\documentclass{book}
%\newcommand{\VolumeName}{Volume 2: Axiom Users Guide}
%\input{bookheader.tex}
\pagenumbering{arabic}
\mainmatter
\setcounter{chapter}{0} % Chapter 1

\usepackage{makeidx}
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\begin{document}
\begin{verbatim}
\start
Date: Wed, 8 Oct 2014 00:24:22 -0500
From: daly@axiom-developer.org
To: axiom-developer@nongnu.org
Subject: [Axiom-developer] permissions
Cc: Kurt Pagani <nilqed@gmail.com>, Jim Wheeler <jim.wheeler@usu.edu>

I am documenting the differential forms issue that recently became
interesting, mostly due to Kurt Pagani's new code and examples.

Often, when I'm trying to understand an area well enough to write
clearly I come across documents and papers authored by others
because, you know, ... google. When that happens I ask for permission
to use their words and, almost always, I get permission.

In this instance, Jim Wheeler (Utah State) is the most recent person
to allow me to use his work.

I make sure that the people get added to the credits list and get
both bibliographic and in-text citations. Unfortunately I haven't
publicly acknowledged people who have generously consented on this
mailing list in the past and, since all of the email exchanges are
"off list", nobody really sees the generosity of our community.
I plan to change that in the future.

Thank you to all who have agreed in the past. Your excellent writing
and clear documentation is what will make Axiom survive.

Tim Daly

\start
Date: Wed, 8 Oct 2014 00:33:18 -0500
From: daly@axiom-developer.org
To: Javier Vazquez-Trejo <javt42@gmail.com>
Subject: [Axiom-developer] thesis on symbolic summation
Cc: Victor Adamchik <adamchik@cs.cmu.edu>, axiom-developer@nongnu.org

Javier,

Thank you for your thesis. It will, of course, take a while for me to
read and understand the details. Eventually I'd like to implement this
in Axiom but it won't be soon.

In case you haven't heard of it, Axiom was a commercial competitor
to Mathematica and Maple, sold by the Numerical Algorithms Group.
NAG withdrew it from the market and it is now open source
(http://axiom-developer.org)

Tim Daly
http://daly.axiom-developer.org

\start
Date: Wed, 8 Oct 2014 10:24:32 -0500
From: daly@axiom-developer.org
To: Victor Adamchik <adamchik@cs.cmu.edu>
Subject: Re: [Axiom-developer] thesis on symbolic summation

>We have tried to implement Karr-Schneider algorithm in Mathematica, 
>however we met two problems that we are not sure how to deal with. If 
>you are interested, Javier can provide more details on this.
>
>Victor

There is no such thing as a simple job.

I'd be interested to know the nature of the problems. Axiom and
Mathematica differ quite radically in their world views so,
depending on the problems, they might not exist in an Axiom setting.
But it still won't be a simple job, I'm sure. 

Is there a set of test cases with expected results I can use?

I'll read the thesis and get back to Javier with questions.

Thanks for the warning.
Tim

\start
Date: Wed, 08 Oct 2014 11:02:21 -0400
From: Victor Adamchik <adamchik@andrew.cmu.edu>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

Tim,

We have tried to implement Karr-Schneider algorithm in Mathematica, 
however we met two problems that we are not sure how to deal with. If 
you are interested, Javier can provide more details on this.

Victor

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Victor S. Adamchik                   Computer Science Department, CMU
Office: GHC 7719                5000 Forbes Ave, Pittsburgh, PA 15213
Phone: (412) 268-8121                 http://www.cs.cmu.edu/~adamchik

\start
Date: Wed, 08 Oct 2014 15:31:28 -0400
From: "William Sit" <wyscc@sci.ccny.cuny.edu>
To: adamchik@andrew.cmu.edu,daly@axiom-developer.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

Didn't Schneider implemented the algorithm in Mathematica 
around 2000?
http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf

William

On Wed, 08 Oct 2014 11:02:21 -0400
  Victor Adamchik <adamchik@andrew.cmu.edu> wrote:
> 
> Tim,
> 
> We have tried to implement Karr-Schneider algorithm in 
>Mathematica, however we met two problems that we are not 
>sure how to deal with. If you are interested, Javier can 
>provide more details on this.
> 
> Victor
> 
> 
> 
> -- 
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
> Victor S. Adamchik                   Computer Science 
>Department, CMU
> Office: GHC 7719                5000 Forbes Ave, 
>Pittsburgh, PA 15213
> Phone: (412) 268-8121 
>                http://www.cs.cmu.edu/~adamchik
> 

William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

\start
Date: Wed, 08 Oct 2014 16:05:57 -0400
From: Victor Adamchik <adamchik@andrew.cmu.edu>
To: William Sit <wyscc@sci.ccny.cuny.edu>
Subject: Re: [Axiom-developer] thesis on symbolic summation

i originally thought that implementing (or reimplementing) would be easy 
as a walk in the park...


Victor


On 10/8/2014 3:31 PM, William Sit wrote:
> Didn't Schneider implemented the algorithm in Mathematica around 2000?
> http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf
>
> William
>
> On Wed, 08 Oct 2014 11:02:21 -0400
>  Victor Adamchik <adamchik@andrew.cmu.edu> wrote:
>>
>> Tim,
>>
>> We have tried to implement Karr-Schneider algorithm in Mathematica, 
>> however we met two problems that we are not sure how to deal with. If 
>> you are interested, Javier can provide more details on this.
>>
>> Victor
>>
>>
>>
>> -- 
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>> Victor S. Adamchik                   Computer Science Department, CMU
>> Office: GHC 7719                5000 Forbes Ave, Pittsburgh, PA 15213
>> Phone: (412) 268-8121                http://www.cs.cmu.edu/~adamchik
>>
>>
>> _______________________________________________
>> Axiom-developer mailing list
>> Axiom-developer@nongnu.org
>> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>
> William Sit, Professor Emeritus
> Mathematics, City College of New York
> Office: R6/291D Tel: 212-650-5179
> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

\start
Date: Wed, 8 Oct 2014 16:07:06 -0400
From: Bill Page <bill.page@newsynthesis.org>
To: William Sit <wyscc@sci.ccny.cuny.edu>
Subject: Re: [Axiom-developer] thesis on symbolic summation
Cc: Victor Adamchik <adamchik@cs.cmu.edu>, adamchik@andrew.cmu.edu,
	Axiom-Developer <axiom-developer@nongnu.org>,
	Javier Vazquez-Trejo <javt42@gmail.com>

http://www.risc.jku.at/research/combinat/software/Sigma/index.php

On 8 October 2014 15:31, William Sit <wyscc@sci.ccny.cuny.edu> wrote:
> Didn't Schneider implemented the algorithm in Mathematica around 2000?
> http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf
>
> William
>
>
> On Wed, 08 Oct 2014 11:02:21 -0400
>  Victor Adamchik <adamchik@andrew.cmu.edu> wrote:
>>
>>
>> Tim,
>>
>> We have tried to implement Karr-Schneider algorithm in Mathematica,
>> however we met two problems that we are not sure how to deal with. If you
>> are interested, Javier can provide more details on this.
>>
>> Victor
>>
>>
>>
>> --
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>> Victor S. Adamchik                   Computer Science Department, CMU
>> Office: GHC 7719                5000 Forbes Ave, Pittsburgh, PA 15213
>> Phone: (412) 268-8121                http://www.cs.cmu.edu/~adamchik
>
> William Sit, Professor Emeritus
> Mathematics, City College of New York
> Office: R6/291D Tel: 212-650-5179
> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
>

\start
Date: Wed, 8 Oct 2014 15:13:48 -0500
From: daly@axiom-developer.org
To: Carsten.Schneider@risc.uni-linz.ac.at
Subject: [Axiom-developer] Karr's Indefinite Summation

Carsten,

I recently ran across a thesis by Javier Vazquez-Trejo on Karr's
indefinite summation, supervised by Victor Adamchik at Carnegie-Mellon.

William Sit mentioned your paper and I see you've implemented the
algorithm in Mathematica. 

I'm interested in developing the Karr's algorithm in Axiom.
Is your Mathematica source code available anywhere?

Tim Daly

\start
Date: Wed, 08 Oct 2014 22:17:53 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: axiom-developer@nongnu.org, Burcin Erocal <burcin@erocal.org>
Subject: Re: [Axiom-developer] thesis on symbolic summation
Cc: adamchik@cs.cmu.edu, javt42@gmail.com

On 10/08/2014 09:31 PM, William Sit wrote:
> Didn't Schneider implemented the algorithm in Mathematica around 2000?
> http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf

Probably more of the theory can be found in his thesis.

http://www.risc.jku.at/publications/download/risc_3017/SymbSumTHESIS.pdf

Burcin Erocal also worked in this field.

http://www.risc.jku.at/publications/download/risc_4320/erocal_thesis.pdf

Burcin, can you give a link to your Sage implementation?

