free working environments for computer algebra - e
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1. for among others several external CASs like Maxima Pari Gp Sage Xcas Giac et al The complete list of packages that can be plugged into TeXmacs can be seen in http www texmacs org In this section we use TeXmacs as an interface to the computer algebra systems Maxima Pari Gp Sage and Xcas Giac and we solve both problems of the previous section that is we a factor the polynomial x 9 in each system and b count the number of primes lt 100 by writing a for loop in the language of each system This way we are able to draw our own conclusions regarding TeXmacs advantages and disadvantages We begin by initiating a session with Maxima To save space the system information that appears after linking TeXmacs with Maxima is not included and the same is true for subsequent CASs in this section The factorization is easily performed 11 factor x 2 9 S01 x 3 x 3 54 msec and so is the counting of the prime numbers lt 100 with the following for loop 12 counter 0 302 O 18 msec 13 for i 1 thru 100 do if primep i then counter counter 1 303 done 76 msec For that it is required to glue the C libraries at the C level not at the Python level or everything must be implemented at the Python level something Sympy is trying to do But even then Python was not designed to be an efficient language for computer algebra systems All assumed installed on the computer and visible from a te2. lt 100 TeXmacs is used as the interface and to start the whole process we initiate an interactive session with Sage with the help of the menu Insert gt Session gt Sage After TeXmacs has been linked to Sage the following system information and Sage prompt appear indicating we are ready to begin note that the red color of the prompt is the only difference between the interface original document and the pdf file produced by LaTeX Sage Version 5 5 Release Date 2012 12 22 Type notebook for the browser based notebook interface Type help for help Sage To factor x 9 in Sage itself we first define the polynomial ring Z x Sage R lt x gt ZZ x 31 msec and then enter the polynomial in the already defined variable x Sage poly x 2 9 60 msec At the site https github com sympy sympy wiki SymPy vs Sage a comparison can be found between Sage and SymPy It is assumed that Sage is installed on the computer and visible from a terminal window 36 OKOMII IOTEPH IE MHCTPYMEHTBI B OBPA30BAHHM N26 2012 r Free Working Environments for Computer Algebra The factors of x 9 are obtained using the function factor of Sage either in the classical way Sage factor poly x 3 x 3 689 msec or in Python syntax Sage poly factor x 3 x 3 11 msec Obviously since factor was imported the first time it was used it runs much fast
3. e MH OPMALIMOHHBIE CHCTEMBI 37 Alkiviadis G Akritas and then enter the polynomial in the already defined variable x Sage polyS singular x 2 9 3 msec The factors of x7 9 are obtained using the function factorize of Singular either through the library Sage singular factorize x 2 9 1 biel 2 xt 3 _ a S233 EZ Tegra galt 13 msec or in Python syntax Sage polyS factorize 1 _LeJ 2 eta Wea Sa 2 Lely 12 msec Here again the factors come with their multiplicities To factor the polynomial x 9 in SymPy we have to import the function factor and the variable x Sage from sympy import factor 3 394 sec Sage from sympy abc import x 63 msec Sage factor x 2 9 x 3 x 3 37 msec or in Python syntax Sage x 2 9 factor x 3 x 3 63 msec Finally to factor the polynomial x 9 in the computer algebra system Gap we have to first define the polynomial ring along with the variable and then to take into consideration its special interface with Sage as described in http www sagemath org doc reference sage interfaces gap html Sage x gap Indeterminate Integers x 1 074 sec Sage poly gap new x 2 9 76 msec Sage poly Factors x 3 x 3 257 msec or 38 OKOMII IOTEPH IE MHCTPYMEHTHI B OBPA30BAHHM N26 2012 r Free Working Environments for Computer Algebra Sage gap Factors poly x 3 x 3 16 msec Next we addre
4. e highlighted with the help of two examples and its advantages disadvantages are explicitly stated at the end In the last section TeXmacs is presented as an interface to other CASs Its main features are highlighted with the help of the same set of examples used in the previous section and its advantages disadvantages are also explicitly stated In both sections below the Output option of TeXmacs Show timings has been enabled to compare the execution times of the examples Note that this article was entirely written in TeXmacs from which a LaTeX version was exported for submission to the Journal l Xcas is the interface for the Giac library All examples in this paper were run on a Compac computer with Ubuntu 12 04 precise 32 bit and a Genuine Intel CPU 585 at 2 16GHz 3 TIpv nogroTosKe K meyaTH CTATbA Obuia CB pCTAHA 3aHOBO B IIPHHATOM B KypHasle CHCTEME B pPCTKH CTATE Alkiviadis G Akritas 2012 MH OPMALIMOHHBIE CHCTEMDI 35 Alkiviadis G Akritas 2 SAGE Sage was started by William Stein in 2004 but is now developed by a worldwide community of contributors it 1s a free open source CAS which integrates many specialized mathematics software into a common interface for which the user needs to know only the programming language Python However Sage is not pure Python since it makes use of many external packages written in various other programming languages C C Fortran Lisp Python etc the list of t
