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http://www.wolfram.com Questions & Answers 1) What does "the world's only fully integrated technical computing system" mean? 2) Who are Mathematica's competitors? 3) How does Mathematica's "smart numerics" differ from numerics in traditional systems? 4) How fast is Mathematica 4? 5) Mathematica is renowned for its symbolic computational power, but how are symbolic computations beneficial in practice? 6) I've read in reviews that Mathematica is very well designed, but how does that help my day-to-day work? 7) Why should I consider such a comprehensive system as Mathematica when I only have a specialized task to do? 8) Can I be sure that results I get out of Mathematica are accurate? 9) Can I share results from Mathematica with colleagues who don't have it? 10) What is MathML? 11) Are there other ways that Mathematica can interact with the internet? 12) What is the .nb document standard? |   | Technical computing systems rank among the most complex off-the-shelf software available today. Given this, it's hardly surprising that comparing capabilities of different products can be a daunting prospect. Even if you select a system that satisfies today's needs, will it meet the demands of tomorrow? This Question and Answer area is designed to help you pick through a few of the key issues that make Mathematica so exceptional. 1) What does "the world's only fully integrated technical computing system" mean?
We've designed Mathematica to be useful throughout as much of a technical person's workday as possible, not just for one or two tasks. Our customers find that using one technical computing system, fully integrated between different aspects, is many times more efficient than using several specialized systems. Mathematica is unique in having taken this approach to assisting technical professionals. back to top 2) Who are Mathematica's competitors?
There is no single competitor to the complete Mathematica environment. Broadly speaking, competitors break into a number of categories: numerical packages, computer algebra systems, typesetting and documentation applications, graphics and statistics systems, traditional programming languages (interface builders), and spreadsheets. Since Mathematica first appeared, a few systems have branched out from their initial purpose of covering one or two of these categories to try to embrace others (e.g computer algebra systems that have added numerical capabilities). Mathematica is unique in integrating these capabilities coherently because only Mathematica's structure has always been designed with this end in mind. back to top 3) How does Mathematica's "smart numerics" differ from numerics in traditional systems?
Our numerics differs in several ways. First, Mathematica automatically chooses the best algorithm for your problem and applies it adaptively: you don't need to be a numerics expert to get reliable results very quickly even with the toughest of problems. Traditional numerics applications force users to manipulate input expressions by hand to get them into the restrictive forms that the applications require for further processing. (For example, the need to transform by hand a higher-order differential equation into a system of first-order equations before entering them into the application). Mathematica's numerics utilizes Mathematica's symbolic capabilities to preprocess input and constructs the appropriate form for its numerical algorithms automatically. Moreover, Mathematica may repeat this strategy at any stage of the computation, choosing from and switching between a broader range of algorithms. Additionally, Mathematica is not just limited to machine-precision arithmetic (usually 16 digits) but can compute with any size or accuracy of number. Moreover, as it's computing, it tracks the precision of your calculations and ensures the accuracy it ascribes to the results is justifiable. back to top 4) How fast is Mathematica 4?
Very fast--considering the intelligent computation it's doing. We optimize Mathematica to be able to take any kind of input efficiently--be it symbolic, machine, or arbitrary precision numbers. We also optimize Mathematica to help you solve your problem from start to finish and as quickly as possible, reducing combined problem specification and computation time rather than just computation time. You don't have to specify how to calculate the problem; Mathematica uses intelligent techniques to get the right answer. That takes a little more computation time, but a lot less of your time. Even if you measure computation time only, Mathematica will often beat custom C code or specialized systems through the use of more sophisticated algorithms, particularly when one top-level function call operates many lower-level calls. With just-in-time compilation and other state-of-the-art techniques, Mathematica 4 spectacularly enhances the speed and efficiency of many kinds of numerical computations compared to previous versions. back to top 5) Mathematica is renowned for its symbolic computational power, but how are symbolic computations beneficial in practice?
