines/packages. The ACM algorithm policy
requires the software to be self-contained with adequate documentation, to
have a test program with sample output, and to be reasonably portable over
a variety of different computers. The TOMS homepage is at: http://www.acm.org/toms
with a searchable table of contents to the algorithm papers and links to the
software. The software is available from ACM, Netlib, and many other sources.
You can obtain information from the ACM Collected Algorithms at the URL site:
http://www.acm.org/calgo. A classification
system is used for indexing the algorithms, and a data base of them is maintained.
The ACM Digital Library is a resource of bibliographic information, citations,
and full-text articles; and it is available at the URL site: http://acm.org/dl.
The bibliographic database is free of charge to visitors by registering, and
access to the full-text database is a pay-per-view subscription service. Many
new and classical mathematical books are available at http://www.siam.org/books/index.htm.
Mathematical journals such as SIAM Journal of Numerical Analysis, SIAM
Journal of Scientific Computing, and many others are available at http://www.siam.org/journals/journals.htm.
Other journals exist for the exchange of software and related information
in particular scientific disciplines. For example, the journal of Computer
Physics Communications publishes papers on the computational aspects of
physics and physical chemistry with refereed computer programs. The Applied
Statistics journal publishes literature on statistical computing with
refereed statistical software. Recently, entirely electronic journals have
appeared with technical articles related to the development of numerical analysis
algorithms. Articles from these journals are available over the Internet for
local printing. One such journal is the Electronic Transactions on Numerical
Analysis at http://etna.mcs.kent.edu
- Mathematics Archive WWW Server
- provides an organized Internet access to a wide variety of mathematical
resources. Primary emphasis is on material that is useful in teaching mathematics
and educational software. http://archives.math.utk.edu
- Mathematics Information Servers
- is an extensive list of mathematics servers on the World Wide Web with
references to many academic departments, electronic journals, sources for
preprints of articles, and information on mathematical software. http://www.math.psu.edu/MathLists
- News Groups
- are used for a wide range of discussions and questions-and-answers postings
in an unmoderated forum. Some USENET news groups for mathematical software
are sci.math.num-analysis, sci.math.research,
- are available on the Internet with announcements and general information
on mathematical software. Examples of these newsletters are
- centers and institutes on numerical analysis and scientific computing exist
throughout the world such as the Texas Institute of Computational and Applied
Mathematics (TICAM) http://www.ticam.utexas.edu
and the Center for Numerical Analysis (CNA) http://www.ma.utexas.edu/CNA
at The University of Texas at Austin http://www.utexas.edu.
In the United States of America, large collections of mathematical software
are available from national laboratories [Argonne (anl), Fermilab, Lawrence
Berkeley (lbnl), Lawrence Livermore (llnl), Los Alamos (lanl), Oak Ridge (ornl),
Pacific Northwest, Sandia, Stanford Linear Accelerator, etc.], national supercomputer
centers [San Diego Supercomputer Center (sdsc), National Energy Supercomputer
Center at Lawrence Berkeley (nesc), etc.] and various government agencies
[National Institute of Standards and Technology (nist), etc.]. Links are available
directly to all of the national laboratories, supercomputer centers, and high
performance research centers.
- with associated mathematical software are widely available. Many numerical
analysis and numerical methods textbooks come with or have associated software.
In these books, one may find for each problem area a general discussion of
analytical mathematics, the presentation of algorithms, and perhaps the actual
implementation of them into computer routines written in one or more computer
languages. The algorithms may be listed in the textbook, or the software may
be available for purchase on a diskette or freely available so that the interested
reader can download it from the Internet. For example, software supporting
this textbook is available by using a Web browser and accessing the files
at the following URL sites: http://www.ma.utexas.edu/CNA/NA3
Here one can download sample computer
programs that are written in various computer programming languages or that
use an advanced mathematical software system. Also available is an up-to-date
list of errata for this textbook. Also, software supporting the elementary
textbook Numerical Mathematics and Computing, 4th Edition by Cheney
and Kincaid is available at the following URL site: http://www.ma.utexas.edu/CNA/NMC4
Comprehensive libraries are large collections of mathematical routines
all written in a uniform style and with high standards of
quality and robustness. Some of these libraries are listed here.