5. er the second time As already mentioned Sage is not reinventing the wheel but when possible uses functions from the other free CASs In this case the Sage manual informs us that the function factor calls the corresponding function of the computer algebra system Pari Gp Moreover since we the users also have direct access to all these CASs we can factor the polynomial x 9 using directly the function factor of Pari Gp without first declaring the polynomial ring Z x Sage pari factor x 2 9 x 3 1 x 3 1 134 msec or in Python syntax Sage f gp factor x 2 9 520 msec Sage f x 3 1 x 3 1 39 msec In the above results along with each factor we obtain its multiplicity 1 In the same spirit we can factor the polynomial x7 9 using the function factor of Maxima Sage maxima factor x 2 9 x 3 x 3 2 26 sec or in Python syntax Sage maxima factor x 2 9 x 3 x 3 68 msec To factor x7 9 in Singular we again first define the polynomial ring Sage R singular ring 0 x 749 msec Sage R characteristic 0 number of vars 1 block 1 gt Ordering 1p E names x 77 BLOCK 2 gt ordering C 174 msec l In http pari math u bordeaux fr we see that Pari is a C library C compatible whereas Gp is an interactive shell giving access to Pari functions easier to use Low level scripts are typically 4 times slower than directly written C cod
6. hese packages can be seen at http www sagemath org linkscomponents html Of interest to us 1s the fact that Sage comes bundled with four well established and time tested CASs Gap Maxima Pari Gp and Singular as well as the Python based SymPy This results in two fold benefits instead of writing functions anew Sage uses the existing functions of these CASs Gap for Group Theory Maxima for Calculus Pari Gp for Number Theory Singular for Commutative Algebra each one of these CASs can be used on its own in this way the user can verify the results obtained in one system by comparing them against the results obtained using another Sage can be downlowded from http www sagemath org and 1s available for Linux MacOSX Solaris and Windows the author found that Debian 1s the easiest package distribution system to deal with The user interface can be a Sage notebook in a web browser in which case Sage connects either locally to one s own Sage installation or to a Sage server on the network acommand line in a terminal and TeXmacs which is not mentioned neither in the Sage main web page nor in the Sage manuals tutorials Inside the Sage notebook or inside TeXmacs one can create embedded graphics typeset mathematical expressions etc In the sequel Sage is demonstrated by a factoring the polynomial x 9 in it as well as in all the CASs that come bundled with it and b counting the number of prime numbers
7. ing Sage installations on Windows it is worth mentioning that there is no Windows port instead Sage provides a VirtualBox that you can run from Windows an arrangement that requires a lot of memory This is no problem if you have a powerful PC however Sage will probably not be able to run on most Windows netbooks or older PCs since a precompiled version of Sage works on a linux netbook 3 TEXMACS TeXmacs is a free wysiwyw what you see is what you want editing platform with special features for scientists It was designed and written by Joris van der Hoeven who was inspired both by the TeX system written by D Knuth and by Emacs written by R Stallman hence the name TeXmacs Today it is supported by a worldwide team of contributors and 1s available for Knoppix Linux MacOSX and Windows for more details see http www texmacs org The software aims to provide a unified and user friendly framework for editing structured documents with different types of content text graphics mathematics interactive content etc The rendering engine uses high quality typesetting algorithms and produces professionally looking documents Included in the software is a a text editor with support for mathematical formulas a small technical picture editor and a tool for making presentations from a laptop New features can be added to the editor using the Scheme extension language Of interest to us is the fact that TeXmacs can be used as an interface