Ralf

\start
Date: Thu, 09 Oct 2014 04:05:39 +0200
From: kp <kp@scios.ch>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

I've recently discovered a thesis called

"Interactive Computer Manipulation of Formal Sums"
by Nivedita Patil
http://www.csd.uwo.ca/~watt/home/students/theses/NPatil2010-msc.pdf

and - just for fun - sketched some functions from the text. There seems
to be a lot of symbolic summation code around as I learned here. Is
there anything related to the 'title' above available in Axiom?
Kurt


Am 08.10.2014 um 22:05 schrieb Victor Adamchik:
> i originally thought that implementing (or reimplementing) would be easy
> as a walk in the park...
> 
> 
> Victor
> 
> 
> On 10/8/2014 3:31 PM, William Sit wrote:
>> Didn't Schneider implemented the algorithm in Mathematica around 2000?
>> http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf
>>
>> William
>>
>> On Wed, 08 Oct 2014 11:02:21 -0400
>>  Victor Adamchik <adamchik@andrew.cmu.edu> wrote:
>>>
>>> Tim,
>>>
>>> We have tried to implement Karr-Schneider algorithm in Mathematica,
>>> however we met two problems that we are not sure how to deal with. If
>>> you are interested, Javier can provide more details on this.
>>>
>>> Victor
>>>
>>>
>>>
>>> -- 
>>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>>> Victor S. Adamchik                   Computer Science Department, CMU
>>> Office: GHC 7719                5000 Forbes Ave, Pittsburgh, PA 15213
>>> Phone: (412) 268-8121                http://www.cs.cmu.edu/~adamchik
>>>
>>>
>>> _______________________________________________
>>> Axiom-developer mailing list
>>> Axiom-developer@nongnu.org
>>> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>>
>> William Sit, Professor Emeritus
>> Mathematics, City College of New York
>> Office: R6/291D Tel: 212-650-5179
>> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
> 
> 
> 
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> https://lists.nongnu.org/mailman/listinfo/axiom-developer


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--------------020509080507060703080002--

\start
Date: Wed, 8 Oct 2014 22:49:24 -0500
From: daly@axiom-developer.org
To: kp <kp@scios.ch>
Subject: [Axiom-developer]  thesis on symbolic summation

Kurt,

I'm currently reading 
Zima, Eugene V.
Accelerating Indefinite Summation: Simple Classes of Summands"

which "presents the history of indefinite summation starting with
classics (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan)
followed by modern classics (Abramov, Gosper, Karr) to the current
implementation in computer algebra system Maple".

It gives a nice overview up to but not including Schneider's work.

Tim

\start
Date: Thu, 09 Oct 2014 04:21:16 -0400
From: "William Sit" <wyscc@sci.ccny.cuny.edu>
To: kp <kp@scios.ch>,axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

Shasta has a Maple interface too.
http://www-sop.inria.fr/cafe/Manuel.Bronstein/sumit/shastadoc/node9.html

William

On Thu, 09 Oct 2014 04:05:39 +0200
  kp <kp@scios.ch> wrote:
> 
> I've recently discovered a thesis called
> 
> "Interactive Computer Manipulation of Formal Sums"
> by Nivedita Patil
> http://www.csd.uwo.ca/~watt/home/students/theses/NPatil2010-msc.pdf
> 
> and - just for fun - sketched some functions from the 
>text. There seems
> to be a lot of symbolic summation code around as I 
>learned here. Is
> there anything related to the 'title' above available in 
>Axiom?
> Kurt
> 
> 
> Am 08.10.2014 um 22:05 schrieb Victor Adamchik:
>> i originally thought that implementing (or 
>>reimplementing) would be easy
>> as a walk in the park...
>> 
>> 
>> Victor
>> 
>> 
>> On 10/8/2014 3:31 PM, William Sit wrote:
>>> Didn't Schneider implemented the algorithm in 
>>>Mathematica around 2000?
>>> http://www.emis.de/journals/SLC/wpapers/s43schneider.pdf
>>>
>>> William
>>>
>>> On Wed, 08 Oct 2014 11:02:21 -0400
>>>  Victor Adamchik <adamchik@andrew.cmu.edu> wrote:
>>>>
>>>> Tim,
>>>>
>>>> We have tried to implement Karr-Schneider algorithm in 
>>>>Mathematica,
>>>> however we met two problems that we are not sure how to 
>>>>deal with. If
>>>> you are interested, Javier can provide more details on 
>>>>this.
>>>>
>>>> Victor
>>>>
>>>>
>>>>
>>>> -- 
>>>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>>>> Victor S. Adamchik                   Computer Science 
>>>>Department, CMU
>>>> Office: GHC 7719                5000 Forbes Ave, 
>>>>Pittsburgh, PA 15213
>>>> Phone: (412) 268-8121 
>>>>               http://www.cs.cmu.edu/~adamchik
>>>>
>>>>
>>>> _______________________________________________
>>>> Axiom-developer mailing list
>>>> Axiom-developer@nongnu.org
>>>> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>>>
>>> William Sit, Professor Emeritus
>>> Mathematics, City College of New York
>>> Office: R6/291D Tel: 212-650-5179
>>> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

\start
Date: Thu, 09 Oct 2014 10:48:31 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

On 10/09/2014 10:21 AM, William Sit wrote:
> Shasta has a Maple interface too.
> http://www-sop.inria.fr/cafe/Manuel.Bronstein/sumit/shastadoc/node9.html

Why interface to Maple if Shasta source code is free?

https://gforge.inria.fr/projects/bronstein-codes/

It would probably be a bit of work to transform it to work with Axiom,
because it's pure Aldor and thus doesn't build on Axiom library code.

Ralf

\start
Date: Thu, 09 Oct 2014 05:15:50 -0400
From: "William Sit" <wyscc@sci.ccny.cuny.edu>
To: Ralf Hemmecke <ralf@hemmecke.org>,axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

Some people already have Maple?

William

On Thu, 09 Oct 2014 10:48:31 +0200
  Ralf Hemmecke <ralf@hemmecke.org> wrote:
> On 10/09/2014 10:21 AM, William Sit wrote:
>> Shasta has a Maple interface too.
>> http://www-sop.inria.fr/cafe/Manuel.Bronstein/sumit/shastadoc/node9.html
> 
> Why interface to Maple if Shasta source code is free?
> 
> https://gforge.inria.fr/projects/bronstein-codes/
> 
> It would probably be a bit of work to transform it to 
>work with Axiom,
> because it's pure Aldor and thus doesn't build on Axiom 
>library code.
> 
> Ralf

William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

\start
Date: Thu, 09 Oct 2014 11:23:51 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: William Sit <wyscc@sci.ccny.cuny.edu>, axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] thesis on symbolic summation

On 10/09/2014 11:15 AM, William Sit wrote:
> Some people already have Maple?

OK. But it will not help if you do not have the shasta executable
locally on your machine. So you need the Shasta and SumIt sources and
the Aldor compiler anyway to interface to maple.

I know that Bronstein used to expose a webinterface to bernina and
shasta, but I guess, it's not running anymore and shasta.m is probably
also not able to access Shasta (at an inria.fr server) via the Internet.

Ralf

\start
From: daly@axiom-developer.org
Date: Mon, 20 Oct 2014 02:30:49 -0500
To: goretkin <notifications@github.com>
Subject: Re: [Axiom-developer] [PATCH] small typo -- FIXED

Gustavo,

Thanks. The patch has been applied and tested and has been pushed.

Tim

\start
Date: Mon, 20 Oct 2014 02:39:07 -0500
From: daly@axiom-developer.org
To: axiom-developer@nongnu.org
Subject: [Axiom-developer] Bernoulli puzzle

Axiom's bernoulli algorithm reads:

  bernoulli n ==
    n < 0 => error "bernoulli not defined for negative integers"
    -- except for 1, all odd terms are zero
    odd? n =>
      n = 1 => -1/2
      0
    -- B is a cache. If we already know the answer, just return it
    l := (#B) :: I
    n < l => B(n)
    -- extend B to hold the new terms we will compute
    -- there are n+1 terms but we may have cached l of them
    concat_!(B, new((n+1-l)::NNI, 0)$IndexedFlexibleArray(RN,0))
    -- compute B(i) i = l+2,l+4,...,n given B(j) j = 0,2,...,i-2
    -- 'by 2' to only compute even entries
    for i in l+1 .. n by 2 repeat
      t:I := 1
      b := (1-i)/2
      for j in 2 .. i-2 by 2 repeat
        t := (t*(i-j+2)*(i-j+3)) quo (j*(j-1))
        b := b + (t::RN) * B(j)
      B(i) := -b/((i+1)::RN)
    B(n)

I'm unable to decode the equations supporting the inner loop
computation (t and b) which support computing the B(i) term.

Since the loop iterates by 2 it looks like someone worked out
the details of the computation taking 2 steps at a time.
Unfortunately I can't figure out how to unwind this into any
of the usual bernoulli formulas.

Does anyone recognize the formula that is being used?
Can you translate it into TeX format?

Tim

\start
Date: Mon, 20 Oct 2014 12:41:56 +0200 (CEST)
From: Waldek Hebisch <hebisch@math.uni.wroc.pl>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] Bernoulli puzzle

Tim Daly wrote:
> 
> Axiom's bernoulli algorithm reads:
<snip>
> 
> I'm unable to decode the equations supporting the inner loop
> computation (t and b) which support computing the B(i) term.
> 
> Since the loop iterates by 2 it looks like someone worked out
> the details of the computation taking 2 steps at a time.
> Unfortunately I can't figure out how to unwind this into any
> of the usual bernoulli formulas.
> 
> Does anyone recognize the formula that is being used?
> Can you translate it into TeX format?