In addition to those listed above, some other general mathematical software libraries
are NSWC, NUMAL, NUMPAC, PORT, Scientific Desk, VECLIB, and more.
is the standard mathematical and statistical subroutine library
developed by Boeing Computer Services.
It contains approximately 250 subroutines for applications in a wide
variety of areas.
BCSEXT is the Boeing Computer Services Extended Library, containing routines
using out-of-core methods for solving large linear algebraic problems.
Applications include large (sparse or dense) linear systems and
eigenvalue problems as well as fast Fourier transform in two and
- is the Core Mathematics LIBrary of the National Institute of Science and
Technology (NIST). It is a collection of approximately 750 high-quality public-domain
Fortran subprograms that are easily transportable. The subroutines in this
library solve many of the standard problems in mathematics and statistics.
It contains mostly externally available software programs such as BLAS, EISPACK,
FISHPACK, FCNPACK, FITPACK, LINPACK, and QUADPACK. http://gams.nist.gov
- CERN Library
- is maintained by the European Laboratory for Particle Physics. This library
is primarily for the support of high-energy physics research, but it contains
many routines for general mathematical use. With some restrictions, the library
is distributed to outside organizations. http://consult.cern.ch
- is an Engineering and Scientific Subroutine Library for use on IBM computers.
It is a state-of-the-art collection of over 450 mathematical routines for
use on a wide range of IBM computers for solving scientific and engineering
applications. The library has been tuned for specific IBM computer architectures
such as workstations and parallel computers. It can be called from applications
written in Fortran, C, or C++. PESSL is the parallel implementation of the
IBM Engineering and Scientific Subroutine Library (ESSL). It consists of scalable
mathematical routines that support parallel processing on IBM computer systems
and clusters of IBM workstations. Parallel ESSL supports the Single Program
Multiple Data (SPMD) programming model using either the Message Passing Interface
(MPI) signal handling library or the MPI threaded library. http://www.ibm.com
- is a library of commonly used mathematical and scientific routines developed
for use on Cray computer systems. For example, it includes routines for linear
algebra, fast Fourier transforms, filtering, packing unpacking, and vector
gather scatter. http://www.netlib.org/scilib
- IMSL Libraries
- are C-coded or Fortran-coded numerical and graphical libraries developed
by Visual Numerics, Inc. These libraries contain a large collection of subroutines
and function subprograms (over 500) that provide access to high-quality implementations
of numerical methods in mathematics and statistics. They have evolved over
more than 30 years. These libraries are available for use on a wide range
of computer platforms with subsets of them available for use on PCs. Other
mathematical software products are available from Visual Numerics, such as
PV-WAVE and Stanford Graphics. http://www.vni.com
- NAG Libraries
- are Fortran77/90-coded or C-coded numerical and statistical libraries for
scientists, engineers, researchers, and software developers with applications
involving mathematics, statistics, and optimization. Developed by the Numerical
Algorithms Group, these software libraries have comprehensive numerical capabilities
and were coded with the collaboration of numerical and statistical specialists.
The largest version of the library has over 1000 routines. Some of this software
has evolved over more than 20 years, producing state-of-the-art products with
robust performance on over 80 computing platforms from PCs to supercomputers.
Also, NAG developed the first fully standard Fortran 90 compiler, and the
computer algebra system Axiom is available from NAG. http://www.nag.com
- is a large collection of Fortran mathematical subprograms distributed by
the Department of Energy (DOE) Energy Science and Technology Center: http://www.energy.gov
or from Netlib http://www.netlib.org/slatec
It is characterized by portability, good numerical technology, good documentation,
robustness, and quality assurance. The primary impetus for this library was
to provide portable, nonproprietary, mathematical software for supercomputers
at a consortium of government research laboratory. The original acronym stood
for the national laboratories involved (Sandia, Los Alamos, Air Force Technical
Exchange Committee). Subsequently, the library committee admitted three additional
national laboratories (Lawrence Livermore, Oak Ridge, Sandia Livermore) plus
the National Energy Supercomputer Center at Lawrence Livermore and the National
Institute of Standards and Technology.