8. nNHDOpMaLNOHHbIC CrnNHCTEMDI B a S m Oe oe Alkiviadis G Akritas YJIK 519 67 FREE WORKING ENVIRONMENTS FOR COMPUTER ALGEBRA AHHOTANHMS B craTbe mpegcTaBJICeHbI ABa CBOOOHO pacnpocTpaHIeMbIX makKeTa C OTKPbITbIM HCXO I HbIM KOJOM JIA MCIHOIb30OBaHHA B KAYECTBE CPE JIA paGOTHI C CACTeEMaMH KOMII IOTEpPHO asIreOpbi OHH MOTYT HCINOJIb3O0BAT CA Kak B CPEMHEM TAK H B ICINEM OOpa30BaHUH a TAKKE YUeHbIMH 13 pa3JIHYH IX OOJIaCTE HayKu B craTbe 10Ka3aHO pellleHHe JIBYX 3414Y B KAXIO H CHCTeM Ha IIpUMepe KOTOPEIX aHasIM3MpYIOTCA MpeUMyIlecTBa H HeOCTATKH MpEACTABJICH HbIX CHCTeM Ku1r0u eBblie C10Ba CHCTeMBI KOMIIbIOTepHOH asireOpbi CAS Sage TeXmacs 1 INTRODUCTION This year marks the 15th anniversary of the Journal Computer Tools in Education Over these years a plethora of new computer tools has been developed for application in education and research Included in these tools is a number of free open source CASs such as Sage Sympy Xcas Giac et al and several Text Editing Platforms with special scientific features such as LyX TeXmacs et al In this article two educational tools are presented as working environments for CASs Sage a computer algebra system that can also serve as an editor and TeXmacs a scientific text editor that can also serve as an interface for several computer algebra systems including Sage In the next section Sage is presented using TeXmacs as an interface The main features of Sage ar
9. rminal window 40 KOMITbIOTEPHbIE VHCTPYMEHTBI B OBPA3OBAHMM N6 2012 r Free Working Environments for Computer Algebra 14 counter 04 25 17 msec Obviously when Maxima is called directly by TeXmacs and not through Sage the for loop is executed without any problems and we obtain the correct answer Next we initiate a session with Pari Again the factorization 1s easily done Pari factor x 2 9 x 3 1 I be j ae 80 msec and so is the for loop to count the prime numbers lt 100 Pari counter 0 32 0 12 msec Pari for i 1 100 if isprime i counter counter 1 54 msec Pari counter 3 25 11 msec Out of curiosity we initiate a session with Sage and through it we call Pari Gp to perform the same for loop and compare times Sage counter pari 0 73 msec Sage for 1 in range 100 if pari itl lt 2sprime Counter counter 1 356 msec Sage counter 25 74 msec Clearly Sage s interface slows things down quite a bit Finally we initiate a session with Giac Here again the factorization 1s as was 1n the previous CASs gt factor x 2 9 x 3 x 3 92 msec and the prime numbers lt 100 are counted in about the same way using the function isprime gt counter 0 0 58 msec gt for j 1 j lt 100 j it isprime j counter counter 1 0 83 msec gt counter 25 83 msec VIH OPMALIMOHHBIE CUCTEMBI 41 Alkiviadis G Akritas Xcas Giac al
10. so known as the Swiss knife for Mathematics is a powerful CAS combined with programming interactive 2 d and 3 d geometry and a spreadsheet that is under active development by Bernard Parisse and supported by a worldwide community of contributors Besides its ability to link to TeXmacs Xcas Giac can also link to the computer algebra systems CoCoA and Pari as well as to several other free open source libraries utilizing all the efficient and fast routines that are available Giac is linked with Pari through the command pari which makes available to Giac all the functions of the Pari library provided they are prefixed by pari So we can have gt pari 55 msec 3 i x 3 1 85 msec From the above examples we can easily draw the following conclusions regarding the advantages and disadvantages of TeXmacs Advantages A first advantage of TeXmacs is that it offers a wider choice of CASs to use than Sage namely it offers most of the packages that come bundled in Sage Sage itself plus additional ones listed in http www texmacs org that are not bundled in Sage The fact that these CASs have to be installed separately is not a problem at all for package distribution systems like Debian Ubuntu Secondly the interface of TeXmacs with each individual CAS 1s direct and not prone to problems that arise using Sage s interface as for example Sage s interface with Maxima that we saw in the previous section Finall
11. ss the second problem and write a small Sage program actually a Sage for loop to count the number of prime numbers lt 100 Sage of course does have the functions prime pi and is prime thatcould be used but we avoid them in order to demonstrate both the advantages and disadvantages of Sage Instead we use the function IsPrime of Gap inside the Sage for loop To know what to expect though we first compute the answer directly using the function primepi of Pari Gp Sage pari primepi 100 25 11 msec or in Python syntax Sage gp primepi 100 ZO 13 msec So there are 25 prime numbers lt 100 And below is the same answer using the Sage for loop with the computer algebra system Gap Sage counter gap 0 1 6 sec Sage for i in range 100 if gap itl IsPrime counter counter 1 671 msec Sage counter 29 463 msec Note that although Maxima does have the function primep to test if an integer is a prime number the system cannot be used in the for loop above because in http www sagemath org doc reference sage interfaces maxima html it is stated that Maxima does not seem to have a non interactive mode which is needed for Sage So the viable candidates for use inside the Sage for loop are Gap and Pari Gp where in the latter isprime tests an integer for primality From the above examples one can easily draw the following conclusions regarding the advantages and disadvantages of Sage Advan
