Subject: Re: help: solve this functional relation
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Date: 25 Nov 1996 21:08:24 GMT
In article <56maqc$dlo@hecate.umd.edu>,
Jason Stratos Papadopoulos wrote:
>Hello. I've run across a problem that has me stumped. How would
>you go about finding a function "f" such that
>
> 1 1
>f ( ----------- ) = j f( --- ) ?
> 2(1+j)(1-j) 1-j
>
...
>Anyway, j is supposed to be close to 1, and an asymptotic series for f is
>
> 2 3 4 5
> (j-1) (j-1) (j-1) 23 (j-1) 263 (j-1)
>f(j) = 1 + ----- - ------ + ------ - --------- + ---------- - ....
> 3 45 189 14175 4677775
>
>Can such a problem even be solved in closed form?
I'm a little confused here; if you take j close to 1 in the top
equation, you'll find yourself evaluating f at very large arguments;
in the final line, you are evaluating f near 1. Is that really your intent?
I also feel something must be amiss here: I don't think there are
any nice nonzero functions satisfying the proposed functional equation.
Functional equations such as this arise frequently in this newsgroup
so if you don't mind I'll generalize your question a little (you can
keep generalizing much more). The typical question here reads
"What function f satisfies f(h(x))= G(x, f(x)) for all x?"
(where the functions h and G are given explicitly).
The first thing to keep in mind is that _the solution is not unique_
(usually). For example, in the original poster's situation, if f is
any function satisfying the functional equation, then any scalar
multiple c.f also satisfies that relation.
The second thing to worry about is that _you need to be clear about
the domain of f_ (or more precisely, the range of x's for which the
purported functional equation is to hold). For example, there may be
many more functions with a complex domain which match the given
requirements, but fewer on the real line. And a restriction on the
domain of f will mean the functional equation holds for fewer x
(giving fewer restrictions on f).
Finally, you need to decide _what you want to assume about continuity_
(and/or differentiability). Assume nothing and you'll get nowhere.
Assume too much and you may find no nontrivial solutions f.
(That is, you must be careful not to "throw out the baby with the
dishwater" as my grandmother used to say.)
Let me clarify this last point. Given any value of x such that both
x and h(x) are (defined and) in the domain of f, the assumed
functional equation will give some information relating f(x) and
f(h(x)). So let us declare x and h(x) to be _equivalent_, and let
" ~ " denote the equivalence relation this generates on the real
number line (or whatever the domain of f is assumed to be). One can
check that this means
x1 ~ x2 iff there exist m,n >= 0 s.t. h^n(x1) = h^m(x2).
Now the crucial observation is this: the given functional equation will
_only_ give some information about the relationships between the
values of f(x) among points x in a single equivalence class.
This is a telling statement in situations like the proposer's problem:
since h is at worst a two-to-one map, it is easily verified that
the equivalence class of x is at most countable, and thus there are
uncountably many equivalence classes. In most cases we have more than
one solutions for the behaviour of f on each equivalence class, so
we obtain uncountably many possible solutions f in toto!
Therefore in order to make some progress we try to assume f is, say,
continuous at x=1. I believe in that case, however, the number of
such functions drops to 1: f must be identically zero.
Let us see what information we do glean about the behaviour of f
on each equivalence class C. Note that C is actually a directed
graph, with an edge from x to y iff h(x) = y. If the (corresponding
undirected) graph is a _tree_, then we can pretty easily determine the
values of f on all of C. Pick a point x0 to be the root of this
tree; define f(x0) to be any value you wish. Then the functional
equation forces f(h(x0)) = G(x, f(x0)) to be a certain value; similarly,
the value of f(h^n(x0)) is forced for any n. For any other point x in C
there exist unique minimal n and m such that h^m(x) = h^n(x0); if
for example m=1 we may then deduce the value of f(x) from the equation
f(h(x)) = G(x, f(x)), which we solve as a single equation in the single
unknown "f(x)". Proceeding by induction we may compute f(x) for those
values of x corresponding to higher values of m.
