Mathematical software

1. Mathematical software Lecture 1

2. Speaker Prof. Dr. Dana Petcu • Working in CS but PhD in Num.Analysis • Working with math. software • Has build an expert system for ODEs http://web.info.uvt.ro/~petcu E-mail: petcu@info.uvt.ro • English level

3. Lecture support • Lecture web page: http://web.info.uvt.ro/~petcu/SOFTMAP.htm • Text – booklet of the lecture: http://web.info.uvt.ro/~petcu/softw/CAS-lectures.zip • After: Handbook in Computer Algebra, 2003 Springer, edited by by Johannes Grabmeier, Erich Kaltofen, Volker Weispfenning • Lecture slides available at: http://web.info.uvt.ro/~petcu/SOFTMAP.htm http://web.info.uvt.ro/~petcu/softw/CAS-lectures.zip • Lecture slides and textbook in Romanian are not the same!

4. Labs • Acquire skills to deal with one system: Maple (www.maplesoft.com) • Examples for other tools available that are free • Maple 4 is installed on the server with license • Maple 10 is the current commercialized version

5. Examination • At the semester’s end • Final grade: mean value of two grades • First grade related to the lectures The students will receive at the last lecture a sheet with questions by pen and pencil; textbooks and lecture slides are allowed. • Second grade related to the labs The students will receive at the last lab a sheet with questions to be solved with Maple; labs exercises are allowed.

6. Lecture structure • Introduction. Short history. CAS (today) • General purpose systems • Applications of CAS (I) • Applications of CAS (II) • On the design of CAS • Topics 1: Mathematical knowledge representation. Coding theory • Topics 2: Exact arithmetic. Number theory. Linear systems. • Topics 3: Polynomials. Discrete structures. Algebraic complexity. Symbolic-numeric • Topics 4: Algebras. Groups. Representation. • Special purpose systems (I) • Special purpose systems (II) • Packages • Composing CAS • Examination

7. Why in UVT-CS curricula? • Special profile of the UVT-CS graduate: not only programming skills but also a medium level of mathematical skills • A CS graduate must promote in education and IT development the use of existing mathematical knowledge stored in software tools • Basic knowledge about a domain at the intersection between CS and Math

8. Why in particular at UVT? • Strong collaboration with RISC-Linz (Research Institute of Symbolic Computation) and Institute e-Austria Timisoara • National and European projects: SCIEnce: http://www.symbolic-computation.org aiming to couple European symb.comp. tools SysThMathEx, PhaseTrans, SEPROPI: http://www.ieat.ro • Conference series at UVT: SYNASC http://synasc07.info.uvt.ro ; International Symposium on Symbolic and Numerical Algorithms for Scientific Computing • Research expertise in Numerical Algs, expert system, kowledge management • Prepare the following generation

9. Several questions Why 2007 is the end of this lecture? • Bologna process • Mathematical software will be used at mathematical courses Why this lecture is different from the one taught in Romanian? • Use the newest textbook in the field Why this different style of lectures, textbook and examination? • Do not learn to reproduce (details), but to “understand the general topics, have a picture”

10. Lecture style • Why “A story”? • “Horizontal” – to link several knowledge skills that were accumulated • “Overview” – to have an overall image, not detailed knowledge • “Learn” – to deal with another world, that of research in IT and neighbor fields • “Different” – it is not an IT company style • “Reality” – find the (fuzzy) answer in chaos

11. Introduction Short history CAS Lecture 1

12. Software - general classification • Software • System software • Operating system • Compilers • C • Java etc • Utilities • File manager Etc • Application software • Word processing • Mathematical software • Numerical software • Symbolic computing software • Simulators etc

13. Mathematical software • Application software dedicated to solve problems described in mathematical form • Numerical software • Refers to the fields of applied mathematics, numerical analysis, numerical calculus, operating systems, hardware issues • Computer algebra • Refers to the fields of applied mathematics, calculus, algebra, symbolic computing, operating systems, hardware issues

14. Solving mathematical problems Mathematical problems • Solvable • Manually or with the computer • Only by computers • By new constructions • Solving procedure exists but cannot be yet implemented • With unknown solving procedure

15. Numeric or Symbolic • Numeric: with approximation • Symbolic: exact • Picture from Romanian slides

16. This lecture series • Dedicated to symbolic computations • In particular systems for algebra

17. Algebra • The terms algebra and algorithm originate from the book Kitab al muhtasar fi hisab al-agabr w’al muqabalah by the Persian science Abu Ja’far Mohammed ibn Musa al-Khorezmi in the 9th century • uses al-gabr and muqabalah to describe symbolic transformations and term reductions respectively, which are performed to solve algebraic equations. • Algorithmic manipulation of symbolic algebraic expressions remained to be the major task of algebra about until the end of the 19th century. • At the beginning of the 19th century, this method was amended and shadowed by developments in abstract algebra, where the main interest is focused on formal investigations of algebraic structures derived from axioms (this structural algebraic method has taken over considerable parts of mathematics nowadays).

18. Interest in automate symbolic computations • Due to accelerated development of computers and digital data processing • Possible to tackle with new applications • Computer algebra evolved as a discipline,linking algorithmic and abstract algebra to methods of computer science, and providing a new methodical tool in the border area between applied mathematics and computer science.

19. Computer Algebra (CA) • Computer Algebra is a subject of science devoted to methods for solving mathematically formulated problems by symbolic algorithms, and to implementation of those algorithms in software and hardware.

20. Characteristics of CA • It is based on the exact finite representation of finite or infinite mathematics objects and structures • Allows for symbolic and abstract manipulation by a computer • Algorithms of computer algebra typically surpass basic number arithmetic; • They extend to computations involving specifically represented algebraic objects like indeterminates, elementary functions, permutations • Instabilities caused by inaccurate input data (well known problem in classic numerical computing) can be detected.

