email: Franz.Locher@Fernuni-Hagen.de

Introduction. New media – PC’s and communication networks – offer a lot of possibilities to reorganize and optimize mathematical teaching and learning. Henceforth the typical situation in mathematical education is the following: Usually a lecture in mathematics is blackboard oriented because a lot of often long and on the first look very hard and cryptic formulas has to be written down, exclaimed and analyzed. The analysis is usually done – more or less completely – by mathematical inspection or by motivation. According to the restrictions in the lecture room – drawing is done by hand, computing by hand or by a pocket calculator – only very small and often rather simple accompanying examples can be demonstrated. A similar situation is found in mathematical textbooks and this has nothing to do with the quality of a textbook, it is problem inherent. Reading a textbook is a very stationary process. The whole information is fixed. At running time – that means at reading time – no epsilon may be added. But this is the advantage of multimedia systems: Examples could be varied by varying specific parameters, information could easily be linked by a browser etc. if adaquate interactivity is realized, if a link structure is incorporated. The realization of a comfortable link structure and of a series of Java applets to demonstrate special nontrivial examples is the purpose of our project Numerix.

The Numerix project. The Institute of Numerical Analysis (= Lehrgebiet Numerische Mathematik) at the FernUniversität Hagen has developped a series of courses and corresponding textbooks which cover the central areas of numerical mathematics as for students of mathematics as for students of computer science and engineering. The following themes and problems are treated:

● errors and error propagation, rounding errors, floating point computation,

● polynomial interpolation by Lagrange and Newton, divided differences, interpolation errors,

● numerical quadrature, interpolatory and repeated rules, Gaussian quadrature,

● Bernstein polynomials, Bernstein-Bézier technique, convexity, interpolation by splines, B-splines and their variation diminishing properties,

● representation of curves and surfaces by B-spline interpolation, tensoring,

● roots of unity, trigonometric interpolation, discrete Fourier transformation, FFT, fast multiplication of long integers via FFT,

● applications of FFT, digital filters, digital images, image processing, data compression,

● Fourier series and transforms, convolution, Gibbs’ phenomenon,

● approximation in normed and unitary spaces, orthogonality, Fourier-Chebyshev expansion,

● Gaussian elimination, pivoting, Cholesky factorization of positive definite matrices, least-squares problem,

● sparse matrices, graphs, bandwidth reduction, symbolic Cholesky factorization, minimum degree algorithm,

● fixed points, Banach’s fixed point theorem, iterative solution of nonlinear equations and of linear systems of equations, Newton’s and secant method,

● linear growth models, logistic equation, bifurcation in discrete dynamical systems, chaos,

● QR-factorization by Householder and by Jacobi-Givens, Gauss-Jordan method, pseudoinverse of a matrix, singular value decomposition, regularization of ill conditioned linear systems, linear statistical models,

● eigenvalues of matrices, Hermitean tridiagonal matrices, Hessenberg form, reduction methods of Householder and Jacobi-Givens, Krylov’s method, Lanczos’ method,

● iterative methods for eigenvalue problems, power method, LR- and QR-method, deflation,

● one-step methods for intial value problems, Euler’s method, explicit Runge-Kutta methods, convergence of one-step methods

● multistep methods, consistency, stability and convergence, discretization error, Adam’s methods, attainable order,

● boundary value problems of PDE’s and their discretization, SOR-methods, Young’s theory, optimal relaxation parameters, model problem.

In the last ten years the following courses and textbooks were produced in our institute – planned, written and printed on the basis of a L^AT_EX manuscript:

● Numerical analysis I/II for mathematicians,

● Numerical analysis for computer scientists,

● Chebyshev polynomials,

● Fourier analysis,

● Mathematical theory of signal processing.

Forthcoming is:

● Mathematical foundations of multimedia.

As it is good practice in mathematical departements, the courses and textbooks were edited in L^AT_EX. By this way texts of optimal graphical quality were produced. But the possibilities offered by the new media are not really used. At this stage the courses are not suited for online studies by the internet or by offline distribution via CD-ROM. To use the very possibilities of the new media, the printed courses have to be converted in a browser adaquate form. Thus first we processed the L^AT_EX files by an automatic L^AT_EX to HTML converter (L^AT_EX 2HTML). But for texts with mathematical formulas the conversion from L^AT_EX to PDF is the better way. PDF texts shown by Acrobat Reader on the screen or printed on paper cannot been distinguished from the L^AT_EX original. Conversely, the HTML version is optically not so good since formulas are handled as pictures, moreover producing hundreds or thousands of such pictures in the digital HTML version. At last we decided to convert the basic texts to PDF and HTML as well but preferring PDF.

The gain of a HTML or PDF version over the printed original of a textbook is the link structure based on the index which is automatically delivered by the converter. By hand the automatically produced link structure can be augmented e.g. by connecting the basic text via index with the glossary or vice versa. At this stage of the reorganization of the material offered to the students we imbedded some additional "low-level" multimedia elements:

● footnotes with pictures and biographic statements,

● audio buttons to transmit the correct pronounciation of greek letters, complicated formulas, foreign names etc.

Now we had arrived at a middle plateau, but the peaks were and are our goal. This means that we started to develop a series of Java applets which allow the user to "play", to "experiment", to "see" or to "hear". Java has the advantage to be platform independent as the source code is translated in byte code which can be interpreted by a usual browser at running time. An applet is devoted to a specific algorithm, to a special mathematical problem, to a basic definition etc. By buttons the user can change the parameters - numbers, vectors, functions, URL’s etc. - he can select out of predefined examples. The following screen shot shows a typical example. In reality you may experiment with some applets on our homepage (http:// www.fernuni-hagen.de/NUMERIK/ )

The development of a series of about 50 applets or more is rather time consuming. In the meanwhile their exist comfortable tools which are a good help during the programming process. But nevertheless a lot of programming has to be done by ourselves. Mention, that a typical applet e.g. Newton’s method from above, is realized by means of 128 kbyte source code.

Conclusion. Beginning with textbooks in printed L^AT_EX oriented form, the Institute of Numerical Analysis at the FernUniversoität Hagen is now developping an interactive and experimental version on the basis of Java applets. By the end of 2001 this programme will be mostly complete. Then it is an attractive offer to students of mathematics and the applied sciences as well which may be used online via communication networks or offline via a CD-ROM.

Acknowledgement. My thanks go the whole staff of my institute. They enabled the realization of this ambitious project.