In general, a fully interactive mathematical software system contains a powerful
set of commands that the user enters at a computer terminal or workstation by
using the keyboard or the mouse to click on an icon. An immediate response is
displayed on the screen. It may be a computation (numerical or symbolic) or
a visual display such as a graph or figure. The programming burden is reduced
since there is no compile-link-execute cycle. The capabilities of interactive
systems can be extended by customizing the set of commands or icons. Interactive
systems are able to integrate nonnumerical tasks with numerical computations.
It seems that graphical and symbolic computing are best done in an interactive
computing environment. For this reason, a recent trend is combining numerical
computing with symbolic computing and graphical visualization into a totally
Computer algebra systems have special capabilities
useful in numerical mathematics.
One such feature
is arbitrary precision or
multiple-precision floating-point arithmetic.
In general, programming languages use the computer hardware
for doing computer arithmetic so that the precision is fixed.
The primary purpose of
computer algebra systems is for exact mathematical calculations
with floating-point computations a secondary capability.
Nevertheless, one obtains a bonus of being able to carry out
arbitrary precision floating-point computations in these systems.
Unless the user specifies otherwise, symbolic systems generally
avoid evaluations that introduce inexact results and
evaluate expressions symbolically (with numbers rendered as rational
fractions having arbitrarily long numerators and denominators or
represented as symbols).
The user can request that floating-point evaluations of numbers
be carried out with arbitrarily long precision.
The following are a sampling of interactive mathematical software
Some of the symbolic algebraic computer systems other than those listed
- is a software package used to solve linear programming problems, including
integer, mixed integer, and network linear programming problems. CPLEX can
be used as an interactive program or as a callable subroutine library. The
CPLEX interactive problem solver allows users to enter, modify, and solve
problems from computer terminals. This is particularly useful if one is solving
a problem once or if one is prototyping a solution method. The callable library
provides access to the CPLEX optimization, utility, problem modification,
query, and file I/O routine directly from C or Fortran programs. http://www.cplex.com
- is an object-based numerical analysis and data visualization software package.
HiQ solves math, science, and engineering problems using a methodology that
combines a worksheet interface, interactive analysis, data visualization,
an extensive mathematics library, and a script-programming language. HiQ is
an interactive problem-solving environment for Macintosh and Power Macintosh
- is an interactive symbolic computation system containing symbolic, numerical,
graphical, and programming capabilities. Maple performs equation solving,
linear algebra, calculus, complex analysis, and more with virtually unlimited
precision. Maple was developed as an interactive system for computer algebraic
manipulations associated with symbolic computing, but many more expanded capabilities
have been added. It is available on a wide range of computers from personal
computers and workstations to supercomputers. http://www.maplesoft.com
- is an interactive software system for numerical, symbolic, and graphical
computing as well as visualizations. The user can integrate Mathematica output
(computations, graphics, animations, etc.) with ordinary text entirely within
the Mathematica system for the preparation of complete electronic documents
to be used as technical reports or presentations. A wide variety of Mathematica
application libraries are available. MathLink is a communication protocol
that allows the exchange of information between Mathematica and other computer
packages such as Matlab or Excel. A programming language based on pattern
matching can be used for extending the capabilities of the Mathematica system.
Mathematica is available on over 20 platforms from PCs to scientific workstations
to large-scale scientific mainframe computers. http://www.mathematica.com
- is a computing environment that provides computation, visualization, and
application-specific tool boxes. Matrix notation is used to produce a matrix
laboratory with a built-in set of commands for standard algorithms involving
numerical computations. A matrix-oriented language may be used for large-scale
computation and data analysis. Interactive 2-D and 3-D graphical capabilities
are available for analysis, transformation, and visualization of data. Matlab
can dynamically link with C or Fortran programs. The package is available
on a wide range of computers such as PCs, workstations, and supercomputers.
Matlab has more than 20 tool boxes available for specialized applications
such as signal and image processing, control system design, frequency domain
identification, robust control design, mathematics, statistics, and data analysis,
neural networks and fuzzy logic, optimization, and splines. Also, symbolic
computing is available with a tool box interface to Maple V. http://www.mathworks.com
- is a high-level language, which is mostly compatible with Matlab, and is
primarily intended for numerical computations. It provides a command line
interface for solving linear and nonlinear problems numerically and for performing
other numerical experiments. Octave has extensive tools for solving common
numerical linear algebra problems such as finding roots of nonlinear equations,
integrating ordinary functions, manipulating polynomials, and integrating
ordinary differential and differential-algebraic equations. It is extensible
and customizable via user-defined functions written in Octave's own language,
or using dynamically loaded modules written in different programming languages.