(Here we're using the fact that G(x, y) is linear in y for the
proposer's problem, so that a unique solution for f(x) exists.
In more general settings of course the equations to be solved do not
have a unique solution for f(x), so we will find more than one
possibility for the function f: C -> R. Or there may
be occasions in which f(h(x)) = G(x, f(x)) admits no solution for f(x);
in these cases we would need to backtrack and see if a different choice
for f(h(x)) would enable a solution.)
More interesting is the case in which C contains some cycles. One can
check that this can only happen if C contains a point x for which
h^n(x) = x for some n. When this occurs we usually have an equation which
limits the possible values of f(x). For example, if x is a fixed
point of h then we must have f(x) = G(x, f(x)). Of course, once the
values of f on the points within cycles have been determined, the
values of f on the rest of C are determined as in the previous paragraphs.
So we see that the cycles under h play a special role. I will look for
some of them in the poster's specific problem. I suppose I should reiterate
that the domain of f is assumed to include all these points, otherwise
the functional equation does not give any further information (that is,
the graph C loses this cycle).
As I remarked at the beginning I'm not sure a mistake in notation hasn't
been made, but I'll take it as is. Rather than write
> 1 1
>f ( ----------- ) = j f( --- )
> 2(1+j)(1-j) 1-j
I would prefer to let x=1/(1-j) so that this equation reads
(*).....f(x^2/(4x-2)) = (x-1)/x f(x)
that is, h(x) = x^2/(4x-2) and G(x,y) = ((x-1)/x) * y .
Now, the fixed points of h are 0 and 2/3. What do we learn here?
From (*) with x=0 we have f(0) = 0*f(0), and so f(0)=0.
Note that h(0)=0 and h(x)=0 implies x=0: the equivalence class of 0
is just { 0 }.
Taking the other fixed point x=2/3 gives f(2/3) = -1/2 * f(2/3), which
requires f(2/3)=0, too. This time the graph C is more complicated; here is
a portion of it:
...-> 29.347... -> 4+sqrt(12) -> 2 -> 2/3 (loops back)
...-> 0.508... -----^ ^
|
...-> 1.349... -> 4-sqrt(12) ---|
...-> 0.794... -----^
Well, since f(2/3)=0, we have 0= f(h(x))=(x-1)/x f(x) for both
x=4 +- sqrt(12); so f vanishes there as well. Similarly, working back over
the graph, we see f(x) = 0 for all x in C.
And now we see more possibilities. The only cycle of length of 2 is the
one containing 1 +- sqrt(1/5) = {x1, x2} , say. Then we have
f(x1) = (x2-1)/x2 f(x2) and f(x2) = (x1-1)/x1 f(x1), so that
f(x1) = (x2-1)/x2 * (x1-1)/x1 * f(x1)= (-1/4) f(x1); again f(x1)=f(x2)=0.
There are two cycles of length 3; on these as well we have
f(2.66)=f(.818)=f(.526)= f(4.27)=f(1.21)=f(.515)=0. I think one can show
that f(x)=0 for all elements in _any_ finite orbit under h, although I
have not carried this out.
In this way, one obtains a large number of points at which f must
vanish. Since for any x > 0.5 there are two points y with h(y)=x,
both with y>0.5, we find that the number of points in the tree doubles
with each lengthening to the left.
So a picture of the behaviour of f begins to emerge. There seems to
be a countable collection of families of graphs such that f(x)=0 for
all x in the graphs; these graphs end in cycles ("on the right") but
go arbitrarily far to the left, splitting in two at each stage.
All the other values of x lie in equivalence classes which are
trees -- roughly as above but with no terminal points. On each tree
f(x) may be chosen arbitrarily at one point x0, and is then determined
everywhere else. (Well, in the equivalence class of 1 we must take x0
to be "to the left of" 1.)
Ah, but you object, what does f "look like"? The answer: it's a mess.
