Mathematik als Schlüssel für Schlüsseltechnologien

Mathematica ist ein Softwaresystem zur Lösung von Problemstellungen in denen Berechnungen und Simulationen aller Art notwendig sind. Mathematica eröffnet dem Nutzer eine nahezu unendliche Vielfalt von mathematischen Funktionen zur Lösung von Aufgaben aus dem Ingenieurwesen, der Finanz- und Versicherungsmathematik, der Mathematik, der Statistik, der Physik, dem "Chemical Computing" und natürlich allen Naturwissenschaften. Dem Anwender gegenüber präsentiert sich Mathematica mit seiner revolutionären - da plattformübergreifend identischen - Benutzeroberfläche, dem Notebookinterface.
Das Notebookinterface ist sowohl die Oberfläche in der Sie Ihre Arbeit verrichten mit integrierter Textverarbeitung und Rechtschreibeprüfung, mit Symbolleisten, vielen mitgelieferten Funktionspaletten und beliebig frei definierbaren Paletten, als auch die Umgebung in der Sie programmieren können. Über die integrierte 4GL-Programmiersprache können Sie die Funktionalität beliebig erweitern und selbst externen Programmcode hinfügen.

Merkmale des Technical Computing

Seit der ersten Version 1988, ist Mathematica die wohl am meisten genutzte Software für Technical Computing.

• Numerische Berechnungen.
• Symbolische Berechnungen (Computer Algebra ist ein Teilbereich).
• 2D- und 3D-Visualisierung.
• Programmierung in unterschiedlichen, anwendungsbedingten Stilen.
• Modellbildung und Simulation.
• Einbindung von bestehendem Programmcode aus FORTRAN und C++.
• Technische Dokumentation in TeX, HTML, und dem eigenen plattformübergreifenden Format.
• Präsentation der Analysen und Ergebnisse.
  
  
  

Einfacher geht es nicht

Mit Mathematica erhalten Sie ein Werkzeug, mit welchem Sie sofort nach der Installation loslegen können, denn alle Funktionen folgen der gleichen Syntax und sind ausführlich im Hilfesystem (über 1600 Buchseiten zu Anwendung und Mathematik) erklärt. Das Hilfesystem ist zudem nicht statisch, sondern erlaubt das direkte Berechnen und Testen der Funktionen im Hilfesystem.
Mit Mathematica erledigen Sie einzelne Aufgaben - beispielsweise das Auswerten von Meßdaten oder die Lösung einer schwierigen Differentialgleichung - genauso, wie Sie komplette Lösungen von der ersten Idee bis zum fertigen Softwaresystem entwickeln.

Einzigartige Rechenpower und einfache Bedienung

Profitieren Sie von der einzigartigen Kombination einer intuitiven Bedienoberfläche mit den legendären Rechenfähigkeiten in Mathematica.

• Vordefinierte Paletten zur Eingabe und Befehlsauswahl
• Programmierbare Benutzeroberfläche
• Interaktives und erweiterbares Hilfesystem
• Editier- und ausführbare Beispiele

Sie klicken mit Hilfe der eingebauten Paletten mathematische Ausdrücke einfach zusammen oder rufen damit Funktionen und Befehle auf. Mit der voll programmierbaren Benutzeroberfläche ermöglichen Sie den Zugriff auf Funktionen oder eigene Pakete beispielsweise durch selbstdefinierte Paletten und Schaltflächen.

Kommunikativ

Mathematica ist kein proprietäres System, sondern ein nach allen Seiten offenes und standardisiertes System, das sich nahezu beliebig anpassen läßt. Es besteht aus mehreren plattformunabhängigen Komponenten, die für den Nutzer meistens unsichtbar im Hintergrund die Arbeit verrichten:

• Der mathematische Kernel (sozusagen das Wissensbuch der Mathematik).
• Das plattformunabhängigen Notebookinterface (die Benutzeroberfläche: identisch für Windows, Macintosh, Linux, Unix).
• MathLink, das Kommunikationsprotokoll zwischen Kernel und der Benutzeroberfläche.
• MathML zur Darstellung von Formeln im WWW.
• J/Link, die Schnittstelle zu Java.
• MathLM, die Lizenzmanagersoftware zur Kontrolle und Steuerung der Zugriffe.

