i. introduction and historical review sándor czirbusz...

Computer Algebra SystemsI. Introduction and Historical Review

Sándor [email protected]

http://compalg.inf.elte.hu/~czirbusz/

Department of Computer AlgebraInformatics Institute, ELTE

D2.711A

2011-2012 spring

http://compalg.inf.elte.hu/~czirbusz/

Table of Contents1 Literature2 What is the Computer Algebra?3 Historical review

Systems, Algorithms, PracticeThe ”ancients”The age of ”preparing”1961-19661966-19711971-19811981-1991Since then...

4 A commercial and a free softwareThe Maple ProjectThe SAGE

5 Genaral and Special Computer Algebra Systems6 Advantages and disadvantages

Literature What is the Computer Algebra? Historical review A commercial and a free software Genaral and Special Computer Algebra Systems Advantages and disadvantages

Literature IBooks

Antal Járai-Attila KovácsKomputeralgebraInformatikai Algoritmusok (editor: Antal Iványi)ELTE Eötvös Kiadó, 2004

Attila KovácsKomputeralgebra a tudományokban és a gyakorlatbanAlk. Mat. Lapok 18, 1998

André HeckIntroduction to MapleSpringer-Verlag, 2003

Geddes-Czapor-LabahnAlgorithms for Computer AlgebraKluwer Academic, 1992

F. WinklerPolynomial Algorithms in Computer AlgebraSpringer,1996


Literature IIInternet

Some interesting web site:

Maple www.maplesoft.com

Sage www.sagemath.org

Mathematica www.wolfram.com LikeWiki http:

//en.wikipedia.org/wiki/Comparison_of_computer_algebra_systemsIt is a comparison

www.maplesoft.com

www.sagemath.org

www.wolfram.com

http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems








Mathematica www.wolfram.com Like

Wiki http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systemsIt is a comparison

www.maplesoft.com

www.sagemath.org

www.wolfram.com









Mathematica www.wolfram.com LikeWiki http:

//en.wikipedia.org/wiki/Comparison_of_computer_algebra_systemsIt is a comparison

www.maplesoft.com

www.sagemath.org

www.wolfram.com





What is the Computer Algebra I

Computer Algebra – or symbolical algebraic systems

Software packages that are capable doing symbolic computation (mostly withthoughtful numerical and graphical abilities). The word ”algebraic” theremeans that operation with symbolic objects are originated in algebra.

The fundamental object of these systems

representing mathematical objects symbolically

doing arithmetics with these objects

The main function of Computer Algebra as scientific discipline

. . . On foundation of this arithmetic searching effective algortithms, analysingand realisationing them to the scientific research and adaptation. [JK]


What is the Computer Algebra IIProperties

1 The programs run over algebraic structures.

2 The results are exact, that is there are not numerical errors.3 Results can be formulas, mathematical objects.



1 The programs run over algebraic structures.2 The results are exact, that is there are not numerical errors.

3 Results can be formulas, mathematical objects.



1 The programs run over algebraic structures.2 The results are exact, that is there are not numerical errors.3 Results can be formulas, mathematical objects.


What is the Computer Algebra IIIExpalanation I

1 It is often practical to algebraically simplify an expression,and then evaluate it.

2 The exact solutions of the simply system

x2 + y2 − 1 = 0

x− y = 0

are the following: {( 1√2, 1√

2), (− 1√

2,− 1√

2)}


What is the Computer Algebra IVExpalanation II

3 We get exact solution if we work with parametric problems:∫x

x2 − adx =

ln |x2 − a|2


Symbolical vs. numerical computations IThe Chebyshev Polynomials of first kind

The Chebyshev Polynomials of first kind:

The recursive definition:

T0(x) = 1;T1(x) = x;