As far as I can tell, the equivalence class containing 2/3, for
example, is dense in the interval (1/2 , oo). Certainly the 1024
points obtained as (h^n)^(-1) (2/3) ( for 0 <= n <= 10) show no sign of
leaving any gap, although there is no upper bound on the magnitude of
the points in the equivalence class.
If this hunch is correct, then we have some bad news: wherever f is
continuous, it must be zero. (This follows since at each of the points in
this equivalence class, f(x)=0.)
In particular, it seems difficult to believe there can be a function f
expressible as a power series near 1, as the poster suggested; this
function must vanish at the points
..., .98987715, .99188524, .99592606, .99795886, 1.0020494,
1.0041074, 1.0082486, 1.0103320, ...
as well, surely, as many points in between.
Summary: Chase through the trees, use continuity if you think it's
appropriate -- and make sure you've expressed the problem correctly.
dave
Subject: Fortran 90/95 Information (November)
From: metcalf@sp060.cern.ch (Michael Metcalf)
Date: Tue, 26 Nov 1996 08:21:28 GMT
*********************************************************************
* Information file, on compilers, tools, books, courses, tutorials, *
* and the standard for the Fortran language. *
* *
* Additional information on Fortran products is available on the *
* WWW at the URL http://www.fortran.com/fortran. *
*********************************************************************
WHAT'S NEW? Since 21 October:
HP announces its optimizing f90 compiler.
Revised Fujitsu and Salford entries.
WHERE CAN I OBTAIN A FORTRAN 90 COMPILER?
Absoft sells its native version of Cray's CF90 for the Power Macintosh
(sales@absoft.com or http://www.absoft.com).
ACE of Holland provides f90 and HPF for Parsytec PowerPC-based machines
(marco@ace.nl, http://www.ace.nl/).
Apogee's f90 compiler is highly optimized for SPARC architectures
(sales@apogee.com or http://www.apogee.com).
Cray Research has a fully-optimizing, native compiler, CF90, that is
being marketed by them for the YMP, J90, C90, T90 and T3E, and by Visual
Numerics for workstations, starting with Suns (craysoft@cray.com or
http://www.cray.com/PUBLIC/product-info/craysoft/Fortran_90.html).
Digital has Digital Fortran 90, a native, optimizing compiler for Digital
UNIX Alpha systems (with HPF and parallel processing as an option), and for
OpenVMS Alpha (with HPF syntax). Versions for Windows NT (Alpha and Intel)
and Windows 95 (Intel) are under development and will have an integrated
development environment - planned for 1997. Fortran 95 support is planned
for mid-1997 (fortran@digital.com or http://www.digital.com/info/hpc/fortran)
EPC has optimizing, native compilers for x86, Sun, RS/6000, SGI and MIPS
(http://www.epc.co.uk, info@epc.com, support@epc.co.uk). HPF is also available.
FORTNER Research (formerly Language Systems Corp) expects to deliver
f90 for Macintoshes at some unspecified date.
Fujitsu is marketing a native Fortran 90 Workbench for Solaris 1.1 and 2.x.
Also HPF. Contact Unicomp (walt@fortran.com), Fujitsu (info@fsc.fujitsu.com
or http://www.adtools.com/lpg/fortranhp.htm).
HP has collaborated with EPC to produce an optimizing compiler for HP-UX and
SPP-UX platforms; see http://www.hp.com/go/hpfortran.
IBM has been shipping its optimizing, native compiler for the RS/6000,
xlf Version 3, since 31 December, 1993. HPF is now available too.
See http://www.software.ibm.com/ap/fortran.
Imagine1 Inc offers F, the subset language for Unix and Windows that they
hope will be the true stepping stone to HPF and at the same time
replace Basic, Pascal and C for teaching purposes. The version for Linux
is free. See http://www.imagine1.com/imagine1 and the book section below.
Lahey has a native LF90 compiler for Windows and DOS (sales@lahey.com or
http://www.lahey.com). Version 3.0 provides an integrated Windows development
environment. There is also elf90, a subset language without old features
like storage association that is designed for teaching, and is very cheap.