Usualy Bernoulli numbers B_n are defined as coefficients of
the following expansion:

t/(exp(t) - 1) = \sum_{n=0}^\infty B_n t^n/n!

Since

t/(exp(t) - 1) + (1/2)*t = t(exp(t) + 1)/(2(exp(t) - 1) = t/(2tanh(t/2))

we have B_0 = 1, B_1 = -1/2 and because t/(2tanh(t/2)) is an
even function for odd n different than 1 we have B_n = 0.

Given this information we can rewrite the defining formula:

t/(exp(t) - 1) = - (1/2)t + \sum_{n=0}^\infty B_{2n}t^{2n}/(2n}!

Multiply both sides by (exp(t) - 1):

t = - t(exp(t) - 1)/2 + 
    (exp(t) - 1)\sum_{n=0}^\infty B_{2n}t^{2n}/(2n}!

Compute coefficient before t^{2n+1} for both sides:

0 =  -1/(2(2n)!) + \sum_{k=0}^{n}B_{2k}/((2k)!(2n+1 - 2k)!)

so

B_{2n}/(2n)! = -( -1/(2(2n)!) + 
               \sum_{k=0}^{n-1}B_{2k}/((2k)!(2n+1 - 2k)!))

Multiplying by (2n+1)!:

B_{2n}(2n+1) = -(-(2n+1)/2 +
               \sum_{k=0}^{n-1}B_{2k}binomial(2n+1, 2k))


Since B_0 = 1 and binomial(2n+1, 0) = 1 the first term of sum
is 1 and we can write:

B_{2n}(2n+1) = -((1 - 2n)/2 + 
                 \sum_{k=1}^{n-1}B_{2k}binomial(2n+1, 2k))

Now, the variable b is used to compute expression in the
parenthesis.  More precisely, table of Bernoulli numbers always
has odd length, to variable l is odd and consequently l + 1
is even.  So i goes trough even numbers, starting from first
unknown position in the tables.  In other words, i in the inner
loop is 2n in the formula above.  Consequently, initialization

   b := (1-i)/2

computes the (1 - 2n)/2 term before sum.  Variable t is used to
compote binomial(2n+1, 2k) using formula

binomial(2n+1, 2k) = (2n + 1 - (2k - 1))(2n + 1 - (2k-2))/
                       ((2k)(2k-1))binomial(2n+1, 2k-2)

and binomial(2n+1, 0) = 1.  Since j in the inner loop is 2k from
above assignment to t implements this formula.

Once experession in parenthesis from formula for B_{2n} is
computed the

   B(i) := -b/((i+1)::RN)

line gets value of Bernoulli number.

PS.  This is essentially using the recursive definition given
in Wikipedia.  Wikipedia calls this inefficient, but AFAICS
due to memoization it is much better than Akiyama-Tanigawa
presented in Wikipedia.

-- 
                              Waldek Hebisch
hebisch@math.uni.wroc.pl 

\start
Date: Mon, 20 Oct 2014 07:19:34 -0500
From: daly@axiom-developer.org
To: Waldek Hebisch <hebisch@math.uni.wroc.pl>
Subject: Re: [Axiom-developer] Bernoulli puzzle

Excellent, thank you.

>PS.  This is essentially using the recursive definition given
>in Wikipedia.  Wikipedia calls this inefficient, but AFAICS
>due to memoization it is much better than Akiyama-Tanigawa
>presented in Wikipedia.

I read several papers on the Bernoulli function. Axiom references
John Brillhart's "On the Euler and Bernoulli polynomials" which
was his Berkeley PhD thesis but I can't find a copy online anywhere.

There were a couple papers which were an efficiency contest between 
Mathematica and Sage. Sage uses Pari's implementation by default
which gives excellent results. Pari uses

\[ |B_n| = \frac{2n!}{(2\pi)^n}\zeta(n) \]

with floats but you have to completely control the precision.

Wolfram published
http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica

Sage claims that bernoulli(10^5) takes about 11 seconds.

Tim

\start
Date: Mon, 20 Oct 2014 14:37:49 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] Bernoulli puzzle

> which gives excellent results. Pari uses
> 
> \[ |B_n| = \frac{2n!}{(2\pi)^n}\zeta(n) \]
> 
> with floats but you have to completely control the precision.

I don't know exactly, but I'd bet that Sage builds on flint2 for the
computation of Bernoulli (implemented (if I am not wrong) by Fredrik
http://fredrikj.net/).

http://flintlib.org/benchmarks.html

Since that is free software, it would make sense to think about using
that library.

Ralf

\start
Date: Mon, 20 Oct 2014 17:03:59 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] Bernoulli puzzle

On 10/20/2014 08:37 AM, Ralf Hemmecke wrote:
>> which gives excellent results. Pari uses
>>
>> \[ |B_n| = \frac{2n!}{(2\pi)^n}\zeta(n) \]
>>
>> with floats but you have to completely control the precision.
> I don't know exactly, but I'd bet that Sage builds on flint2 for the
> computation of Bernoulli (implemented (if I am not wrong) by Fredrik
> http://fredrikj.net/).
>
> http://flintlib.org/benchmarks.html
>
> Since that is free software, it would make sense to think about using
> that library.
>
> Ralf
>
>
A little off topic; but I have developed an alternate way of dealing 
with polynomial sequences like
Bernoulli polynomials that are generated by generating functions. It 
involves casting the sequences in
matrices and apply Pascal Matrices and Umbral calculus.  It makes some 
known relations obvious
and casts a different viewpoint on others.
It might allow some kind of Polynomial sequence algebra or some such.  
It does have the advantage of
automatically converting some (actually  most) sequences to others by 
symbolic/parametrized methods.
If anybody is interested let me know and I will write up the application 
to Bernoulli polynomials as a special case.

Ray

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana


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    <br>
    <div class="moz-cite-prefix">On 10/20/2014 08:37 AM, Ralf Hemmecke
      wrote:<br>
    </div>
    <blockquote cite="mid:5445021D.20500@hemmecke.org" type="cite">
      <blockquote type="cite">
        <pre wrap="">which gives excellent results. Pari uses

\[ |B_n| = \frac{2n!}{(2\pi)^n}\zeta(n) \]

with floats but you have to completely control the precision.
</pre>
      </blockquote>
      <pre wrap="">
I don't know exactly, but I'd bet that Sage builds on flint2 for the
computation of Bernoulli (implemented (if I am not wrong) by Fredrik
<a class="moz-txt-link-freetext" href="http://fredrikj.net/">http://fredrikj.net/</a>).

<a class="moz-txt-link-freetext" href="http://flintlib.org/benchmarks.html">http://flintlib.org/benchmarks.html</a>

Since that is free software, it would make sense to think about using
that library.

Ralf


</pre>
    </blockquote>
    <big>A little off topic; but I have developed an alternate way of
      dealing with polynomial sequences like<br>
      Bernoulli polynomials that are generated by generating functions.
      It involves casting the sequences in<br>
      matrices and apply Pascal Matrices and Umbral calculus. It makes
      some known relations obvious <br>
      and casts a different viewpoint on others. <br>
      It might allow some kind of Polynomial sequence algebra or some
      such. It does have the advantage of<br>
      automatically converting some (actually most) sequences to others
      by symbolic/parametrized methods.<br>
      If anybody is interested let me know and I will write up the
      application to Bernoulli polynomials as a special case.<br>
      <br>
      Ray</big><br>
    <pre class="moz-signature" cols="72">-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana
</pre>
  </body>
</html>

--------------060204000503090605090905--

\start
Date: Mon, 20 Oct 2014 17:20:36 -0500
From: daly@axiom-developer.org
To: Raymond Rogers <raymond.rogers72@gmail.com>
Subject: Re: [Axiom-developer] Bernoulli puzzle

Ray,

>A little off topic; but I have developed an alternate way of dealing
>with polynomial sequences like Bernoulli polynomials that are
>generated by generating functions. It involves casting the sequences
>in matrices and apply Pascal Matrices and Umbral calculus.  It makes
>some known relations obvious and casts a different viewpoint on
>others.  It might allow some kind of Polynomial sequence algebra or
>some such.  It does have the advantage of automatically converting
>some (actually most) sequences to others by symbolic/parametrized
>methods.  If anybody is interested let me know and I will write up the
>application to Bernoulli polynomials as a special case.

That would be an interesting generalization. Axiom implements several
number theory algorithms with generating functions. If they were all
just "cover calls" to a common method it would be useful.

If you were to write something like that I'm begging you to write 
a fair amount of natural language explanation. I lost a whole weekend
trying to reverse-engineer the bernoulli code so I can document it. 
Without Waldek's help I'd still be struggling. Please consider that,
although YOU understand the code you write, nobody else does. Few
people, myself included, have heard of Unbral calculus.