Tk(x) = 2 ∗ x ∗ Tk−1(x)− Tk−2(x), if k >= 2


Symbolical vs. numerical computations IIThe Chebyshev Polynomials of first kind

Concrete polynomials for some k

k Tk(x)0 11 x2 2 ∗ x2 − 13 4 ∗ x3 − 3 ∗ x4 8 ∗ x4 − 8 ∗ x2 + 1


Symbolical vs. numerical computations IIIThe Chebyshev Polynomials of first kind

A FORTRAN program to generating Chebyshev Polynomials

REAL T(5)READ(5,1) XT(1)=1.0T(2)=XWRITE(6,2) T(1),T(2)DO 10 N=3,5T(N)=2.0*X*T(N-1)-T(N-2)WRITE(6,2) T(N)10 CONTINUESTOP

1 FORMAT(F5.2)2 FORMAT(F9,4)END


Symbolical vs. numerical computations IVThe Chebyshev Polynomials of first kind

Generating Chebyshev Polynomials in ALTRANPROCEDURE MAINALGEBRAIC(X:4) ARRAY(0:4) TINTEGER NT(0)=1T(1)=XWRITE T(0),T(1)DO N=2,4T(N)=2*X*T(N-1)-T(N-2)WRITE T(N)DOENDEND


Symbolical vs. numerical computations VThe Chebyshev Polynomials of first kind

The same in MAPLET0 := 1;T1 := x;for n from 2 to 4 doTn := expand(2 ∗ x ∗ Tn−1 − Tn−2)end do;


Systems, Algorithms, Practice


These three partly independent factors formed the evolution ofComputer Algebra Systems.

Sytems Programming languages, programmingmethodology.

Algorithms ”Itself the Computer Algebra”Practice Systems adapting the algorithms.

And do not forget the hardware.






Algorithms ”Itself the Computer Algebra”

Practice Systems adapting the algorithms.And do not forget the hardware.






Algorithms ”Itself the Computer Algebra”Practice Systems adapting the algorithms.

And do not forget the hardware.


The ”ancients”

There were life before the computers (?)

The followings play crucial role in Computer Algebra nowadaystoo:

Euclidean algortithmSearching the greatest common divisor (not only fornumbers), inverting in finite fields.

Chinese reamainder theorem”Inverting” homomorphisms.Newton interpolationSolving equations.


The ”ancients”



Euclidean algortithmSearching the greatest common divisor (not only fornumbers), inverting in finite fields.Chinese reamainder theorem”Inverting” homomorphisms.

Newton interpolationSolving equations.


The ”ancients”



Euclidean algortithmSearching the greatest common divisor (not only fornumbers), inverting in finite fields.Chinese reamainder theorem”Inverting” homomorphisms.Newton interpolationSolving equations.


The age of ”preparing”

-1960

The first swallow1953. H. G. Kahrimanian (Temple University), in assembly onUNIVAC I: some symbolic differentiation.

The first high level languages: FORTRAN(1958),Algol(1960)Lisp : John McCarthy, 1958, M.I.T, published in 1960It is nowaday active too : AutoCad AutoLisp, Emacs,SchemeWikipédia

http://hu.wikipedia.org/wiki/Lisp



-1960


The first high level languages: FORTRAN(1958),Algol(1960)

Lisp : John McCarthy, 1958, M.I.T, published in 1960It is nowaday active too : AutoCad AutoLisp, Emacs,SchemeWikipédia




-1960


The first high level languages: FORTRAN(1958),Algol(1960)Lisp : John McCarthy, 1958, M.I.T, published in 1960It is nowaday active too : AutoCad AutoLisp, Emacs,SchemeWikipédia



1961-1966

The early years

The first systems

M.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.PM –polynomial manipulationMATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1961-1966

The early years

The first systemsM.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.

FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.PM –polynomial manipulationMATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1961-1966

The early years

The first systemsM.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)

Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.PM –polynomial manipulationMATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1961-1966

The early years

The first systemsM.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.

PM –polynomial manipulationMATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1961-1966

The early years

The first systemsM.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.PM –polynomial manipulation

MATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1961-1966

The early years

The first systemsM.I.T, 1961: SAINT (Symbolic Automatic INTegration)programmed in LISP, and built on heuristical algorithms.FORMAC: IBM 1962-1964programmed in FORTRAN, could manipulating elementaryfunctions (polynomials too)Bell Laboratories, ALPAKassembly routines, callable from FORTRAN.PM –polynomial manipulationMATHLAB - the first interactive programbased on LISP, handling polynomials and rationalexpressions.


1966-1971

The phase of ripening I

SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.

1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.MATLAB-68Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.1970 REDUCE-2programmed in the Algol-like RLISP.ALTRAN originated on ALPAK.


1966-1971


SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.

MATLAB-68Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.1970 REDUCE-2programmed in the Algol-like RLISP.ALTRAN originated on ALPAK.