12. t the user s manual opens up The choice of the CAS itself depends on the user s needs The argument in favor of Sage is that by learning one programming language Python the user can utilize several CASs whereas to program in the other CASs the user has to learn one programming language for each one of them However as we saw in the examples of the previous section the syntaxes of all these languages are related and it 1s not hard to switch from one to the other Moreover like with natural languages when circumstances demand it one easily becomes a polyglot l For more details see A tutorial on the Maxima plug in by Andrey Grozin which can be found in http www texmacs org strongly recommended Abstract Two free open source packages are presented for use as working environments for Computer Algebra Systems CASs They can be used in Secondary and Higher Education as well as by scientists in various fields In this paper a set of two problems 1s solved in both systems revealing their advantages and disadvantages Keywords Computer Algebra Systems CAS Sage TeXmacs Alkiviadis G Akritas University of Thessaly Haun aBTO pbl 2012 Dept of Communication and Computer Our auth ors 9012 Engineering Volos Greece ag akritas gmail com MH OPMALIMOHHBIE CHCTEMBI
13. tages The main advantage of Sage is the ability to use the various CASs packaged with it by learning just one general purpose language Python Moreover Sage can connect to optional CASs like Magma Maple and Mathematica all commercial and Giac free Also another advantage is the ease with which one can get the archive of files tarball and compile all the packages simultaneously as opposed to finding and compiling packages individually Disadvantages Obviously it is quite impressive to be able to call different packages in the same program by integrating these packages into a common Python interface Sage gets a lot of functionality without having to reinvent the wheel However this very interface can also be a bane as1n the case of Maxima not been able to run the Sage for loop above Moreover due to this interface at times it is impossible to take advantage of the very efficient and fast routines that exist in some libraries To wit Consider the C library LinBox which is one of the packages loaded with Sage and where there is an e cient implementation of large integer determinants that can be called from Sage However if Sage needs to call Maxima to compute say the antiderivative of a function and VIH OPMALIMOHHBIE CUCTEMBI 39 Alkiviadis G Akritas during this computation Maxima needs to compute a large determinant then Maxima will never be able to call that efficient implementation in LinBox Finally regard
14. y if the user already has experience with a certain CAS then s he can go on and use it right away with TeXmacs as an interface without having to worry about learning Python first Disadvantages The only disadvantage that one might attribute to TeXmacs 1s the fact that one cannot use it as an editor for the Greek language however Greek letters are available in math mode Hopefully this situation will be remedied in the future gt pari TaCvOn x 4 CONCLUSIONS From the previous discussion it is obvious that for the packages discussed in this paper we have the following dependencies Giac Maxima Pari Gap Maxima l eiknacs siren Plugin Sage p Singular Sympy Therefore if your favorite CAS has a TeXmacs plug in and you need a powerful editor then TeXmacs is the interface to use Itis worth noting that after TeXmacs session with the corresponding CAS has been initiated along with the system information there appear instructions on how to That is unlike Sage Giac glues the C libraries at the C level 42 KOMITbIOTEPHbIE VHCTPYMEHTBI B OBPA3OBAHMM N6 2012 r Free Working Environments for Computer Algebra open the CAS s manuals and tutorials in case you need them The only exception is Maxima where only the system information appears however simultaneously a big question mark appears without any warning at the right end of the menu bar and it is by pressing this question mark tha
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