Tim

\start
Date: Tue, 21 Oct 2014 12:25:45 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] Bernoulli puzzle

Tim,
     Yes I learned some time ago that maintenance can be devastating.  I 
am a little at odds
with your theory but not too much; and I will try to conform to your 
styles.  Along that
vein: can you point me to the current tools you recommend and examples 
of how you are
currently doing documentation?   They seem to have drifted a little over 
the years.

BTW: my ideal is "The Elements of Programming Style," by Kernighan and 
Plauger.
After a particularlly upsetting run in, with myself, regarding "clever" 
coding I found this to
be an exceptional guide.  Clarity is typically more important than 
cleverness or one or two
machine cycles.

"although YOU understand the code you write, nobody else does. "
Even when I was younger and clever that wasn't true after six months 
:):)  I guess I might
be schizoid but after six months and then looking at some "clever" math 
or programing I typically
have to start from scratch (really sad).

Ray

On 10/20/2014 06:20 PM, daly@axiom-developer.org wrote:
> Ray,
>
>> A little off topic; but I have developed an alternate way of dealing
>> with polynomial sequences like Bernoulli polynomials that are
>> generated by generating functions. It involves casting the sequences
>> in matrices and apply Pascal Matrices and Umbral calculus.  It makes
>> some known relations obvious and casts a different viewpoint on
>> others.  It might allow some kind of Polynomial sequence algebra or
>> some such.  It does have the advantage of automatically converting
>> some (actually most) sequences to others by symbolic/parametrized
>> methods.  If anybody is interested let me know and I will write up the
>> application to Bernoulli polynomials as a special case.
> That would be an interesting generalization. Axiom implements several
> number theory algorithms with generating functions. If they were all
> just "cover calls" to a common method it would be useful.
>
> If you were to write something like that I'm begging you to write
> a fair amount of natural language explanation. I lost a whole weekend
> trying to reverse-engineer the bernoulli code so I can document it.
> Without Waldek's help I'd still be struggling. Please consider that,
> although YOU understand the code you write, nobody else does. Few
> people, myself included, have heard of Unbral calculus.
>
> Tim
>
>

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana


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    Tim,<br>
     Yes I learned some time ago that maintenance can be
    devastating. I am a little at odds <br>
    with your theory but not too much; and I will try to conform to your
    styles. Along that<br>
    vein: can you point me to the current tools you recommend and
    examples of how you are<br>
    currently doing documentation? They seem to have drifted a little
    over the years.<br>
    <br>
    BTW: my ideal is "The Elements of Programming Style," by Kernighan
    and Plauger.<br>
    After a particularlly upsetting run in, with myself, regarding
    "clever" coding I found this to<br>
    be an exceptional guide. Clarity is typically more important than
    cleverness or one or two <br>
    machine cycles.<br>
    <br>
    "although YOU understand the code you write, nobody else does.
    "<br>
    Even when I was younger and clever that wasn't true after six months
    :):) I guess I might <br>
    be schizoid but after six months and then looking at some "clever"
    math or programing I typically<br>
    have to start from scratch (really sad).<br>
    <br>
    Ray<br>
    <br>
    <span style="color: rgb(37, 37, 37); font-family: sans-serif;
      font-size: 14px; font-style: normal; font-variant: normal;
      font-weight: normal; letter-spacing: normal; line-height:
      22.3999996185303px; orphans: auto; text-align: start; text-indent:
      0px; text-transform: none; white-space: normal; widows: auto;
      word-spacing: 0px; -webkit-text-stroke-width: 0px; display: inline
      !important; float: none; background-color: rgb(255, 255, 255);"></span>
    <div class="moz-cite-prefix">On 10/20/2014 06:20 PM,
      <a class="moz-txt-link-abbreviated" href="mailto:daly@axiom-developer.org">daly@axiom-developer.org</a> wrote:<br>
    </div>
    <blockquote
      cite="mid:201410202220.s9KMKa4n018670@axiom-developer.org"
      type="cite">
      <pre wrap="">Ray,

</pre>
      <blockquote type="cite">
        <pre wrap="">A little off topic; but I have developed an alternate way of dealing
with polynomial sequences like Bernoulli polynomials that are
generated by generating functions. It involves casting the sequences
in matrices and apply Pascal Matrices and Umbral calculus.  It makes
some known relations obvious and casts a different viewpoint on
others.  It might allow some kind of Polynomial sequence algebra or
some such.  It does have the advantage of automatically converting
some (actually most) sequences to others by symbolic/parametrized
methods.  If anybody is interested let me know and I will write up the
application to Bernoulli polynomials as a special case.
</pre>
      </blockquote>
      <pre wrap="">
That would be an interesting generalization. Axiom implements several
number theory algorithms with generating functions. If they were all
just "cover calls" to a common method it would be useful.

If you were to write something like that I'm begging you to write 
a fair amount of natural language explanation. I lost a whole weekend
trying to reverse-engineer the bernoulli code so I can document it. 
Without Waldek's help I'd still be struggling. Please consider that,
although YOU understand the code you write, nobody else does. Few
people, myself included, have heard of Unbral calculus.

Tim


</pre>
    </blockquote>
    <br>
    <pre class="moz-signature" cols="72">-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana
</pre>
  </body>
</html>

--------------060809020906050509040201--

\start
Date: Tue, 21 Oct 2014 14:50:00 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not as
	streams but as sumations

On 10/21/2014 04:39 PM, Raymond Rogers wrote:
> Let's think about this.
> I think the question can be interpreted as a formating/display problem
> for the underlying code.

If you believe so... Let me state it a bit differently what
datastructure comes out of the series command. Or rather what is stored
inside the data structure.

Here an example:

(1) -> Z==>Integer; Q==>Fraction Z;
                                                                    Type:
Void
(2) -> f(q:Q):Q == truncate(abs(q*cos(q::Float))+1)*(q/2+1)

    Function declaration f : Fraction(Integer) -> Fraction(Integer) has
       been added to workspace.
                                                                    Type:
Void
(3) -> s: Stream Q := stream(f, 0)
    Compiling function f with type Fraction(Integer) -> Fraction(Integer
       )

              3 7 15 31 63 127 255 511
    (3)  [0,1,-,-,--,--,--,---,---,---,...]
              2 4  8 16 32  64 128 256
                                               Type:
Stream(Fraction(Integer))
(4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)

    (4)
          3  2   7  3   15  4   31  5   63  6   127  7   255  8   511  9
      x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + --- x  + --- x
          2      4       8      16      32       64      128      256
    +
      1023  10      11
      ---- x   + O(x  )
       512
                          Type:
UnivariateFormalPowerSeries(Fraction(Integer))

The only thing that s (the Stream) knows is a function object f (if you
like you can use f=random). It cannot look inside f except to ask for
more values. Remember AXIOM code is compiled.

If from that you would somehow be able to extract a pattern, great, but
I doubt.

> Of course I am not that familiar with the Axiom internals.

Well, you might come from another CAS. AXIOM *is* different. It is in
general *not* dealing with expression trees. So what you have in mind
(adding presentation) does not work for Stream of Series.

That is not saying that it is impossible, but rather that the Stream and
PowerSeries domains are not appropriate for what you have in mind. (And
at the moment I don't think there is anything like what you want in Axiom.)

Ralf

\start
Date: Tue, 21 Oct 2014 15:05:20 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: Ralf Hemmecke <ralf@hemmecke.org>, axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not as
	streams but as sumations

On 10/21/2014 11:27 AM, Ralf Hemmecke wrote:
> On 10/21/2014 04:39 PM, Raymond Rogers wrote:
>> Let's think about this.
>> I think the question can be interpreted as a formating/display problem
>> for the underlying code.
> If you believe so... Let me state it a bit differently what
> datastructure comes out of the series command. Or rather what is stored
> inside the data structure.
>
> Here an example:
>
> (1) -> Z==>Integer; Q==>Fraction Z;
>                                                                     Type:
> Void
> (2) -> f(q:Q):Q == truncate(abs(q*cos(q::Float))+1)*(q/2+1)
>
>     Function declaration f : Fraction(Integer) -> Fraction(Integer) has
>        been added to workspace.
>                                                                     Type:
> Void
> (3) -> s: Stream Q := stream(f, 0)
>     Compiling function f with type Fraction(Integer) -> Fraction(Integer
>        )
>
>               3 7 15 31 63 127 255 511
>     (3)  [0,1,-,-,--,--,--,---,---,---,...]
>               2 4  8 16 32  64 128 256
>                                                Type:
> Stream(Fraction(Integer))
> (4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)
>
>     (4)
>           3  2   7  3   15  4   31  5   63  6   127  7   255  8   511  9
>       x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + --- x  + --- x
>           2      4       8      16      32       64      128      256
>     +
>       1023  10      11
>       ---- x   + O(x  )
>        512
>                           Type:
> UnivariateFormalPowerSeries(Fraction(Integer))
>
> The only thing that s (the Stream) knows is a function object f (if you
> like you can use f=random). It cannot look inside f except to ask for
> more values. Remember AXIOM code is compiled.
>
> If from that you would somehow be able to extract a pattern, great, but
> I doubt.
>
>> Of course I am not that familiar with the Axiom internals.
> Well, you might come from another CAS. AXIOM *is* different. It is in
> general *not* dealing with expression trees. So what you have in mind
> (adding presentation) does not work for Stream of Series.
>
> That is not saying that it is impossible, but rather that the Stream and
> PowerSeries domains are not appropriate for what you have in mind. (And
> at the moment I don't think there is anything like what you want in Axiom.)
>
> Ralf
>
> PS: BTW, why didn't you continue on the mailing list?
I made a mistake :)  I have forwarded it.
  Everything seems reasonable; stream seems to
have invented an iterator but I guess that's implicit (tacit?) in the 0 
of stream(f, 0).  And series() seems to have
invented x and the number of terms it is going to display but these 
things are not real killers.
I will attempt an explicit translation (say something that could be done 
by awk) later.