1966-1971


SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.MATLAB-68

Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.1970 REDUCE-2programmed in the Algol-like RLISP.ALTRAN originated on ALPAK.


1966-1971


SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.MATLAB-68Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.

1970 REDUCE-2programmed in the Algol-like RLISP.ALTRAN originated on ALPAK.


1966-1971


SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.MATLAB-68Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.1970 REDUCE-2programmed in the Algol-like RLISP.

ALTRAN originated on ALPAK.


1966-1971


SIN (Symbolic INtegration) 1966-1967, MITalready used symbolical integration, based on LISP.1968, Stanford University, REDUCE, based on LISP,interactive, mainly for calculations of physics.MATLAB-68Symbolic Mathematical Laboratory 1967graphical output to CRT terminal, network capable.1970 REDUCE-2programmed in the Algol-like RLISP.ALTRAN originated on ALPAK.


1966-1971

The phase of ripening II

a rebuilt PM: SAC-1 (then SAC-2, then SAC/ALDES)

CAMAL (CAMbridge ALgebra system)programmed in BCPL, for celestial mechanics and theoryof relativity.SCRATCHPAD: ingtegrates the knowledge of many erliersystems1971 MACSYMA: calculating limits, symbolic integrationthis was the ”big promise”


1966-1971


a rebuilt PM: SAC-1 (then SAC-2, then SAC/ALDES)CAMAL (CAMbridge ALgebra system)programmed in BCPL, for celestial mechanics and theoryof relativity.

SCRATCHPAD: ingtegrates the knowledge of many erliersystems1971 MACSYMA: calculating limits, symbolic integrationthis was the ”big promise”


1966-1971


a rebuilt PM: SAC-1 (then SAC-2, then SAC/ALDES)CAMAL (CAMbridge ALgebra system)programmed in BCPL, for celestial mechanics and theoryof relativity.SCRATCHPAD: ingtegrates the knowledge of many erliersystems

1971 MACSYMA: calculating limits, symbolic integrationthis was the ”big promise”


1966-1971


a rebuilt PM: SAC-1 (then SAC-2, then SAC/ALDES)CAMAL (CAMbridge ALgebra system)programmed in BCPL, for celestial mechanics and theoryof relativity.SCRATCHPAD: ingtegrates the knowledge of many erliersystems1971 MACSYMA: calculating limits, symbolic integrationthis was the ”big promise”


1971-1981

The phase of practice

Go out from laboratories: to reaserchers and engineers.REDUCE: the mostly wide spread because its portability.

MACSYMA: powerful algorithmic developingAppeared some special CAS-s: SAC/ALDES, SHEEP,TRIGMAN, SCHOONSCHIPSome light from east: Kiev, Cybernetical University:ANALITIK


1971-1981


Go out from laboratories: to reaserchers and engineers.REDUCE: the mostly wide spread because its portability.MACSYMA: powerful algorithmic developing

Appeared some special CAS-s: SAC/ALDES, SHEEP,TRIGMAN, SCHOONSCHIPSome light from east: Kiev, Cybernetical University:ANALITIK


1971-1981


Go out from laboratories: to reaserchers and engineers.REDUCE: the mostly wide spread because its portability.MACSYMA: powerful algorithmic developingAppeared some special CAS-s: SAC/ALDES, SHEEP,TRIGMAN, SCHOONSCHIP

Some light from east: Kiev, Cybernetical University:ANALITIK


1971-1981


Go out from laboratories: to reaserchers and engineers.REDUCE: the mostly wide spread because its portability.MACSYMA: powerful algorithmic developingAppeared some special CAS-s: SAC/ALDES, SHEEP,TRIGMAN, SCHOONSCHIPSome light from east: Kiev, Cybernetical University:ANALITIK


1981-1991

PC-s, new programming languages

University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.

SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.DERIVEfor PC-s; only an interactive environmentCayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.FORM, MACULAY,LiE, PARI

And they dropped out: Naturally this in not complete



1981-1991


University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.

MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.DERIVEfor PC-s; only an interactive environmentCayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.FORM, MACULAY,LiE, PARI




1981-1991


University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.

DERIVEfor PC-s; only an interactive environmentCayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.FORM, MACULAY,LiE, PARI




1981-1991


University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.DERIVEfor PC-s; only an interactive environment

Cayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.FORM, MACULAY,LiE, PARI




1981-1991


University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.DERIVEfor PC-s; only an interactive environmentCayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.