Aufgrund der offenen Standards ist es möglich, den mathematischen Kernel an nahezu jede andere Software oder Benutzeroberfläche anzudocken. So sind z.B. Zusatzmodule zur direkten Anbindung an Microsoft Excel, Microsoft Access, Origin, LabView, Oracle... verfügbar. Selbst dem Internet gegenüber zeigt sich der Kernel via J/Link offen und kann als webMathematica die unternehmensweite Mathematiklösung oder einfach die Black-Box im Hintergrund von web-basierten Analysen und Graphen seine Arbeit verrichten. Und reicht die Leistung eines Computers einmal nicht aus, dann gestattet Ihnen der Einsatz des Zusatzmodules "Parallel Computing Toolkit" die Verteilung/Parallelisierung von Berechnungen.
Sie sind jetzt verwirrt durch die vielen Standards! Das ist die globale Erfolgsstrategie von Mathematica - nicht geschlossen sondern offen für die Welt, eben ...

Von der Mathematik zur Lösung

Mathematik gibt Antwort auf entscheidende Probleme aus diversen Bereichen des Ingenieuralltags, von der Physik bis zur Finanzwirtschaft. Antworten, die mit gesundem Menschenverstand alleine oft nicht zu finden sind. Wirkliche Probleme haben nicht nur vier oder fünf Parameter, sondern hunderte. Zu deren Lösung brauchen wir die Mathematik - und zwar mehr, als ein Ingenieur, ein Finanzwirt oder der Mathematiker im allgemeinen auswendig kennt. Dies hat zur Folge, daß in der Praxis häufig mit vereinfachten Modellen, unscharfen Definitionen oder gar mit Trial-and-Error gearbeitet wird. Selbstverständlich wollen wir zur Lösung eines praktischen Problems nicht erst ein Mathematikstudium absolvieren. Deshalb benötigen wir elektronische Helfer, die uns das Expertenwissen liefern. Mathematica ist ein solcher elektronischer Helfer. Eines der Entwicklungsziele war die Abschaffung aller Unannehmlichkeiten im Umgang mit der Mathematik, die sich z.B. aus umständlichem und oft fehlerträchtigem Ableiten, Integrieren und Umformen komplizierter Gleichungen ergeben.
Mathematica befähigt den Anwender, die Tiefen der Mathematik auszuschöpfen, zu visualisieren und zu lösen; ohne Papier, Bleistift, Taschenrechner oder komplexe Software.

Eine Alternative zur reinen Numerik

Als Alternative zu herkömmlichen, rein numerischen Methoden, wie Finiten Elementen, gewinnt die Computer-Algebra gestützte Simulation in zahlreichen Gebieten an Bedeutung. Mathematica beherrscht beide Methoden und erlaubt den direkten Vergleich der Ansätze in einem System! Aus diesem Grund hat sich Mathematica in den letzten zwei Jahren, aus dem universitären und wissenschaftlichen Bereich kommend, mehr und mehr in der Forschung und Entwicklung im industriellen Einsatz bewährt. Projekte, wie die Entwicklung einer Radbahn für die olympischen Spiele in Atlanta, der Einsatz in der MIR Raumstation oder im Risikomanagement der Finanzwelt, zeigen diesen Trend.
Die integrierte hochentwickelte Programmiersprache ermöglicht Rapid Prototyping, d.h. direktes Umsetzen von Ideen und Algorithmen in "computerverständliche" Form.

Ein breites Anwendungsspektrum

Der Anwendungsvielfalt von Mathematica sind damit kaum Grenzen gesetzt, weswegen es nicht einfach ist, Mathematica zu beschreiben. Für die einen ist es ein Mathematiksystem, für die anderen ein Computer-Algebra-System (CAS). Für den Programierer ein effizientes Tool, das alle modernen Programmierstile vereint, für den Geologen ein Visualisierungssystem und für den Ingenieur ein Berechnungsengine.
Diese Liste könnte unendlich fortgesetzt werden. Spezialisten rund um den Globus komprimieren aus diesem Grunde ihr Expertenwissen zu kommerziellen Anwendungspaketen - entworfen und programmiert in Mathematica - und bieten dieses Know-How als Zusatzmodule an. Exemplarisch greifen wir hier einige heraus:

• Control Systems Professional
• Experimental Data Analyst
• Fuzzy Logic
• Technical Trader
• Scientifc Astronomer, Optica
• Industrial Thermics
• Operations Research
• MathTensor, ..die Liste erweitert sich ständig.