In fact, the elf90 compiler itself can be downloaded free from the Web site.
Microsoft has released its Fortran Powerstation V4.0 that includes f90
for Windows NT 3.51 and Windows 95 (fortran@microsoft.com or
http://www.microsoft.com/fortran). It is a 32-bit compiler with
optimizations for Pentium and 486.
Microway NDP Fortran 90 for 386/486 and Pentium is available
(nina@microway.com).
NAG provides a compiler for most unix platforms, VMS and PCs (including
Linux). This was the first f90 compiler, in 1991. An optimizing version
produced in collaboration with ACE (see above) for Suns is also
available. The NAGWare f90 Tools are a suite of Fortran 90 tools derived
from the same technology as the NAGWare f90 compiler (infodesk@nag.com,
infodesk@nag.co.uk or http://www.nag.co.uk/).
NA Software supplies Fortran 90 Plus on PCs (including Windows 95 and
Linux), Sparc, and T800 transputers. There is a cheap student version
available. They also supply an F77 to f90 syntax convertor, LOFT90,
and as well as HPF (http://www.nasoftware.co.uk/home.html).
NEC has released a native, optimizing Fortran90 compiler,
FORTRAN90/SX, with an automatic vectorization and parallelization
capability, for its supercomputer SX series (sx-4@sxsmd.ho.nec.co.jp).
PSR's VAST/f90 compiler for unix, VMS and Convex includes a vectorizer.
PSR also supplies VAST/77to90, to convert FORTRAN 77 programs into
Fortran 90 syntax, as well as HPF (info@psrv.com or http://www.psrv.com/)
ParaSoft has a compiler (f90-info@parasoft.com, or
http://www.parasoft.com/f90.html).
PGI has a Fortran 90/HPF compiler for SGI, IBM SP2, HP/Convex, etc.
(sales@pgroup.com or http://www.pgroup.com/). It supplies HPF to Cray
and Intel.
Salford Software markets a PC version of the NAG compiler, also for
Windows 95 and NT (http://www.salford.ac.uk/ssl/ss.html or
sales@salford-software.com). A very cheap student version is available.
SGI has the MIPSpro Fortran 90 64-bit compiler, version 6.2. It can
be configured with an optional MIPSpro Power Fortran 90 Accelerator
(PFA90) to optimize Fortran 90 code for SGI's multiprocessor systems
(http://www.sgi.com/Technology/TechPubs/lib/0620bom.html).
SofTech has a licence to sell its own versions of DEC's HPF/f90 compiler.
Sun has released an f90 compiler based on Cray's CF90, initially for
Solaris 2 (tel. 1-800-SUNSOFT or URL
http://www.sun.com/sunsoft/Products/Developer-products).
OTHER USEFUL PRODUCTS
FORCHECK is a static analyzer for Fortran programs. It analyses both the
individual program units and the whole program. It optionally verifies the
syntax for conformance to the Fortran 90 standard, and provides warnings
on undefined and unreferenced syntax items, inconsistent argument lists,
and much more. FORCHECK generates documentation, such as
cross-reference tables. See http://www.medfac.leidenuniv.nl/forcheck.
FORGE90 and an HPF processor from APR (support@apri.com or
http://www.infomall.org/apri/) are available.
HPF is apparently available not only as listed above, but also from CDAC,
Hitachi, Intel, Motorola, Meiko, NEC, Transtech and Thinking Machines.
A source form convertor, convert.f90, is obtainable by ftp
from jkr.cc.rl.ac.uk in the directory /pub/MandR. Latest version is 1.4.
A graphics interface, f90gl, is obtainable at http://math.nist.gov/f90gl.
NAG (see above) and IMSL (now Visual Numerics, mktg@houston.vni.com)
offer f90 versions of their maths libraries that take
full advantage of the language's library building capabilities.
An f90 mode is included in the official Emacs distribution
(GNU Emacs-19.28/XEmacs-19.13 or later).