FORM, MACULAY,LiE, PARI




1981-1991


University of Waterloo : MAPLE (Geddes, Gonet) firstly inB, then C.SMP –Symbolic Maipulation ProgramStephen Wolfram, portable, rule-based.MATHEMATICA, 1988: the first system with graphicalabilities + GUI !Stephen Wolfram, based on C, self language is from SMP.DERIVEfor PC-s; only an interactive environmentCayley , GAP (Group Algorithms And Programming)group theory, by the model of Mathematica and Maple.FORM, MACULAY,LiE, PARI




Since then...

Popularisation

The spreading of internet is a breakthrough: fast streamingof information, the webmathematic is available toeverybody.

Many free system besides the commercial systems. (viavLinux, GPL)The Computer Algebra marched into the education, on theone hand like a powerful tool(calculus, differentialequations, linear algebra), on the other hand,in its ownright.The adaptable mathematics increased: logic,combinatorics, geometry, universal algebra


Since then...

Popularisation

The spreading of internet is a breakthrough: fast streamingof information, the webmathematic is available toeverybody.Many free system besides the commercial systems. (viavLinux, GPL)

The Computer Algebra marched into the education, on theone hand like a powerful tool(calculus, differentialequations, linear algebra), on the other hand,in its ownright.The adaptable mathematics increased: logic,combinatorics, geometry, universal algebra


Since then...

Popularisation

The spreading of internet is a breakthrough: fast streamingof information, the webmathematic is available toeverybody.Many free system besides the commercial systems. (viavLinux, GPL)The Computer Algebra marched into the education, on theone hand like a powerful tool(calculus, differentialequations, linear algebra), on the other hand,in its ownright.

The adaptable mathematics increased: logic,combinatorics, geometry, universal algebra


Since then...

Popularisation

The spreading of internet is a breakthrough: fast streamingof information, the webmathematic is available toeverybody.Many free system besides the commercial systems. (viavLinux, GPL)The Computer Algebra marched into the education, on theone hand like a powerful tool(calculus, differentialequations, linear algebra), on the other hand,in its ownright.The adaptable mathematics increased: logic,combinatorics, geometry, universal algebra


The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.

From 1984: attended by Watcom.1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.1990: the crucial – Gonnet emerged from project.Milesotens: Maple V, 8., 10, and the last: 15.Authentical pages

http://www.cs.uwaterloo.ca/~kogeddes/papers/Banquet06/Banquet06.html


The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.From 1984: attended by Watcom.

1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.1990: the crucial – Gonnet emerged from project.Milesotens: Maple V, 8., 10, and the last: 15.Authentical pages



The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.From 1984: attended by Watcom.1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.

1990: the crucial – Gonnet emerged from project.Milesotens: Maple V, 8., 10, and the last: 15.Authentical pages



The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.From 1984: attended by Watcom.1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.1990: the crucial – Gonnet emerged from project.

Milesotens: Maple V, 8., 10, and the last: 15.Authentical pages



The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.From 1984: attended by Watcom.1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.1990: the crucial – Gonnet emerged from project.Milesotens: Maple V, 8., 10, and the last: 15.

Authentical pages



The Maple Project

Brief History

Produced at1980, in the Waterlo University by K.O.Geddes, Gaston Gonnet, Morven Gentleman. The aimwere: making en effective CAS on litle computers.From 1984: attended by Watcom.1988: Geddes and Gonnet founded the ”Waterloo MapleSoftware, Inc.”, Maple became to the leading CAS.1990: the crucial – Gonnet emerged from project.Milesotens: Maple V, 8., 10, and the last: 15.Authentical pages



The Maple Project

Jellemzõk

The kernel programmed in C

After version 10 the visualisation made in Java.Own programming language an Algol–like language; someof the matheamtical packages made in it.Two-way connection with Java, C, Fortran, Visual basic.It works in client–server mode.Is not cheaper


The Maple Project

Jellemzõk

The kernel programmed in CAfter version 10 the visualisation made in Java.

Own programming language an Algol–like language; someof the matheamtical packages made in it.Two-way connection with Java, C, Fortran, Visual basic.It works in client–server mode.Is not cheaper


The Maple Project

Jellemzõk

The kernel programmed in CAfter version 10 the visualisation made in Java.Own programming language an Algol–like language; someof the matheamtical packages made in it.