1. Notebook Document System

Mathematica notebooks provide a complete technical document system with typeset math, sound, graphics, and animations.
Whether you are creating a report, an academic paper, courseware, or an electronic book or just want to keep a record of your work, Mathematica notebooks are the ideal medium for all of your technical projects. They are the main interface to all Mathematica computations and let you combine all of your calculations, code, results, and graphics into one interactive technical document.
Notebooks are platform independent and combine interactive typeset mathematical expressions, formatted text, hyperlinks, graphics, animations, sound, and fully customizable buttons and palettes. Mathematica's user interface includes such word-processing capabilities as spell checking (with a large technical vocabulary) and automatic hyphenation.
You can send notebooks by email or put them on a website or an FTP site without affecting their quality, use them to create high-quality printouts or sophisticated on-screen presentations, or translate them to other document formats such as HTML, TeX, and MathML, part of the new XML standard.

Mathematica’s integrated notebooks allow you to keep your analysis, raphics, report, and presentation together in one unified document.

2. Complex Analysis

High-speed numerics and high-level commands make complex analysis quick and easy.
Mathematica comes with a wide range of high-level statistics and data analysis functions as well as powerful import, export, and connectivity functionality, making even complex analysis of large data sets quick and easy. Mathematica's record-breaking speed for numerical linear algebra also makes processing large data sets faster than ever.
Mathematica includes import and export filters for over 70 popular file formats, including XML. Mathematica can also connect to databases with JDBC or .NET mechanisms through J/Link and .NET/Link. Mathematica's connection tools also allow you to easily build and access online data feeds, other data acquisition software such as LabView, and web services.
Once you have read your data files into Mathematica, you can apply sophisticated analysis or visualization techniques or use Mathematica's computational power to build complex models. Mathematica comes with fast tools for data manipulation, descriptive statistics of uni- and multivariate data, generalized linear and nonlinear fitting, multidimensional interpolation, convolution, correlation, regression, ANOVA, hypothesis testing, and visualization and statistical plotting tools.
Additional packages for specialized analysis including time series, digital processing, neural networks, and signal processing are available from Wolfram Research as well as independent developers.

MathLink lets you connect to other applications and set up alternative interfaces to Mathematica.

3. Volumes of Knowledge

Put the world's largest collection of mathematical knowledge at your fingertips.
Mathematica takes the most extensive collection of computation and visualization tools you'll find anywhere and puts them right on your desktop. Mathematica contains and surpasses the knowledge of thousands of mathematical tables, hundreds of reference books, and dozens of software systems. Yet Mathematica is faster to use, more accurate, and better integrated than any of them. All of the components you need to pursue a solution are built into Mathematica, from the basic functions like Sin, Log, and Eigenvalues to powerful superfunctions such as Solve, Integrate, and Simplify.

Plotting Functions and Visualization
Mathematica provides many flexible plotting options for visualizing your results: Plot, Plot3D, ContourPlot, DensityPlot, ArrayPlot, ParametricPlot, MoviePlot, MoviePlot3D, LogPlot, LogLogPlot, PolarPlot, ImplicitPlot, ListPlot, ScatterPlot3D, and many other variations. Yet these plotting routines represent only a subset of Mathematica's extensive graphics and visualization capabilities.

Automatic Numeric-Precision Control
Mathematica keeps track of the precision of its numerical results automatically throughout each calculation and adjusts its internal algorithms as needed to provide the precision you require.

More than 87,000 facts about mathematical functions...

4. Typesetting

Fully typeset input and output are interactive.
In addition to working with pure-text input and output, Mathematica works with typeset expressions. Both text and mathematical expressions can be formatted in any typeface, size, or style. Mathematical expressions are also "live," and you can use them as input or can make instant modifications. This feature allows you to work with mathematical expressions that are familiar from textbooks and to input formulas and parse results far more quickly than you can in any other program.