For make files, a perl5 script, which behaves like an X11 makedepend
program (it edits an existing Makefile) and recursively searches
include files for more dependencies, is available from Kate Hedstrom:
ftp://ahab.rutgers.edu/pub/perl/sfmakedepend
http://marine.rutgers.edu/po/perl.html
For a makemake perl script: http://www.fortran.com/fortran/makemake.html.
WHAT BOOKS ARE AVAILABLE?
English:
Advanced Scientific Computing - Wille, Wiley, 1995, ISBN 0471-95383-0.
Fortran 90 - Meissner, PWS Kent, Boston, 1995, ISBN 0-534-93372-6.
Fortran 90 - Counihan, Pitman, 1991, ISBN 0-273-03073-6.
Fortran 90 and Engineering Computation - Schick and Silverman, John
Wiley, 1994, ISBN 0-471-58512-2.
Fortran 90, A Reference Guide - Chamberland, Prentice Hall PTR, 1995,
ISBN 0-13-397332-8.
Fortran 90/95 Explained - Metcalf and Reid, Oxford University Press,
1996, ISBN 0-19-851888-9, about $33. This book is a complete, audited
description of the Fortran 90 and Fortran 95 languages in a more
readable style than the standards themselves. It incorporates all X3J3
and WG5's interpretations and has a complete chapter on Fortran 95.
It has seven Appendices, including an extended example program that is
available by ftp and solutions to exercises, as well as an Index.
US e-mail orders may be sent to: orders@oup-usa.org. The Fortran 90
version is also available in French, Japanese and Russian (see below).
Fortran 90 for Scientists and Engineers - Brian D. Hahn, Edward
Arnold, 1994, ISBN 0-340-60034-9.
Fortran 90 Handbook - Adams, Brainerd, Martin, Smith and Wagener,
McGraw-Hill, 1992, ISBN 0-07-000406-4.
Fortran 90 Language Guide - Gehrke, Springer, London, 1995,
ISBN 3-540-19926-8.
Fortran 95 Language Guide - Gehrke, Springer, London, 1996,
ISBN 3-540-76062-8.
Fortran 90 Programming - Ellis, Philips, Lahey, Addison Wesley,
Wokingham, 1994, ISBN 0-201-54446-6.
Fortran Top 90-Ninety Key Features of Fortran 90 - Adams, Brainerd,
Martin and Smith, Unicomp, 1994, ISBN 0-9640135-0-9.
Introducing Fortran 90 - Chivers and Sleightholme, Springer-Verlag
London, 1995, ISBN 3-540-19940-3.
Introduction to Fortran 90/95, Algorithms, and Structured Programming,
Part 1: Introduction to Fortran 90, Part 2: Algorithms and Fortran 90.
R. Vowels: 93 Park Drive, Parkville 3052, Victoria, AUSTRALIA,
(rav@goanna.cs.rmit.edu.au). $41 Aust, ISBN 0-9596384-8-2.
Introduction to Fortran 90 for Scientific Computing - Ortega, Saunders
College Publishing, 1994, ISBN 0-030010198-0.
Numerical Recipes in Fortran 90: The Art of Parallel Scientific
Computing, Volume 2 of Fortran Numerical Recipes - Press, Teukolsky,
Vetterling and Flannery, Cambridge U. Press, ISBN 0-521-57439-0, 1996.
Code can be downloaded (purchased) from http://nr.harvard.edu/nr/store.
A CDROM is also availble (see Web site).
Programmer's Guide to Fortran 90, third edition - Brainerd, Goldberg
and Adams, Springer, 1996, ISBN 0-387-94570-9.
Programming in Fortran 90 - Morgan and Schonfelder, Alfred Waller/
McGraw-Hill, Oxfordshire, 1993, ISBN 1-872474-06-3.
Programming in Fortran 90 - I.M. Smith, Wiley, ISBN 0471-94185-9.
Schaum's Outline of Theory and Praxis -- Programming in Fortran 90 -
Mayo and Cwiakala, Mc Graw Hill, 1996. ISBN 0-07-041156-5.