GNU Octave is free software, which may be redistributed or modified under
the terms of the GNU General Public License as published by the Free Software
- is an interactive computer algebra system designed for general algebraic
computations of interest to mathematicians, scientists, and engineers. While
it is used by many as an algebraic calculator for problems that can be done
by hand, the main aim of REDUCE is to support calculations that are feasible
only by a computer solution. http://www.uni-koeln.de/REDUCE/
Intermediate Libraries and Software Packages
There are many Intermediate Libraries and
software packages available for solving
mathematical problems of various types.
Some examples of just a few of them,
which are primarily public-domain software,
are the following:
- is a mathematical library that performs curve-fitting and surface-fitting
using splines under tension. Featues include a variety of dimensions for base
and target spaces, approximation of data with apparent poles, and the aproximation
of integrals, derivatives, and lengths of the approximating curves. http://www.netlib.org/fitpack
- are packages of subroutines for solving large sparse linear systems of
equations by iterative methods. Features of the software include automatic
parameter estimation and implementaton of accurate termination criteria. These
packages can be employed to solve linear systems arising from either finite
difference or finite element modeling of ellpitc partial differential equations.
ITPACKV is a vectorized software package with basic iterative schemes such
as Jacobi, successive overrelaxation (SOR), symmetric SOR, and reduced system
with convergence accelerated by either Chebyshev or conjugate gradient procedures.
The routines in NSPCG also involve various acceleraton techniques, including
conjuage gradient, Chebyshev acceleration, and generalized conjugate gradient
methods for nonsymmetric systems, in conjunction with preconditioners (or
basic iterative methods). These packages were developed as part of the ITPACK
Project of the Center for Numerical Analysis at The University of Texas at
- contains routines for solving systems of simultaneous linear equations,
least-squares solutions of linear systems of equations, eigenvalue problems,
and singular-value problems. The associated matrix factorizations (LU, Cholesky,
QR, SVD, Schur, generalized Schur) are also provided, as are related computations
such as reordering of the Schur factorizations and estimating condition numbers.
Dense and banded matrices are handled, but not general sparse matrices. In
all areas, similar functionality is provided for real and complex matrices,
in both single and double precision. http://netlib2.cs.utk.edu/lapack/index.html
- is a library of routines for solving systems of nonlinear equations and
nonlinear least squares problems. One can either choose whether to use routines
that calculate the Jacobian automaticaly or to supply routines for the Jacobian.
- is a finite element code that solves systems of linear and nonlinear time-dependent,
steady-state, and eigenvalue partial differential equations in general two-dimensional
regions and in three-dimensional boxes. It can handle problems in such areas
as elasticity, diffusion, heat condution, potential energy, and fluid mechanics.
It has an interative interface and extensive graphical output capabilities.
- is a collection of subroutines for the solution of ordinary differential
equations, with initial values given. These routines are particularly useful
for solving the ODE systems that arise when the method of lines is used to
solve time-dependent partial differential equations. They were developed at
Lawrence Livermore National Laboratory (LLNL) and the homepage of this package
- is a Portable Extensible Toolkit for Scientific Computing for solving partial
differential equations and related problems. It is a suite of data structures
and routines for building large scale applications consisting of parallel
linear and nonlinear equation solvers and unconstrained minimization modules.
Argonne National Laboratory (ANL) supports this project and is the primary
distribution site: http://www-fp.mcs.anl.gov/petsc
- is a collaborative effort involving several institutions [Oak Ridge National
Laboratory, Rice University, University of California (Berkeley and Los Angeles),
University of Illinois, and University of Tennessee (Knoxville)] and comprises
four components: dense and band matrix software (ScaLAPACK), large sparse
eigenvalue software (PARPACK and ARPACK), sparse direct systems software (CAPSS
and MFACT), and preconditioners for large sparse iterative solvers (ParPre).