Mathematica lets you work with live evaluatable typeset expressions.

5. Symbolic and Numeric Computations

Perform symbolic as well as numeric computations.
Every function in Mathematica is implemented as completely as possible, handling the widest range of numeric and symbolic inputs. Mathematica knows how to evaluate functions to any precision anywhere in the complex plane. Along with supporting numerical inputs, Mathematica supports the world's largest collection of symbolic transformation rules, allowing sophisticated manipulation and reduction of formulas.
Pass the function Sin an exact input such as , and the result comes back exactly as . Pass it the numerical approximation 0.261799, and it returns the numerical result 0.258819. Try a complex value, and you get a complex result. Since all of these cases are handled simply by calling the function Sin, you won't need to memorize a different function name for each kind of argument.

Here Mathematica solves a nonlinear differential equation numerically.

Here is the parametric plot of the solution.

Here is a symbolic sum.

6. Graphics

Choose from over 50 styles of graphics, or create your own.
Mathematica provides over 50 built-in graphics types for visualizing your results, including a variety of 2D and 3D plots, contour and density graphics, and a full complement of specialized business and statistical plots. Mathematica also lets you generate animations and sounds with simple commands.
However, these plotting routines represent only a subset of Mathematica's extensive graphics and visualization capabilities. Mathematica also comes with a graphics language that lets you customize graphics to your exact specifications or even create your own graphic types from a large set of built-in primitives.

7. Application Development

Develop applications with Mathematica
In many cases you want not only to publish your results but also to make your Mathematica applications available to others--coworkers in your organization, customers, or colleagues around the world.
Mathematica's combination of computational sophistication and programmability makes it ideal for prototyping and developing complete applications. Because it provides a high-level environment, you can concentrate on what's unique to your work instead of spend time coding generic, low-level functionality. Once your application is finished, Mathematica offers numerous ways to rapidly deploy it in the way that is most efficient for your purpose.

Mathematica Notebooks and Packages
The most direct way of allowing others to use your Mathematica programs is to send your notebooks or packages to them. All Mathematica documents and programs are fully platform independent, so you do not have to worry about portability issues or incompatibilities. You can even add a point-and-click user interface, using either Mathematica buttons and palettes or Java, so that end users never have to work with the command line.

Interactive Web and Intranet Sites
With very little effort almost any Mathematica program can be turned into an MSP, an interactive web application running on a webMathematica server. In many cases, the process requires only a few steps--for example, saving the notebook as HTML, extracting the code, and then adding a few simple Mathlet tags. The resulting web application can be used from any web browser and through an interface; no Mathematica knowledge is required. You can also easily create more-advanced user interfaces using any number of standard web-development tools and languages such as JavaScript, JSP, or PHP.

Mathematica as a Software Component
With Mathematica's J/Link and MathLink API, you can also deploy your Mathematica application as part of a Java, .NET, or C/C++ program right out of the box. Additional products from Wolfram Research and independent developers provide prebuilt links to Visual Basic, scripting languages, and Microsoft Excel, which enable the products to interface with Mathematica and a variety of application packages.

Mathematica Link for Excel provides two-way communication between Mathematica and Excel.

8. Programmable Palettes

Programmable palettes let you have instant access to sophisticated functionality.
Mathematica comes with a collection of ready-to-use palettes that give you instant access to many of the built-in functions with one click. Because Mathematica is so flexible, you can also easily create your own palettes in seconds.
Put the functions and symbols you use most often on a single palette, or make notebooks interactive by including custom buttons in them. You can even add the palettes you use most often to a menu for quick access or can send them via email to your colleagues.
Since you can run any Mathematica function or program from a button, you can build complete interfaces to your Mathematica packages or courseware--making Mathematica an even more productive environment in which to work.

Mathematica’s palettes and buttons provide a simple but fully customizable point-and-click interface.