Ray

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana


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    <br>
    <div class="moz-cite-prefix">On 10/21/2014 11:27 AM, Ralf Hemmecke
      wrote:<br>
    </div>
    <blockquote cite="mid:54467B66.3010202@hemmecke.org" type="cite">
      <pre wrap="">On 10/21/2014 04:39 PM, Raymond Rogers wrote:
</pre>
      <blockquote type="cite">
        <pre wrap="">Let's think about this.
I think the question can be interpreted as a formating/display problem
for the underlying code.
</pre>
      </blockquote>
      <pre wrap="">
If you believe so... Let me state it a bit differently what
datastructure comes out of the series command. Or rather what is stored
inside the data structure.

Here an example:

(1) -&gt; Z==&gt;Integer; Q==&gt;Fraction Z;
                                                                   Type:
Void
(2) -&gt; f(q:Q):Q == truncate(abs(q*cos(q::Float))+1)*(q/2+1)

   Function declaration f : Fraction(Integer) -&gt; Fraction(Integer) has
      been added to workspace.
                                                                   Type:
Void
(3) -&gt; s: Stream Q := stream(f, 0)
   Compiling function f with type Fraction(Integer) -&gt; Fraction(Integer
      )

             3 7 15 31 63 127 255 511
   (3)  [0,1,-,-,--,--,--,---,---,---,...]
             2 4  8 16 32  64 128 256
                                              Type:
Stream(Fraction(Integer))
(4) -&gt; p:=series(s)$UnivariateFormalPowerSeries(Q)

   (4)
         3  2   7  3   15  4   31  5   63  6   127  7   255  8   511  9
     x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + --- x  + --- x
         2      4       8      16      32       64      128      256
   +
     1023  10      11
     ---- x   + O(x  )
      512
                         Type:
UnivariateFormalPowerSeries(Fraction(Integer))

The only thing that s (the Stream) knows is a function object f (if you
like you can use f=random). It cannot look inside f except to ask for
more values. Remember AXIOM code is compiled.

If from that you would somehow be able to extract a pattern, great, but
I doubt.

</pre>
      <blockquote type="cite">
        <pre wrap="">Of course I am not that familiar with the Axiom internals.
</pre>
      </blockquote>
      <pre wrap="">
Well, you might come from another CAS. AXIOM *is* different. It is in
general *not* dealing with expression trees. So what you have in mind
(adding presentation) does not work for Stream of Series.

That is not saying that it is impossible, but rather that the Stream and
PowerSeries domains are not appropriate for what you have in mind. (And
at the moment I don't think there is anything like what you want in Axiom.)

Ralf

PS: BTW, why didn't you continue on the mailing list?
</pre>
    </blockquote>
    <big>I made a mistake :)  I have forwarded it.<br>
       Everything seems reasonable; stream seems to <br>
      have invented an iterator but I guess that's implicit (tacit?) in
      the 0 of stream(f, 0).  And series() seems to have<br>
      invented x and the number of terms it is going to display but
      these things are not real killers.<br>
      I will attempt an explicit translation (say something that could
      be done by awk) later.<br>
      <br>
      Ray</big><br>
    <pre class="moz-signature" cols="72">-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana
</pre>
  </body>
</html>

--------------050007080003000103060209--

\start
From: "William Sit" <wyscc@sci.ccny.cuny.edu>
To: Raymond Rogers <raymond.rogers72@gmail.com>,
Date: Tue, 21 Oct 2014 17:05:55 -0400
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not
	as	streams but as sumations

If I follow the question Raymond Rogers raised (I did not 
receive all the email for some reason), and Ralf's reply 
below, I don't think Axiom is that different, and the 
problem that Stream cannot "look inside" the parameter f 
is not because Axiom is compiled. Rather it is the nature 
that f is passed as a function, which by definition, has a 
source domain and a target domain, and computes values and 
nothing more. If the aim is to "know" more about f inside 
Stream (or a calling function/domain), then one should 
change the property of the parameter in Stream and hence 
create a new domain, say call StreamMore, where the 
function f is wrapped into a domain, say 
FunctionDomain(f), and inside FunctionDomain(f) you have 
auxiliary functions such as isFunctionRandom?(f), and any 
other properties of f that you would like StreamMore to 
access or query.

This is not different from OOP where the properties of f 
are provided by methods, like f.isFunctionRandom?.

In general, when a parameter is passed to another function 
(or a domain like Stream), only its categorical properties 
(here the definition of what a set function is, with 
source and target, and the production of values, but not 
details of the computation itself) are known from inside 
the calling function (or calling domain). Such 
"restrictions" are universal in computer programming 
languages.

At least, that's my limited understanding.

William

On Tue, 21 Oct 2014 14:50:00 -0400
  Raymond Rogers <raymond.rogers72@gmail.com> wrote:
> 
> On 10/21/2014 04:39 PM, Raymond Rogers wrote:
>> Let's think about this.
>> I think the question can be interpreted as a 
>>formating/display problem
>> for the underlying code.
> 
> If you believe so... Let me state it a bit differently 
>what
> datastructure comes out of the series command. Or rather 
>what is stored
> inside the data structure.
> 
> Here an example:
> 
> (1) -> Z==>Integer; Q==>Fraction Z;
>                                                           
>         Type:
> Void
> (2) -> f(q:Q):Q == 
>truncate(abs(q*cos(q::Float))+1)*(q/2+1)
> 
>    Function declaration f : Fraction(Integer) -> 
>Fraction(Integer) has
>       been added to workspace.
>                                                           
>         Type:
> Void
> (3) -> s: Stream Q := stream(f, 0)
>    Compiling function f with type Fraction(Integer) -> 
>Fraction(Integer
>       )
> 
>              3 7 15 31 63 127 255 511
>    (3)  [0,1,-,-,--,--,--,---,---,---,...]
>              2 4  8 16 32  64 128 256
>                                               Type:
> Stream(Fraction(Integer))
> (4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)
> 
>    (4)
>          3  2   7  3   15  4   31  5   63  6   127  7 
>  255  8   511  9
>      x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + 
>--- x  + --- x
>          2      4       8      16      32       64 
>     128      256
>    +
>      1023  10      11
>      ---- x   + O(x  )
>       512
>                          Type:
> UnivariateFormalPowerSeries(Fraction(Integer))
> 
> The only thing that s (the Stream) knows is a function 
>object f (if you
> like you can use f=random). It cannot look inside f 
>except to ask for
> more values. Remember AXIOM code is compiled.
> 
> If from that you would somehow be able to extract a 
>pattern, great, but
> I doubt.
> 
>> Of course I am not that familiar with the Axiom 
>>internals.
> 
> Well, you might come from another CAS. AXIOM *is* 
>different. It is in
> general *not* dealing with expression trees. So what you 
>have in mind
> (adding presentation) does not work for Stream of 
>Series.
> 
> That is not saying that it is impossible, but rather 
>that the Stream and
> PowerSeries domains are not appropriate for what you 
>have in mind. (And
> at the moment I don't think there is anything like what 
>you want in Axiom.)
> 
> Ralf

William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/

\start
Date: Tue, 21 Oct 2014 18:46:25 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] Bernoulli puzzle

Tim,
     Attached is what I meant by strict matrix equivalence.  Thus a 
property expressed in terms
of the coefficient array for one normalized Orthogonal Polynomial has a 
corresponding property for all the
normalized Orthogonal Polynomials.
     This is just a side-effect of the technique.  The real part for 
generating functions is that things
like Appell sequence Orthogonal Polynomials coefficient arrays can be 
obtained by replacing t in:
g(x,t)=sum(a_0(x)t^0,a_1(x)t^1 ....a_n(x)t^n  ..)
with the creation matrix   t'.
g(x,t')
directly.
The other sequences Associated and Sheffer have very similar processes.

Ray

On 10/20/2014 06:20 PM, daly@axiom-developer.org wrote:
> Ray,
>
>> A little off topic; but I have developed an alternate way of dealing
>> with polynomial sequences like Bernoulli polynomials that are
>> generated by generating functions. It involves casting the sequences
>> in matrices and apply Pascal Matrices and Umbral calculus.  It makes
>> some known relations obvious and casts a different viewpoint on
>> others.  It might allow some kind of Polynomial sequence algebra or
>> some such.  It does have the advantage of automatically converting
>> some (actually most) sequences to others by symbolic/parametrized
>> methods.  If anybody is interested let me know and I will write up the
>> application to Bernoulli polynomials as a special case.
> That would be an interesting generalization. Axiom implements several
> number theory algorithms with generating functions. If they were all
> just "cover calls" to a common method it would be useful.
>
> If you were to write something like that I'm begging you to write
> a fair amount of natural language explanation. I lost a whole weekend
> trying to reverse-engineer the bernoulli code so I can document it.
> Without Waldek's help I'd still be struggling. Please consider that,
> although YOU understand the code you write, nobody else does. Few
> people, myself included, have heard of Unbral calculus.
>
> Tim
>
>

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana


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Content-Disposition: attachment;
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--------------040603050706050503080107--

\start
Date: Wed, 22 Oct 2014 00:09:15 -0500
From: daly@axiom-developer.org
To: Raymond Rogers <raymond.rogers72@gmail.com>
Subject: Re: [Axiom-developer] documentation standards?