Two-way connection with Java, C, Fortran, Visual basic.It works in client–server mode.Is not cheaper


The Maple Project

Jellemzõk

The kernel programmed in CAfter version 10 the visualisation made in Java.Own programming language an Algol–like language; someof the matheamtical packages made in it.Two-way connection with Java, C, Fortran, Visual basic.

It works in client–server mode.Is not cheaper


The Maple Project

Jellemzõk

The kernel programmed in CAfter version 10 the visualisation made in Java.Own programming language an Algol–like language; someof the matheamtical packages made in it.Two-way connection with Java, C, Fortran, Visual basic.It works in client–server mode.

Is not cheaper


The Maple Project

Jellemzõk

The kernel programmed in CAfter version 10 the visualisation made in Java.Own programming language an Algol–like language; someof the matheamtical packages made in it.Two-way connection with Java, C, Fortran, Visual basic.It works in client–server mode.Is not cheaper


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.

Motivation: Who believe such results of software, whichcan not check?Author: William Stein, University of Washingtonversion 0.1: 24. february, 2005.version 1.0: february, 2006nowaday version 4.8.2


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.Motivation: Who believe such results of software, whichcan not check?

Author: William Stein, University of Washingtonversion 0.1: 24. february, 2005.version 1.0: february, 2006nowaday version 4.8.2


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.Motivation: Who believe such results of software, whichcan not check?Author: William Stein, University of Washington

version 0.1: 24. february, 2005.version 1.0: february, 2006nowaday version 4.8.2


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.Motivation: Who believe such results of software, whichcan not check?Author: William Stein, University of Washingtonversion 0.1: 24. february, 2005.

version 1.0: february, 2006nowaday version 4.8.2


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.Motivation: Who believe such results of software, whichcan not check?Author: William Stein, University of Washingtonversion 0.1: 24. february, 2005.version 1.0: february, 2006

nowaday version 4.8.2


The SAGE

History

Mission: a viable free alternative to Magma, Mathematica,and Maple.Motivation: Who believe such results of software, whichcan not check?Author: William Stein, University of Washingtonversion 0.1: 24. february, 2005.version 1.0: february, 2006nowaday version 4.8.2


The SAGE

Properties

The base: Python (object orientated)

Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.Built–in WikiVersion controllMay upgrade from the WebThere are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free software

Command–line (ipython) and GUI (within Web–browser)interface.Built–in WikiVersion controllMay upgrade from the WebThere are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.

Built–in WikiVersion controllMay upgrade from the WebThere are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.Built–in Wiki

Version controllMay upgrade from the WebThere are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.Built–in WikiVersion controll

May upgrade from the WebThere are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.Built–in WikiVersion controllMay upgrade from the Web

There are many free servers.


The SAGE

Properties

The base: Python (object orientated)Integrates aprroximatelly 150 free softwareCommand–line (ipython) and GUI (within Web–browser)interface.Built–in WikiVersion controllMay upgrade from the WebThere are many free servers.


Genaral and Special Computer Algebra Systems

General systemsVery big mathematical apparat with lazyness

Special systemsOptimized for special matheamtical areas

SCHOONSCHIP, CAMAL, SHEEP, STENSORSpeciális matematikai területekhttp://www.computeralgebra.nl/systems_and_packages/per_purpose/special/systems.html





SCHOONSCHIP, CAMAL, SHEEP, STENSOR

Speciális matematikai területekhttp://www.computeralgebra.nl/systems_and_packages/per_purpose/special/systems.html





SCHOONSCHIP, CAMAL, SHEEP, STENSORSpeciális matematikai területek

http://www.computeralgebra.nl/systems_and_packages/per_purpose/special/systems.html





SCHOONSCHIP, CAMAL, SHEEP, STENSORSpeciális matematikai területekhttp://www.computeralgebra.nl/systems_and_packages/per_purpose/special/systems.html


Advantages and disadvantages

Advantages

Quick computation of large quantity mathematical computations.

“Infallability”

It “knows” to integrate.

Support of teaching mathematics in several levels.

Accelerating the research, new research areas.

Disadvantages

Exponentially growing the size of numbers and expressions because of exactarithmetic.

Psychological

We can handle the numerical problems better with "orthodox" computer methods.

Complexity (complexity<-> efficiency)

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