9. Special-Purpose Interfaces

Create special-purpose interfaces using Java, .NET, or C/C++.
Mathematica allows you to create complete document-centric and graphical user interfaces. You can build buttons and palettes, input forms and dialogs, and even fully interactive documents using nothing but built-in Mathematica functions. Moreover, your programs can generate any of these interface elements on the fly.
Automatically generate reports with completely cross-referenced hyperlinks. Create a survey that adapts itself to the answers given by the user. Make self-modifying palettes. The possibilities are endless, and the programs and interfaces created are platform independent.
There are many additional ways to generate custom user and programmatic interfaces for Mathematica. For example, Mathematica now comes with J/Link and a Java Runtime Environment preinstalled, allowing you to use AWT or Swing components to create a Java-based graphical user interface to Mathematica that will run seamlessly on all platforms for which Mathematica and Java are available.
The Windows version of Mathematica also includes .NET/Link for full integration with the Microsoft .NET Framework. With .NET/Link, Mathematica users can load any .NET object into Mathematica and extend it. .NET/Link also provides an easy way to call any DLL or COM object from within Mathematica.

Mathematica lets you easily create graphical user interfaces.

10. Programming Language

Program in the uniquely productive Mathematica language.
Whether you call them simulations, models, or algorithms, representing your concepts in Mathematica is easy. There's hardly a distinction between interactive and programmed calculations in Mathematica. You can build intricate calculations piece by piece. Specify a definition for an expression. Look up a formula and add it as a Mathematica transformation rule. Add more rules for other cases or for related formulas. The intuitive nature of Mathematica lets you build surprisingly sophisticated calculations easily and incrementally.
Mathematica includes a modern, wide-ranging, and highly versatile language that doesn't force you into a single style of programming. Just as a spoken language gives you many ways to express each idea, Mathematica provides many different programming paradigms.
Your code reflects your style of specifying the problem, which can make the command much shorter and easier to read. This unique flexibility makes switching to Mathematica from other programming languages easy--and cost effective. Even those who haven't programmed before can write powerful programs without extensive training.

Concentrate on your ideas.
Mathematica takes care of the programming infrastructure. There is no need to predeclare variable types or dimensions of lists and arrays, to direct memory management, or to compile your programs.
Common procedures such as sorting, searching, handling files, and manipulating data are built in and remove peripheral code from your routines. This feature helps to make typical Mathematica programs only 5 to 10 percent the size of those created in traditional languages or numerical systems and greatly shrinks development time.

Choose your programming style.
Mathematica handles problems of any scale and complexity equally well; it's more than a simple scripting language. One key feature is dynamic arrays of arbitrary size and dimension; optional compilation is another. By providing multiple paradigms and the world's most powerful pattern-matching engine, Mathematica lets you choose the most effective programming style for your problem. You don't have to work around the limitations of a restrictive language.
With such a variety of programming approaches, it's easy to see why Mathematica has become the language of choice for technical professionals around the world. Add it all together: Mathematica makes you many times more productive.

Mathematica's high-level programming constructs let you build sophisticated programs more quickly than ever before.

The Mathematica programming language lets you choose your programming style.

"f is the built-in factorial function."

"f of any integer x is the factorial of x."

"The factorial of each positive integer is that number times the factorial of the preceding integer. And the factorial of 1 is 1."

"The factorial of x is the product of all the integers from 1 to x."

11. Interactive Help Browser

All documentation is available through the interactive Help Browser.
The Mathematica Help Browser includes the complete documentation for all functions in Mathematica and the entire text of The Mathematica Book as fully indexed Mathematica notebooks with advanced search capabilities and comprehensive hyperlinks.
The Help Browser also contains thousands of interactive examples that demonstrate the use of Mathematica functions, its general capabilities, and the best way to take advantage of them.

Modify and evaluate examples in the Help Browser.
Unlike any other software, Mathematica enables users to modify and evaluate expressions directly within the Help Browser. The online material for the majority of built-in functions includes several examples to be evaluated or altered, providing a particularly helpful aid for those who learn best by example. User modifications of material within the Help Browser are not permanent, however. If you accidentally delete an example or section of help text, you need only to exit and reenter that page to restore the original information.
However, the material presented in the Help Browser is not fixed permanently to include only what is provided with the Mathematica installation. Help Browser information is stored as a Mathematica notebook. Thus, you can create help notebooks that become fully integrated with other Help Browser information, including the insertion of new entries into the Help Browser's master index.

Mathematica's sophisticated Help Browser lets you interact with thousands of pages of documentation, useful examples, and demos.