>     Yes I learned some time ago that maintenance can be devastating.
>I am a little at odds with your theory but not too much; and I will
>try to conform to your styles.  Along that vein: can you point me to
>the current tools you recommend and examples of how you are currently
>doing documentation?  They seem to have drifted a little over the
>years.




All of this is optional guidance. The goal is to improve Axiom so that
it can be maintained, modified, and extended by the next generation.
I hope it can eventually be used as primary teaching material for the
next generation of computational mathematicians.

The documentation is "self-organizing" as more of it gets added so
it is likely to continue to "drift". There isn't any "style" to
conform to as you will be one of the first people to generate docs.

Tools? Emacs. latex. Quicktime for videos. latexdraw for diagrams.
graphviz for graphs. I try to minimize using new latex packages so
users do not have to install anything.




There are three kinds of documentation currently being used in Axiom.

  1) literature references (all books have a biblio section now)

  2) domain specific ideas, equations, special terms, etc (Volume 10.1)

  3) "WHY did you write code that way" (in code-containing vols such as
     5: interpreter, 9: compiler, 10.2: categories, 10.3: domains,
     10.4: packages, 10.5: numerics, etc.) directly associated with
     the code.





The first is literature references. I have a collection of papers
for some of the references in the bibliography book (the way to tell
is to look for the keyword entry 

   paper = "NameYY.pdf"

in the bibtex entry). When I find a copy of a paper I try to include
the abstract from the paper. Ralf found a style that allows the
abstract to be included in the bibliography section of the books. That
makes it easier to know if a reference might contain something useful.

The down side of literature references is that obtaining some of
them can be expensive (e.g. Luke's 2 volumes were $150) and some
of them simply are not available (e.g Brillhart's PhD thesis).

URLs to online copies would be really appreciated. I know it is asking
too much but you could really make my day by including the abstracts
so I don't have to hunt down the papers :-)





The second, more specific documentation, is a page or so of theory so
that a beginner has some clue about the maths that your code depends
on (e.g. Umbral calculus). It doesn't have to have a lot of background
but it ought to be slanted toward an overview you'd give to a mathematician
who is new to your subject. The likely audience is a mathematician trying
to write new code or using your domain. Clarity and brevity matter.

Write the ideas (e.g. a dhmatrix is used to perform rotations,
translations, and scaling) rather than the heavy theory (e.g.
theorems, lemmas, proofs, etc). Include references so readers have a
gateway to important literature if they want more.

This second form of documentation will be a separate chapter or
section or be merged into some existing chapter or section of Volume
10: Axiom Algebra Theory. This book is intended to be an overview of
the subjects that Axiom covers so that people can understand where to
look for things and what might be available in Axiom.

An excellent example would be Weisstein's pages (DO NOT COPY THEM), e.g
http://mathworld.wolfram.com/BernoulliPolynomial.html
Realistically speaking, ANYTHING that gives an Axiom user a clue about
your domain would be most welcome.

Save the user from doing a literature crawl to figure out why you use
a matrix argument to construct Quaternions when none of the usual
presentations even mentions the idea.




The third form of documentation is "Why did you write this code
the way you wrote it?". This is NOT intended to be a line by line
explanation but rather to explain things which are painfully 
obvious to your PhD committee but nobody else. 

A recent example is the Bernoulli code. If the author had said

  "This implements the formula: \sum_{p=1}^n ....
   where the outer loop counter is p. Since odd Bernoulli numbers 
   are always 0 I collapsed the inner loop to only compute even
   numbered terms. So the inner loop equations for "t" and "b"
   are derived from the above formula into a more efficient form by
        .... (show how the t and b formulas used came from
              the formula above so the clever code makes sense)...

It is usually the case that, being an expert in a domain, you know
the most efficient, most clever, most compact ways of writing code.
But nobody else has a clue. Give them a clue WHY you wrote the
clever parts the way you did. 

Give the maintainers a clue about what you might depend on. When the
algebra gets reorganized and your code breaks the maintainer has to
"re-derive" WHY you were cleverly groveling through the Lisp
representation of another domain.

The "maintainer" could be YOU a few years from now and you probably
won't remember why
    CDR(devaluate(u.1)$Lisp)$Lisp::List(Any)
was such a clever hack.



I know all of that seems like a lot but actually it mostly amounts to
a ratio of a few lines of words for each line of code. A one page Spad
domain would likely generate two literature references, a page of
"intro-theory" and a couple paragraphs of "WHY" for the clever parts.



For the future....

Think of it as generating slides for a talk about your domain.

I'm starting to include videos and pondering per-domain presentations.
Think of them as "TED Talks" for Axiom users and developers.
http://axiom-developer.org/axiom-website/videos.html

The plan is to include hyperlinks from the books to the videos.

Tim

\start
Date: Wed, 22 Oct 2014 00:53:26 -0500
From: daly@axiom-developer.org
To: Raymond Rogers <raymond.rogers72@gmail.com>
Subject: Re: [Axiom-developer] documentation standards?

re: For the future.... (thankfully, a 30 year horizon)

Just so you know, I'm looking at two new, long term efforts, both
of which are invisible if you don't read the source code.



The first effort involves proving Axiom algorithms correct. See Volume 13
which is the long-term place to collect the ideas and practices. This is 
still in the early stages and ideas are still forming around my literature 
readings. Axiom's function signatures really help here.

The primitive change to the source code involves adding things like
loop invariants. For instance, when processing a polynomial you might
see this code:

  while p ^= 0
    ....
    p := reductum p
    ...

Since reductum removes the first term of the polynomial p we know that
the loop will eventually terminate. If we had function properties, we
can automate combining the function property of reductum with the loop
invariant to yield the termination proof, if we had loop invariants.

(It's a form of the old saying 
   "If we had ham we could have ham and eggs, if we had eggs")

I am looking at 3 layers of proof technology. One to move from Spad to
Lisp (Coq?), one to move from Lisp to C (GCL generates C code), and
one to move from C to machine (based on my prior research work on
Intel instruction semantics using Conditional Concurrent Assignments).

I'm also looking at Leslie Lamport's TLA+ language for code decoration. 
It seems to fit our needs reasonably well. 





The second effort is to consider "cost" metrics. Starting from the
ground (Lisp) functions such as +, *, etc. compute a "cost" of a
simple Spad function. Compute the O() cost and decorate the function
with this information.  Find the callers of the function and compute
the O() cost. Recurse.

Ideally this would be computed dynamically so that if a lower level
algorithm is changed it changes the cost of the callers. 

This effort may start to appear in domains such as Integer in the
near future, time permitting.


Tim
(planning to join the NASA Mars team so I can take advantage of
the longer years on Mars.)

\start
Date: Wed, 22 Oct 2014 10:21:43 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: Ralf Hemmecke <ralf@hemmecke.org>, axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not as
	streams but as sumations

Ralf,  Attached is an example of the analysis I was thinking about.
In theory it adds nothing to the information content; but in practice:
The human race has been developing, arguing, and changing the 
mathematical notation for a long time.  The present incarnation is not 
perfect but I think as a vechile of communication it is preferable to 
any particular human language with its deliberate ambiguities.  I say 
deliberate because it must accommodate poetry and background 
associations.  For instance "closed set"  is supposed to bring to mind a 
certain type of set but does not contain the true definition.  
Especially the definition that is used in point-set topology.

Although I haven't found it yet; if Axiom is to be a mathematical 
language it must have a definition along the lines of BNF or my favorite 
the railroad tracks/yard of Pascal.  This diagram/definition can be read 
"automatically" and a parser generated for use by any old program that 
has is compatible with it.  I would presume there would be c programs 
that read the output of yacc and produce target sources and  
executables.  Personally I would aim for grep or awk as the final 
program they are runtime ASCII programs.  But I have a penchant for 
simplicity.

