Ministerul Educatiei al Republicii MoldovaUniversitatea de Stat din Moldova

Departamentul Informatică

Lucrare de laborator nr.1la disciplina Sisteme Algebrice

Tema: Computer Algebra Systems

Efectuat : Purice Cristina, gr. Ia31

Verificat: Valeriu Ungureanu

Chişinău 2015

Contents1.About Computer Algebra System.............................................................................................................3

1.1 General information.........................................................................................................................3

1.2 History...............................................................................................................................................4

2.Symbolic Manipulations...........................................................................................................................5

3.Types of expressions.................................................................................................................................5

4.Additional capabilities..............................................................................................................................6

5. Mathematics used in computer algebra systems....................................................................................7

6.References................................................................................................................................................7

1.About Computer Algebra System

1.1 General information

A computer algebra system (CAS) is a software program that allows computation

over mathematical expressions in a way which is similar to the traditional manual computations

of mathematicians and scientists. The development of the computer algebra systems in the

second half of the 20th century is part of the discipline of "computer algebra" or "symbolic

computation", which has spurred work in algorithms over mathematical objects such

as polynomials.

Computer algebra systems may be divided in two classes: the specialized ones and the general

purpose ones. The specialized ones are devoted to a specific part of mathematics, such as number

theory, group theory, or teaching of elementary mathematics.

General purpose computer algebra systems aim to be useful to a user working in any scientific

field that requires manipulation of mathematical expressions. To be useful, a general purpose

computer algebra system must include various features such as

a user interface allowing to enter and display mathematical formulas

a programming language and an interpreter (the result of a computation has commonly an

unpredictible form and an unpredictible size; therefore user intervention is frequently

needed)

a simplifier, which is a rewrite system for simplifying mathematics formulas

a memory manager, including a garbage collector, needed by the huge size of the

intermediate data, which may appear during a computation

an arbitrary-precision arithmetic, needed by the huge size of the integers that may occur

a large library of mathematical algorithms

The library must cover not only the needs of the users, but also the needs of the simplifier. For

example, the computation of polynomial greatest common divisors is systemically used for the

simplification of expressions involving fractions.

This large amount of required computer capabilities explains the small number of general

purpose computer algebra systems. The main ones

are Axiom, Macsyma, Magma,Maple, Mathematica and Sage.

1.2 History

Computer algebra systems began to appear in the 1960s, and evolved out of two quite different

sources—the requirements of theoretical physicists and research into artificial intelligence.

A prime example for the first development was the pioneering work conducted by the later

Nobel Prize laureate in physics Martinus Veltman, who designed a program for symbolic

mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in

1963.

Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964

at MITRE within an artificial intelligence research environment. Later MATHLAB was made

available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities.

Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB

("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory")

which is a system for numerical computation built 15 years later at the University of New

Mexico, accidentally named rather similarly.

The first popular computer algebra systems were muMATH, Reduce, Derive (based on

muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively

being maintained. As of today, the most popular commercial systems

are Mathematica[1] and Maple, which are commonly used by research mathematicians, scientists,

and engineers. Freely available alternatives include Sage (which can act as a front-end to several

other free and nonfree CAS).

In 1987, Hewlett-Packard introduced the first hand held calculator CAS with the HP-28 series,

and it was possible, for the first time in a calculator,[2] to arrange algebraic expressions,

differentiation, limited symbolic integration, Taylor series construction and a solver for algebraic

equations. In 1999, the independently developed CAS Erable for theHP 48 series became an

officially integrated part of the firmware of the emerging HP 49/50 series, and a year later into

the HP 40 series as well, whereas the HP Prime adopted the Xcas system in 2013.

The Texas Instruments company in 1995 released the TI-92 calculator with a CAS based on the

software Derive; the TI-Nspire series replaced Derive in 2007. The TI-89 series, first released in

1998, also contains a CAS.

CAS-equipped calculators are not permitted on the ACT, the PLAN, and in some

classrooms[3] though it may be permitted on all of College Board's calculator-permitted tests,

including the SAT, some SAT Subject Tests and the AP Calculus, Chemistry, Physics,

and Statistics exams.

2.Symbolic Manipulations

The symbolic manipulations supported typically include:

simplification to a smaller expression or some standard form, including automatic

simplification with assumptions and simplification with constraints

substitution of symbols or numeric values for certain expressions

change of form of expressions: expanding products and powers, partial and full factorization,

rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as

exponentials, transforming logic expressions, etc.

partial and total differentiation

some indefinite and definite integration (see symbolic integration), including

multidimensional integrals

symbolic constrained and unconstrained global optimization

solution of linear and some non-linear equations over various domains

solution of some differential and difference equations

taking some limits

integral transforms

series operations such as expansion, summation and products

matrix operations including products, inverses, etc.

statistical computation

theorem proving and verification which is very useful in the area of experimental

mathematics

optimized code generation

In the above, the word some indicates that the operation cannot always be performed.

3.Types of expressions

The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include real, integer, complex, interval, rational, and algebraic.

4.Additional capabilities

Many also include:

a programming language, allowing users to implement their own algorithms

arbitrary-precision numeric operations

exact integer arithmetic and number theory functionality

Editing of mathematical expressions in two-dimensional form

plotting graphs and parametric plots of functions in two and three dimensions, and animating

them

drawing charts and diagrams

APIs for linking it on an external program such as a database, or using in a programming

language to use the computer algebra system

string manipulation such as matching and searching

add-ons for use in applied mathematics such as physics, bioinformatics, computational

chemistry and packages for physical computation

Some include:

graphic production and editing such as computer generated imagery and signal

processing as image processing

sound synthesis

Some computer algebra systems focus on a specific area of application; these are typically

developed in academia and are free. They can be inefficient for numeric operations compared

to numeric systems.

5. Mathematics used in computer algebra systems

Symbolic integration via e.g. Risch algorithm

Hypergeometric summation via e.g. Gosper's algorithm

Limit computation via e.g. Gruntz's algorithm

Polynomial factorization via e.g., over finite fields, Berlekamp's algorithm or Cantor–

Zassenhaus algorithm.

Greatest common divisor via e.g. Euclidean algorithm

Gaussian elimination

Gröbner basis via e.g. Buchberger's algorithm; generalization of Euclidean algorithm and

Gaussian elimination

Padé approximant

Schwartz–Zippel lemma and testing polynomial identities

Chinese remainder theorem

Diophantine equations

Quantifier elimination over real numbers via e.g. Tarski's method/Cylindrical algebraic

decomposition

Landau's algorithm

Derivatives of elementary and special functions. (e.g. See Incomplete Gamma function.)

Cylindrical algebraic decomposition

6.References

1.  Interview with Gaston Gonnet, co-creator of Maple, SIAM History of Numerical Analysis and Computing, March 16, 2005

2. Jump up^ Nelson, Richard. "Hewlett-Packard Calculator Firsts". Hewlett-Packard. Archived from the original on 2010-07-03.

3. Jump up^ ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test