The F Programming Language - Metcalf and Reid, Oxford University Press,
1996, ISBN 0-19-850026-2, about $33. This book is the definitive
description of the F programming language - a carefully crafted subset
of Fortran 90 that is highly regular and stripped of Fortran's older,
dangerous features, but retains the powerful array language, data
abstraction and pointers. It has six Appendices, including an extended
example program that is available by ftp and solutions to exercises, as
well as an Index. US orders may be sent to: orders@oup-usa.org.
Upgrading to Fortran 90 - Redwine, Springer-Verlag, New York, 1995,
ISBN 0-387-97995-6.
Chinese:
Programming Language Fortran 90 - He Xingui, Xu Zuyuan, Wu Qingbao and
Chen Mingyuan, China Railway Publishing House, Beijing,
ISBN 7-113-01788-6/TP.187, 1994.
Dutch:
Fortran 90 - W.S. Brainerd, Ch.H. Goldberg, and J.C. Adams, translated
by J.M. den Haan, Academic Service, 1991, ISBN 90 6233 722 8.
French:
Fortran 90; Approche par la Pratique - Lignelet, Se'rie Informatique
E'ditions, Menton, 1993, ISBN 2-090615-01-4.
Fortran 90. Les concepts fondamentaux, the translation of "Fortran 90
Explained" M. Metcalf, J. Reid, translated by M. Caillet and B. Pichon,
AFNOR, Paris, ISBN 2-12-486513-7.
Fortran 90; Initiation a` partir du Fortran 77 - Aberti, Se'rie
Informatique E'ditions, Menton, 1992, ISBN 2-090615-00-6.
Les specificites du Fortran 90, DUBESSET, M. et VIGNES, J.,
editions Technip, 1993. ISBN 2-7108-0652-5
Manuel complet du langage Fortran 90, et guide d'application,
LIGNELET, P., S.I. editions, Jan. 1995. ISBN 2-909615-02-2
Manuel Complet du Langage FORTRAN 90 et FORTRAN 95, Calcul intensif et
Genie Logiciel (MASSON Editions, Paris; ISBN: 2-225-85229-4).
Programmer en Fortran 90, DELANNOY, C., Eyrolles, 1992.
ISBN 2-212-08723-3
Traitement des donnees numeriques avec Fortran 90, OLAGNON M., Masson,
1996, ISBN 2-225-85259-6.
Savez-vous parler Fortran, AIN, M., Bibliotheque des universites
(de Boeck), 1994. ISBN 2-8041-1755-3
STRUCTURES DE DONNEES (et leurs algorithmes) EN FORTRAN 90/95, P. Lignelet,
Les Editions MASSON (Paris, Milan, Barcelone ISBN: 2-225-85373-8).
German:
Fortran 90 - B.Wojcieszynski and R.Wojcieszynski, Addison-Wesley,
1993, ISBN 3-89319-600-5.
Fortran 90: eine informelle Einfu"hrung - Heisterkamp,
BI-Wissenschaftsverlag, 1991, ISBN 3-411153-21-0.
Fortran 90, Lehr- und Arbeitsbuch fuer das erfolgreiche Programmieren -
W.S. Brainerd, C.H. Goldberg, and J.C. Adams, translated by
Peter Thomas and Klaus G. Paul, R. Olbenbourg Verlag, Muenchen, 1994,
ISBN 3-486-22102-7.
Fortran 90 Lehr- und Handbuch - T. Michel, BI-Wissenschaftsverlag,
1994.
Fortran 90 Referenz-Handbuch: der neue Fortran-Standard - Gehrke,
Carl Hansen Verlag, 1991, ISBN 3-446163-21-2.
Programmierung in Fortran 90 - Schobert, Oldenburg, 1991.
Programmieren in Fortran - Erasmus Langer, Springer-Verlag,
Wien New York, 1993. ISBN 3-211-82446-4, 0-387-82446-4.
Software Entwicklung in Fortran 90 - U"berhuber and Meditz, Springer
Verlag, 1993, ISBN 0-387-82450-2.