Ray

On 10/21/2014 11:27 AM, Ralf Hemmecke wrote:
> On 10/21/2014 04:39 PM, Raymond Rogers wrote:
>> Let's think about this.
>> I think the question can be interpreted as a formating/display problem
>> for the underlying code.
> If you believe so... Let me state it a bit differently what
> datastructure comes out of the series command. Or rather what is stored
> inside the data structure.
>
> Here an example:
>
> (1) -> Z==>Integer; Q==>Fraction Z;
>                                                                     Type:
> Void
> (2) -> f(q:Q):Q == truncate(abs(q*cos(q::Float))+1)*(q/2+1)
>
>     Function declaration f : Fraction(Integer) -> Fraction(Integer) has
>        been added to workspace.
>                                                                     Type:
> Void
> (3) -> s: Stream Q := stream(f, 0)
>     Compiling function f with type Fraction(Integer) -> Fraction(Integer
>        )
>
>               3 7 15 31 63 127 255 511
>     (3)  [0,1,-,-,--,--,--,---,---,---,...]
>               2 4  8 16 32  64 128 256
>                                                Type:
> Stream(Fraction(Integer))
> (4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)
>
>     (4)
>           3  2   7  3   15  4   31  5   63  6   127  7   255  8   511  9
>       x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + --- x  + --- x
>           2      4       8      16      32       64      128      256
>     +
>       1023  10      11
>       ---- x   + O(x  )
>        512
>                           Type:
> UnivariateFormalPowerSeries(Fraction(Integer))
>
> The only thing that s (the Stream) knows is a function object f (if you
> like you can use f=random). It cannot look inside f except to ask for
> more values. Remember AXIOM code is compiled.
>
> If from that you would somehow be able to extract a pattern, great, but
> I doubt.
>
>> Of course I am not that familiar with the Axiom internals.
> Well, you might come from another CAS. AXIOM *is* different. It is in
> general *not* dealing with expression trees. So what you have in mind
> (adding presentation) does not work for Stream of Series.
>
> That is not saying that it is impossible, but rather that the Stream and
> PowerSeries domains are not appropriate for what you have in mind. (And
> at the moment I don't think there is anything like what you want in Axiom.)
>
> Ralf
>
> PS: BTW, why didn't you continue on the mailing list?

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana


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--------------090308010807010307000001--

\start
Date: Wed, 22 Oct 2014 12:47:05 -0400
From: Tim Daly <daly@axiom-developer.org>
To: Raymond Rogers <raymond.rogers72@gmail.com>
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not
	as	streams but as sumations

Raymond,

If I understand your email correctly you seem to be proposing a way
to parse (and convert to latex) the OUTPUT of Axiom. There might be
a better way. 

Axiom's architecture is something like

  parse -> execute -> generate lisp data structure -> Charybdis print

Charybdis is a very old lisp to teletype output formatter from the 60s.

What it seems you want to do (and what Ralf might want to consider) is
to grab the lisp s-expression just before it gets handed off to print.
>From that you can do a lot of things such as compute the likely size
of the expression (for line splits) or grab individual terms of a 
polynomial or series.

For example, do

  -> )lisp (trace |recordAndPrint|)
  -> 3*x^2+5

   1> (|recordAndPrint| (1 |x| (2 0 . 3) (0 0 . 5))
        (|Polynomial| (|Integer|)))

        2
  (1) 3x  +5
                                   Type: Polynomial(Integer)

You can look up the internal format of POLY(INT), which is
a list of Terms and, knowing that, process each term. 

Tim

\start
Date: Wed, 22 Oct 2014 12:14:15 -0500
From: daly@axiom-developer.org
To: Raymond Rogers <raymond.rogers72@gmail.com>
Subject: [Axiom-developer] CHARYBDIS paper

The CHARYBDIS paper is:

@article{Mill68,
  author = ""Millen, Jonathan K.",
  title = "CHARYBDIS: A LISP program to dispaly mathematical expressions
           on typewriter-like devices",
  year = "1968",
  journal = "Interactive Systems for Experimental and Applied Mathematics",
  publisher = "M. Klerer and J. Reinfelds, eds., Academic Press, New York",
  pages = "79--90",
  abstract = "
    CHARYBDIS (for CHARacter-compoxed sYmBolic DISplay) is a Lisp program
    to display mathematical expressions on typewriter-like devices such as
    line printers, teletypes, and scopes which display lines of characters,
    as well as typewriters."
}

\start
Date: Wed, 22 Oct 2014 20:28:35 -0500
From: daly@axiom-developer.org
To: Kurt Pagani <kp@scios.ch>
Subject: Re: [Axiom-developer] documentation standards?

Kurt,

Your documentation is outstanding. 

Tim

\start
Date: Wed, 22 Oct 2014 20:46:40 -0500
From: daly@axiom-developer.org
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] documentation standards?

Kurt Pagani has posted an example of documentation that really
raises the bar for what would constitute "well documented" Spad code.
Not everyone will go this far but this shows what could be done.

Imagine what a pleasure it would be to work in a system that had
this level of documentation for all of the code.

I've put a copy at
http://daly.axiom-developer.org/DiffForm.pdf

Axiom is literate so the code segments will end up being wrapped
with \begin{chunk}...\end{chunk} which makes it possible to extract
the actual code and compile it with the rest of the system.

Beautiful work, Kurt.

Tim

\start
Date: Thu, 23 Oct 2014 08:50:36 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] [Axiom-mail] manipulating series not
 as	streams but as sumations

On 10/21/2014 11:05 PM, William Sit wrote:
> I don't think Axiom is that different, and the problem that Stream
> cannot "look inside" the parameter f is not because Axiom is
> compiled. Rather it is the nature that f is passed as a function,
> which by definition, has a source domain and a target domain, and
> computes values and nothing more.

The point I wanted to make is that, in fact, Axiom has exactly that view
while other systems don't have it, because there everything is an
expression tree that can always be extracted from the parameter.

=======================================
Mathematica 9.0 for Linux x86 (64-bit)
Copyright 1988-2013 Wolfram Research, Inc.

In[1]:= f = Function[x, Sin[x]*x]


Out[1]= Function[x, Sin[x] x]

In[2]:= f[2]


Out[2]= 2 Sin[2]

In[3]:= g[fun_] := {fun[0], fun[1], fun[2], FullForm[fun]}


In[4]:= g[f]


Out[4]= {0, Sin[1], 2 Sin[2], Function[x, Times[Sin[x], x]]}
======================================
    |\^/|     Maple 12 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc.
2008
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> f := x -> x*sin(x);
                              f := x -> x sin(x)

> g := proc (fun) print(fun); {fun(0), fun(1)} end proc;
             g := proc(fun) print(fun); {fun(0), fun(1)} end proc

> g(f);
                                 x -> x sin(x)

                                  {0, sin(1)}

> print(g);
                proc(fun) print(fun); {fun(0), fun(1)} end proc
======================================

As William said, Axiom would only allow to print f if f itself (or
rather its type) exports a function that lets you look inside or prints
the (original) expression of f. Like in OOP, if there is no such method,
there is no way to get the information.

Well, Tim might argue that everything is still in Lisp, but Lisp is not
the inherent feature of SPAD, but rather an implementation detail of
SPAD, i.e. talking about SPAD should not have any need to talk on a Lisp
level.

Ralf

\start
Date: Thu, 23 Oct 2014 03:49:10 -0500
From: daly@axiom-developer.org
To: Ralf Hemmecke <ralf@hemmecke.org>
Subject: [Axiom-developer] accessing the input function form in Axiom

Ralf,

Axiom does keep the information you're looking for.

(1) -> f(x) == 3*x^2+5
                                                Type: Void
(2) -> )history )show 1
    [1] f(x) == 3*x^2+5

If you want a Spad "cover" you could create the package

)abbrev package HISTORY History
....

which exported a function to access the command line history
(using lisp function calls unfortunately but it would maintain
the illusion that Spad "isn't really lisp" outside the package).

You could keep it "at the Spad level" by using the file system.

)history )save foo
  Edit foo.input to see the saved input lines
  The saved history file is foo.axh

The "foo.input" file looks like

------------------------------------------------------------------------
f(x) == 3*x^2+5
------------------------------------------------------------------------

which you could search to find "f(x) == 3*x^2+5".

For more detail about the function you could access foo.axh/index.kaf
which is a "keyed access file". You could extend GETDATABASE
which knows how to manipulate .kaf files. The format is (was)
explained in the source code. Anyway, the foo.axh/index.kaf file:

------------------------------------------------------------------------
335                 ("f(x) == 3*x^2+5" (% (|value| (|Void|) . "()"))
                     (|f|
                      (|value| (|FunctionCalled| |f|) SPADMAP
                       (|#1| + (* 3 (^ |#1| 2)) 5))
                      (|recursive|) (|alias| |x|) (|isInterpreterFunction| . T)
                      (|isInterpreterRule|)))
(("1" 0 20))
------------------------------------------------------------------------

where 335 is a byte offset to the dictionary entry list (("1" 0 20))
which means that user input (1) is 20 bytes into the file

so you seek 20 bytes from the start of the file for the object
which is a list object whose CAR is the string input you seek.

(Sorry for the lisp)

Tim

\start
Date: Thu, 23 Oct 2014 10:56:00 +0200
From: Ralf Hemmecke <ralf@hemmecke.org>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] accessing the input function form in Axiom

On 10/23/2014 10:49 AM, daly@axiom-developer.org wrote:
> Axiom does keep the information you're looking for.
> 
> (1) -> f(x) == 3*x^2+5
>                                                 Type: Void
> (2) -> )history )show 1
>     [1] f(x) == 3*x^2+5

Yes. But as William already said, it's not exported by the function
itself. One has to look elsewhere.