Symbolic computations provide a general solution to your problem for a wide range of cases rather than just at selected points. That means you can look at the form of your model, plot it, test it numerically, and continue to transform it symbolically without loss of accuracy. Additionally, some techniques are only applicable symbolically. For these reasons, you ideally want to maintain your solution in a symbolic form as far into a computation as possible. Keeping computations symbolic has traditionally meant doing them by hand--far less productive than doing numerical computations by computer. Not only does Mathematica's symbolic capability change this, it greatly enhances the sophistication of symbolic calculations that can be done at all. This is achieved by combining the world's largest collection of mathematical knowledge with the ability to process far larger expressions more quickly than can be processed by hand. Moreover, the ability to mix symbolic manipulation, programmed computation, and numerical substitutions seamlessly is more powerful than any of these individual abilities. The realization of the advantages of symbolic computation have led engineering and financial analysis industry experts to predict widespread use of symbolic computations in their industries. back to top 6) I've read in reviews that Mathematica is very well designed, but how does that help my day-to-day work?
First, it makes it quicker to learn, since understanding a few simple rules provides an understanding of how a large number of functions operate. Rather than learning the operation of each function individually, you only have to learn a few simple rules and then you can predict many details of how other functions will operate. Second, outputs from calculations are structured correctly to be used immediately as input for other calculations--even when the latter calculation is of a very different nature. Third, you can be confident that work you do today will continue to be compatible with future versions of Mathematica; careful design makes later changes unnecessary, even when the underlying algorithms are improved. It takes time to appreciate the usefulness of Mathematica's meticulous design. But the more heavily you rely on your technical computing system, the more crucial good design becomes. Wolfram Research is unique among technical computing products in ascribing this level of importance to design. back to top 7) Why should I consider such a comprehensive system as Mathematica when I only have a specialized task to do?
If your work involves just one well-defined task which fits precisely into one of the traditional categories of software (numerics, algebra, graphics, etc.) and will never change, you might find it acceptable to use a specialized product. But this situation is very rare! Usually, you'll want other capabilities too--if not immediately, then after a little while. At some point your specialized system simply may not be capable of doing what you want, giving you stark choices: abandon what you wanted to do with it, or move to another system and invest more time and money. That's why so many people start with Mathematica. And those who don't often end up switching to Mathematica later for these reasons. But there are several other reasons to use Mathematica for specialized tasks. First, its comprehensive capabilities help it to give you the right answers. For example, you might only want to use Mathematica's numerics, but internally Mathematica uses symbolic computation to optimize the numerical answers it gives you. Second, by providing an all-in-one environment, Mathematica allows standardization in technical computing tools across an organization. back to top 8) Can I be sure that results I get out of Mathematica are accurate?
You can be far more sure than by applying published mathematical theorems, and far more sure than with the vast majority of technical computing products. Wolfram Research's software quality assurance program is one of the most sophisticated in the industry. Every week, throughout the development process, Mathematica is subjected to an extensive battery of manual and automated testing, including comparisons of nearly a half million computations (chosen from books of tables, bug reports, documented behavior, and other Wolfram Research generated tests) with known results. Mathematica's ability to solve many problems in a variety of different ways (thereby allowing self-checking), automatic numerical precision control, and testing by the more than 1,000,000 people who've used Mathematica contribute further to its robustness. Because Mathematica is now viewed as a standard for reliability, many major companies specifically test their products against it. back to top 9) Can I share results from Mathematica with colleagues who don't have it?
Yes, in several ways. Mathematica notebooks can be viewed and printed, cells opened and closed, and animations run using MathReader--a utility free to individuals. There is full compatibility between platforms. Mathematica 4 notebooks can also be saved as HTML, TeX, and RTF, among other formats. Mathematica 4 is the first commercial software application able to save mathematical formulas as MathML--the new standard for mathematical typesetting on the web. back to top 10) What is MathML?
MathML is the new standard agreed by the W3C web consortium for presenting mathematical typesetting on the web. Wolfram Research was a key contributor to this standard which was heavily influenced by Mathematica's typesetting system, introduced in Version 3. back to top 11) Are there other ways that Mathematica can interact with the internet?
You can link a special Mathematica kernel to web pages to provide calculation ability on your web site. You can find an example at The Integrator, which usesa Mathematica kernel server at Wolfram Research. back to top 12) What is the .nb document standard?
The .nb document standard, used by Mathematica, MathReader, and other products, is an environment for technical documents. It combines text, graphics, formulas, algorithms, hyperlinks, and buttons in one platform-independent document. A Windows user can mail a .nb document to a Mac, Unix, and Linux user, each using a different .nb-supporting product, and the notebook will function identically. This makes the .nb standard an ideal information conduit for technical communication--a common means of communication for technical professionals everywhere. back to top | |