Japanese:
Fortran 90 Explained - Metcalf and Reid, translated by H. Nisimura,
H. Wada, K. Nishimura, M. Takata, Kyoritsu Shuppan Co., Ltd., 1993,
ISSN 0385-6984.
Russian
An Explanation of the Fortran 90 Programming Language (translation of
Fortran 90 Explained - Metcalf and Reid), translated P. Gorbounov,
Mir, Moscow, 1995, ISBN 5-03-001426-8. Available also from
Petr.Gorbounov@cern.ch.
Swedish
Fortran 90 - en introduktion - Blom, Studentlitteratur, Lund, 1994,
ISBN 91-44-47881-X.
WHERE CAN I OBTAIN COURSES, COURSE MATERIAL OR CONSULTANCY?
Copyright but freely available course material is available
on the World Wide Web from the URLs:
Manchester Computer Centre:
http://www.hpctec.mcc.ac.uk/hpctec/courses/Fortran90/F90course.html
or via ftp: ftp.mcc.ac.uk, in the directory /pub/mantec/Fortran90.
The University of Liverpool:
http://www.liv.ac.uk/HPC/HPCpage.html.
CERN: http://wwwcn.cern.ch/asdoc/f90.html or via anonymous ftp
from asisftp.cern.ch in the directory cnl as the file f90tutor.ps.
In French: Support de cours Fortran 90 IDRIS - Corde & Delouis (from
ftp.ifremer.fr, file pub/ifremer/fortran90/f90_cours_4.ps.gz).
A course on HPF is freely available from Edinburgh: http://
www.epcc.ed.ac.uk/epcc-tec/course-packages/HPF-Package-form.html
Courses are available from:
Walt Brainerd, a member of X3J3, also on HPF (walt@fortran.com);
Tom Lahey (sales@lahey.com).
PSR (see above);
CETech, Inc. (also on HPF)
8196 SW Hall Blvd., Ste. 304, Beaverton, Oregon 97008, USA.
Phone: (503)644-6106 Fax: (503)643-8425 (cetech@teleport.com).
European companies offering courses and conversion consultancy are:
IT Independent Training Limited,
2 Windlebrook Green, Bracknell, Berkshire, UK
tel. +44 1344 860172 fax. +44 1344 867992
Salford Software (see above);
Simulog, attn. Mr. E. Plestan,
1 rue James Joule, F-78286 Guyancourt Cedex, France
tel: +33 1 30 12 27 80 fax: +33 1 30 12 27 27
e-mail: plestan@simulog.fr
Allgemeiner Software Service
Prinz-Otto Str.7c, D-85521 Ottobrunn, Germany
tel: +49-89-6083758 Fax: +49-89-6083758
e-mail: 100722.746@compuserve.com
URL: http://www.wp.com/AllSoftServe
WHERE CAN I FIND THE STANDARD?
Fortran 90 was adopted as an International Standard by ISO in July, 1991,
as ISO/IEC 1539:1991, and is obtainable for 185 Swiss francs from
ISO Publications, 1 rue de Varembe, Case postale 56
CH-1211 Geneva 20, Switzerland
Fax. + 41 22 734 10 79
It may also be obtained from national member bodies such as
ANSI, 1430 Broadway, New York, N.Y. 10018
(where it is also known as ANSI X3.198-1992), or in electronic PostScript
or ASCII form from Unicomp (walt@fortran.com) at a cost and
under conditions agreed by ISO.
Corrigenda 1 and 2 were published by ISO in 1993 and 1995, respectively,
and are available from them (cost about 30 Swiss francs). Corrigendum 3
was approved for publication in 1996.
A Russian translation of the standard (translator S.G.Drobyshevich)
is available from the editor, Alla Gorelik (gorelik@applmat.msk.su).
*****
This information is compiled on a 'best-effort' basis and is without
prejudice. It may be freely copied and disseminated. Corrections and
additions are solicited.
Mike Metcalf
(metcalf@cern.ch)
Version of 12 November, 1996.