> For more detail about the function you could access foo.axh/index.kaf
> which is a "keyed access file". You could extend GETDATABASE
> which knows how to manipulate .kaf files. The format is (was)
> explained in the source code. Anyway, the foo.axh/index.kaf file:
> 
> ------------------------------------------------------------------------
> 335                 ("f(x) == 3*x^2+5" (% (|value| (|Void|) . "()"))
>                      (|f|
>                       (|value| (|FunctionCalled| |f|) SPADMAP
>                        (|#1| + (* 3 (^ |#1| 2)) 5))
>                       (|recursive|) (|alias| |x|) (|isInterpreterFunction| . T)
>                       (|isInterpreterRule|)))
> (("1" 0 20))
> ------------------------------------------------------------------------
> 
> where 335 is a byte offset to the dictionary entry list (("1" 0 20))
> which means that user input (1) is 20 bytes into the file
> 
> so you seek 20 bytes from the start of the file for the object
> which is a list object whose CAR is the string input you seek.

Yes, I belief that all this is possible, but it sounds like what one can
do in Python, namely look inside an object although it doesn't export
the respective method. I call that cheating and not proper programming.

In other words "cheating" is the default in Mathematica and Maple. I
like Axiom better, because it doesn't allow for easy cheating.

Ralf

\start
Date: Thu, 23 Oct 2014 10:12:31 -0400
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] CHARYBDIS paper

I can't find it online except for ACM who would charge for Babbage's papers.

Ray
On 10/22/2014 01:14 PM, daly@axiom-developer.org wrote:
> CHARYBDIS: A LISP program to dispaly mathematical expressions
>             on typewriter-like device

-- 
The primary use of conversation is to satisfy the impulse to talk
George Santanyana

\start
Date: Fri, 24 Oct 2014 17:07:34 -0500
From: daly@axiom-developer.org
To: Antonio Duran <duran@us.es>, Mario Perez <mperez@unizar.es>,
	Juan Varona <jvarona@unirioja.es>
Subject: [Axiom-developer] Trusting computer algebra systems

Gentlemen,

I'm Tim Daly, lead developer on Axiom, an open source computer algebra
system.

I just finished reading your paper "The Misfortunes of a Trio of
Mathematicians Using Computer Algebra Systems. Can We Trust in Them?"

You mentioned the "black box" aspect of Mathematica and Maple.
I wonder if you've tried using Axiom which is fully open source.
http://axiom-developer.org

I will try to decode the MMA notebooks you posted so I can run
your examples in Axiom.

Tim

\start
Date: Sun, 26 Oct 2014 03:13:34 +0000
From: Juan Luis Varona Malumbres <jvarona@unirioja.es>
To: "daly@axiom-developer.org" <daly@axiom-developer.org>
Subject: Re: [Axiom-developer] Trusting computer algebra systems

Dear Tim,

(Sorry for my English)

Thanks a lot for your message.

I heard about Axiom for the first time in 1994, when Andrew Tonks gave a ta=
lk about Axiom in my University.=20

But I use Mac computers and Axiom was not available for them.=20
Actually, I have seen today in http://axiom-developer.org/axiom-website/dow=
nload.html=20
that a mac version has been released in August 2014.
(although http://axiom-developer.org/axiom-website/faq.html does not mentio=
n a Mac version)

I have downloaded it and try to use it as a terminal binary.
But it has been impossible for me to do anything. I have used some of the b=
inary
files in MACOSX/bin and I have gotten only errors.

For instance (I copy and paste):
----
/Users/jvarona/Downloads/axiom/mnt/MACOSX/bin/axiom=20
AXIOM variable is not set
assuming AXIOM =3D /usr/local/axiom/mnt/linux
The directory for Axiom, /usr/local/axiom/mnt/linux, does not exist.
Goodbye.
----
or
----
/Users/jvarona/Downloads/axiom/mnt/MACOSX/bin/AXIOMsys=20
This function is obsolete -- use SET-STARTING-HOLE-DIVISOR instead
                        AXIOM Computer Algebra System=20
                        Version: Axiom (August 2014)
                Timestamp: Monday July 28, 2014 at 04:12:34=20
---------------------------------------------------------------------------=
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
   Visit http://axiom-developer.org for more information
---------------------------------------------------------------------------=
   >> System error:
   Cannot open the file /Users/daly/axiom/mnt/MACOSX/algebra/interp.daase.

   >> System error:
   The tag |top_level| is undefined.
----

In any case, to use Axiom in terminal mode is, of course, much more ugly an=
d much more complicate than to use a program with a GUI interface.

I have read in axiom web about the use of a browser as GUI (similar to sage=
, I suppose). Is it already available?

Yours,

Juan Luis

> Gentlemen,
> I'm Tim Daly, lead developer on Axiom, an open source computer algebra
> system.
> I just finished reading your paper "The Misfortunes of a Trio of
> Mathematicians Using Computer Algebra Systems. Can We Trust in Them?"
> You mentioned the "black box" aspect of Mathematica and Maple.
> I wonder if you've tried using Axiom which is fully open source.
> http://axiom-developer.org
> I will try to decode the MMA notebooks you posted so I can run
> your examples in Axiom.
> Tim

Juan Luis Varona
jvarona@unirioja.es

http://www.unirioja.es/cu/jvarona/

\start
From: u1204 <daly@axiom-developer.org>
To: Juan Luis Varona Malumbres <jvarona@unirioja.es>
Date: Sun, 26 Oct 2014 20:14:13 -0400
Subject: Re: [Axiom-developer] Trusting computer algebra systems
Cc: duran@us.es, axiom-developer@nongnu.org, mperez@unizar.es,
	jvarona@unirioja.es

I'm on the road at the moment.
I have MMA on a machine in my office. I will try your workbook
and then see if I can recreate it in Axiom.

You must have installed Axiom locally.
If you have the top level of Axiom in the directory 
   /home/me/axiom
type
   cd /home/me/axiom
   export AXIOM=`pwd`/mnt/MACOSX
   export PATH=$AXIOM/bin:$PATH
   axiom

Tim

\start
Date: Mon, 27 Oct 2014 00:54:59 +0000
From: Juan Luis Varona Malumbres <jvarona@unirioja.es>
To: u1204 <daly@axiom-developer.org>
Subject: Re: [Axiom-developer] Trusting computer algebra systems

Dear Tim,

Thanks, in this way, I can see axiom working in my mac:
---
ArTeXano:home jvarona$ cd /Users/jvarona/axiom=20
ArTeXano:axiom jvarona$ export AXIOM=3D`pwd`/mnt/MACOSX
ArTeXano:axiom jvarona$ export PATH=3D$AXIOM/bin:$PATH
ArTeXano:axiom jvarona$ axiom
This function is obsolete -- use SET-STARTING-HOLE-DIVISOR instead
                        AXIOM Computer Algebra System=20
                        Version: Axiom (August 2014)
                Timestamp: Monday July 28, 2014 at 04:12:34=20
---------------------------------------------------------------------------=
--
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
   Visit http://axiom-developer.org for more information
---------------------------------------------------------------------------=
--
=20
   Re-reading interp.daase
   Re-reading operation.daase
   Re-reading category.daase
   Re-reading browse.daase
(1) ->=20
(1) -> 5-3
(1) ->=20
   (1)  2
                                                       Type: PositiveIntege=
r
---

Another thing is to learn to use it...

By the moment, I can do only small computations.
The X11 window is ugly, but I can use it as a help.

Yours,

Juan Luis

> I'm on the road at the moment.
> I have MMA on a machine in my office. I will try your workbook
> and then see if I can recreate it in Axiom.
>=20
> You must have installed Axiom locally.
> If you have the top level of Axiom in the directory=20
>   /home/me/axiom
> type
>   cd /home/me/axiom
>   export AXIOM=3D`pwd`/mnt/MACOSX
>   export PATH=3D$AXIOM/bin:$PATH
>   axiom
> Tim

\start
Date: Tue, 28 Oct 2014 11:06:57 -0400
From: Camm Maguire <camm@maguirefamily.org>
To: gcl-devel@gnu.org,maxima-discuss@lists.sourceforge.net,
	axiom-developer@nongnu.org,kaufmann@cs.utexas.edu
Subject: [Axiom-developer] GCL 2.6.12 is released

Greetings!  The GCL team is happy to announce the release of version
2.6.12, the latest achievement in the 'stable' (as opposed to
'development') series.  Please see http://www.gnu.org/software/gcl for
downloading information.

This release was largely motivated to accelerate closure construction
and compiler hash table performance for ACL2(h).  Along the way,
support was added for ACL2 static-consing, making GCL at present one
of two lisps offering this optimization for very large and heavy jobs.
This release also features a TAGS target in make, a variety of
compiler speed improvements, an overhaul and standardization of the
error, conditions, and restart systems, protection of system and fork
from SIGPROF, and getc/putc from SIGINT, faster l2 list functions and
nani/address, proper handling of missing objdump and generic TMP
environment variables, processing of mismatched fast-link calls slowly
without error, ppc64le and aarch64 fixes, support for the
si::*link-list* fast-linking diagnostic, more robust segfault
trapping, and full pathname fixes to the unradomize/re-exec
mechanism.

Take care,
-- 
Camm Maguire			     		    camm@maguirefamily.org


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