21. Characteristics of CA • continues traditional algorithmic computing, incorporates elements of more recent structural mathematics. • CA is by no means a substitute for creativity and mathematical knowledge; it can assist in tapping the vast mathematical resources, • and is therefore an essential tool of experimental mathematics.

22. Impact on education • Hand-held computer algebra tools were developed by Casio and Texas Instruments – they offer the opportunity to rid the curriculum of technical ballast for theoretically lesser inclined users of mathematics (engineers, computer scientists, economists, medics, biologists, etc.) • Enables to teach moreapplication oriented material, and can considerably improve comprehension of mathematical methods, their potential and their limitations, by non-mathematicians • There is a danger that computation by hand and related skills are not acquired to the same extent as it used to be before the introduction of the computer to the classroom This could go far that future generations of users of mathematics could become incapable of solving even simple problems without the help of computer (e.g., how to compute decimal approximations of square roots).

23. Lectures A typical CA curriculum is usually divided into two parts. • 1st level: impart the necessary skills for proper utilization of CAS (CA systems). • 2nd level: address anyone who is required to have an in-depth knowledge of the structure of CAS, especially of their underlying algorithms, in order to be able to use them in critical applications.

24. Impact on innovation/research • CAS have significantly influenced scientific research in many fields, among them mathematics, computer science, physics, chemistry, engineering. • The use of CAS has become a standard tool for experiments in advanced research that help to find, support, or refute conjectures. • CAS become absolutely indispensable in the design and update of tables of mathematical, physical or other scientific objects.

25. Roles of CAS in science & Engin. • can support straight-forward activities like • carrying out a manipulation of mathematical expressions for testing conjectures, • deriving solution properties, • or recording the steps to avoid hand-calculation mistakes. • can be intricate, like • designing and building a computer algebra system for a specialized task in one’s research area (e.g. Nobel Prize in physics was awarded in 1999 for a theory verified by calculations using the Schoonship computer program which, using symbols, performed algebraic simplifications of complicated expressions. • can be in-between, • taking a scientific computational task at hand and studying how current algorithms and their implementations can be customized and improved to solve the problem.

26. CAS & mathematics • computer algebra supported experiments have been used to support or refute well-known conjectures and to gain additional insight in these problems • like conjectures in group rings. • applied in treating a large number of cases in proofs of general theorems • like the four-color problem • contributed significantly to establish and update large mathematical databases • such as integral tables, tables of special functions, and the atlas of finite groups. • contributed to a new upswing in algorithmic methods not only as tools, but as new objects worthy of mathematical study; • this concern both the design, verification, and complexity analysis of computer algebra algorithms, as well as non-algorithmic structural mathematics (a typical example is the theory of Groebner bases).

27. CAS & CS CA helped to the development of several tools and methods, among them: • list processing, • abstract data types, • parametric polymorphism, • constraint logic programming, • dynamical memory management, • user interfaces and interfaces between systems, • mathematical knowledge representation and management.

28. CAS & physics • Physics is one of the oldest application field of computer algebra • Many of the early CAS were designed of strongly influenced by physicists in order to suit their research needs Eg. Formac, Schoonship, Camal, Macsyma, Reduce

29. CAS & engineering • CA is recognized as a valuable tool supplementing and extending numerical software. • Well known examples are: • the inverse kinematic problem and • the path planning problem in robotics – these are typical instances of parametric problems, where the symbolic-algebraic approach may offer a closed from solution in comparison to purely numerical solutions of specific problem instances • Other areas where CA methods have a major impact include: • computer aided geometric design, • mechanics, • flow dynamics, • thermodynamics and combustion, • signal processing.

30. Future: optimization • For a great number of tasks new or faster algorithms have to be found, more proficient implementations, special purpose systems, or even hardware solutions have to be developed  special attention has to be paid to optimal utilization of existing hardware and computing environment. • In the near future it is expected the growth of general purpose systems, as well as their efficiency and ease of use. • Despite the progress in hardware sector, the general purpose systems will generally not be able to provide the required power for complex computing tasks  special software packages, or specialized CASs will have to be designed, and used. • In order to keep the necessary work at a minimum, it becomes imperative to think about means of standardization and the opening up of CASa to facilitate the integration of customized program parts, and to allow data transfer between systems. • The OpenMath initiative is devoted to this idea and attempts to create a standard for communication between computer algebra systems.

31. Future: integration • Furthermore, it is desirable to have at least basic algorithms in standardized form, maybe even implemented on the processor level; examples in this direction are packages for long integer arithmetic. • CAS will increasingly have access to numerical packages and integrate them into the system. However, currently one can see also the development of system independent libraries of data, data types and programs as well as stand-alone programs using symbolic code • It is also expected that CAS will employ the network computing facilities in an Internet environment; • there are already examples of CA servers in the Internet devoted to certain topics of CA where each user can depose specific problems to be solved there. • CA plugins will allow standard browser to be an interface for solving symbolic problems.

32. Future: development issues • To support the process of enlarging the user base of the mathematical knowledge by means of CA, mathematicians have to • take over a growing no. of more general consulting positions and • get involved in mathematical modeling of complicated problems. • Another requirement is the comparison of systems and algorithms under various points of view; • the variety of topics and themes of CA will not allow to have standard benchmarks as in numerics. • to develop benchmarks for different subthemes, test suites and parameterized problem settings are important tasks to be solved in the future. • Continued development and maintenance of systems take up an extent which make financing viable only through cost sharing by users through service or license fees. • Experimental and specialized packages and systems, on the other hand, will be essentially free of charge.