Symbol-Crunching Programs: Mainframe Analytical Power Reaches The Desktop

A new breed of mathematical software brings computing power formerly reserved for mainframes to personal computers, allowing scientists with desktop machines to pose and answer problems of greater complexity than ever before. Those taking advantage of the increased availability of such sophisticated analytical support for their work include--perhaps expectedly--many physicists and engineers, but also biological researchers studying such phenomena as fish population cycles and the foraging habits of bees. Since the early days of room-sized digital machines, scientists have been putting computers to work crunching numbers for their research--the "grunt" work associated with long data columns. In the 1960s, several research groups began to extend the realm of computers from numerical calculations to symbolic manipulations--that is, the higher-level math that uses symbols to represent constants, variables, and mathematical operations. The programs that resulted provided scientists with versatile new research tools, but they required mainframe computers to run, making them effectively unavailable to many researchers. Now, with the capabilities of personal computers expanding each year, versions of several symbolic computation programs originally written to run on mainframe platforms are available on the desktop. The more popular programs offer a truly interactive mathematical environment, which includes not only symbolic manipulation capability, but also numerical computation, flexible graphics, and programming language for the user. The programs allow researchers to more easily glean insights from mathematical relationships among data and to concentrate on the intellectual content of a problem. Stephen Buchmann, for example, is a research entomologist at the United States Department of Agriculture's Carl Hayden Bee Research Center in Tucson, Ariz., and an adjunct professor of entomology at the University of Arizona, Tucson. He uses the best-selling symbolic computation program, Mathe- matica, from Wolfram Research Inc. of Champaign, Ill., running on a Macintosh, to analyze the foraging habits of honey bees--as many as 50,000 workers per colony--as they make from five to 15 daily "sorties" each from their hives. "We collect data from a number of sources," Buchmann says. "One would be feeding stations [where we count] genetically or paint-marked bees to see how far they go. Another would be pollen traps, where the pollen loads on the hind legs of returning bees get knocked off. We microscopically identify the genera and species of plants they were foraging from. Then we combine those [data] with aerial and topographic maps to figure out minimum and maximum distances [for areas] where the bees could have collected that particular food." Using Mathematica and an algebraic expression, Buchmann is able to estimate the number of bees at a given distance from the hive at any time. The program provides him with a three-dimensional graphical display that can help in modeling foraging patterns. Buchmann can also assign false colors to the overlapping foraging areas between colonies. One discovery from this information: If colonies are placed too closely in a large apiary, the bees will compete for the flowers and less honey will be produced. "We use Mathematica," he says, "as a powerful visualization tool on, let's say, a day's worth of data, or a week's, or even a whole year's, to ask, `Where were most of the bees?' As it turns out, it's counterintuitive, because the foraging distribution looks like a donut, or a torus, with not much foraging in close to the hive and most of it out at an intermediate distance. Then it drops off exponentially." Working with a colleague, Buchmann has also been looking at the adaptive, or optimal, geometry of the honeycomb. "Mathematica is widely used in mathematics, computer sciences, and engineering," says Buchmann, "but most biologists have never heard of it. My audiences [at professional meetings] are always taken aback by the figures and wonder how I produced them." Donald Pereira is another biological researcher putting symbolic computation to work in his specialty. Pereira is a fisheries research biologist with the Minnesota Department of Natural Resources in St. Paul. Like Buchmann, he uses Mathe-matica on a Macintosh, in this case to analyze fish population data. "I'm processing biological data on fish populations to quantify cyclic behavior and dynamics of fish stocks," says Pereira. He says he is also looking at correlations between cycles in different species. Pereira's explanation of the work he is doing demonstrates one way in which, increasingly, powerful desktop mathematical analysis is supporting life sciences research. "I'm doing time-series analysis in the frequency domain," he says, "so that the primary basis of it is using the Fourier transformation to do a harmonic decomposition of the time series, to quantify the amplitude of [population] cycles at various frequencies. "In other words," he adds, "species A might have a significant [population] cycle of about five years, species B might have a significant cycle of about seven years, and species C might have a significant cycle also at five years. The next thing to do would be to see if the five-year cycles of species A and species C might be correlated." He emphasizes that the mathematical correlations he discovers do not necessarily imply causal relationships in the real world. "We're using this as an exploratory technique," he says. "We would use the results to direct further field studies to try to determine causality." Pereira says he's "fiddled" with other packages capable of doing the time-series analyses, but he prefers Mathematica because he considers it a more complete software tool than other programs. Users say Mathematica's success is based on the fact that it was the first program for personal computers to effectively integrate computer algebra systems, graphing systems, numerical analysis, and word processing into a seamless whole. Mathematica's programming language even allows users to extend its 950 functions in many simulated languages, such as C, FORTRAN, and Pascal. But Mathematica is not the only mathematical software program on the market, although it may be the common reference point against which the others are measured. There are several other respected programs, as well. Among these are Maple (Waterloo Maple Software, Waterloo, Ontario), Reduce (Calcode Systems, Venice, Calif.), Macsyma (Macsyma Inc., Arlington, Mass.), and Derive: A Mathematician's Assistant (Soft Warehouse, Honolulu, Hawaii). Each has its virtues and adherents. Not many comparative reviews have been written of symbolic computation software. The learning curve for these programs is so steep that one is apt to stick with whatever program one begins with, unless forced to change for some very compelling reason, so that there are few users who consider themselves expert in more than one of these systems. Mathematica originator Stephen Wolfram has suggested that a two-year research project to accurately evaluate these programs should be conducted. But this has yet to be done. The Stanford Linear Accelerator Center (SLAC) is one of the few places that has run any of these programs side by side on the same platform. Researchers there ran problems regarding research on string theory using Mathematica and Maple. Originally, the idea was to use the programs to check each other, but Mathematica faltered. "When it comes down to big problems that might take hours or days to complete, Maple completes the problems and Mathematica just dies," says Marvin Weinstein, physicist and senior scientist at the SLAC. "In a number of problems, Maple runs on an order of magnitude faster than Mathematica. And, it's more sparing of memory. "Mathematica's user interface is a little more intuitive. It has more elegant syntax for doing certain things in its current release, and the graphics are glitzier. But Maple has it all over Mathematica when it comes to user support and power." SLAC has become a licensed site for Maple. Reduce and Macsyma are thought to be better for certain symbolic manipulations than the other programs. "To this day, Reduce probably takes traces of things faster than any other language around, but it's not flexible," says Weinstein. "It's very good for a subset of problems, but arcane in how you use it." Reduce also has no graphics and lacks some elementary math functions, such as the capability of simplifying radicals. Macsyma's developers claim superiority to both Maple and Mathematica in performing Laplace transform and inverse transform operations. They also point to Macsyma's more than 200 worker-years of development, suggesting the program is more debugged than others. The program originated at the Massachusetts Institute of Technology more than 20 years ago. "The grandfather of these programs is Macsyma," says George Rublein, an associate professor of mathematics at the College of William and Mary, Williamsburg, Va. "It's older, but I dare say they still think of themselves as competitive in this racket." Rublein says that Macsyma, despite being very powerful, was limited for many years by the fact that the program would run only on a proprietary machine, sold with the program. (A version for personal computers has now been out for several years.) "If you wanted to run Macsyma, you had to buy their machine," says Rublein. In April of this year, a new group took over the program from long-time owner Symbolics Inc. and set up a company called Macsyma Inc. to further develop the software and market it. An improved personal-computer version of the program is due out soon. Another popular program, Derive: A Mathematician's Assistant, is limited by a relatively minimal programming language, say some users, and is used more as a teaching tool than a research tool. It's important, says Stanford's Weinstein, to note that the symbolic computation systems do make mistakes. "The use of all these programs is a bit of an art form," says Weinstein. "They allow you to automate the process and do all the things you would do by hand to check for errors. But the person who thinks these programs are automatic, self-checking things that will avoid all errors, [so that] one doesn't have to check the results, is a fool." For example, Weinstein says if you ask the symbolic computation programs to calculate the electrical potential inside a spherical shell, none will correctly report that the electrical field comes out zero. There are other problems that the symbolic computation programs have as a group, according to Richard Pavelle, a mathematician who helped to develop Macsyma. Pavelle lists a few of the common shortcomings: Basic commands are readily forgotten, user interfaces are poor, commands don't work as advertised, and answer-checking can be difficult. "You have to use a program every day for several years, and even then you may not be a power user," says Pavelle. "For each of these systems, there are probably only a half-dozen people in the world who know them inside and out." Still, power users or not, more and more different kinds of researchers are finding ways to put these powerful mathematical programs to work for them. As a case in point, Walter Stewart, a scientist at the National Institutes of Health, is widely known for his support of charges that a paper coauthored by Nobel laureate David Baltimore (Cell, 45:247, 1986) misrepresented research data. Those charges led to controversial hearings and the eventual resignation of Baltimore as president of Rockefeller University. The disputed paper reported that transplanted genes seem to produce new antibodies in mice. The laboratory data included two sets of measurements for 129 hybridomas. To assess the accuracy of the study, Stewart wanted to examine the 16,000 ways that the two data sets could be arranged. In 1986, using software available at the time, Stewart spent three months analyzing the Baltimore study. He was, however, unable to extract enough evidence from the data to substantiate the allegations. Two years later, Stewart wrote a program using elementary group theory in Mathematica and reanalyzed the data. According to Stewart, Mathematica provided him with evidence to support claims that data were misrepresented. The intricate calculations this time around took less than four hours. The family of symbolic computation programs for personal computers all offer expanded capabilities for researchers without access to mainframes. And, for mathematicians, they each appear to have particular strengths. But Mathematica does dominate the marketplace, and seems to be more successful in making inroads into such research domains as the life sciences, areas that might be thought of as less exclusively mathematical in orientation. One reason, certainly, might be the sophisticated and aggressive marketing of the program's parent, Wolfram Research Inc. But another reason, perhaps, is the program's functional breadth. "There are many jobs that Mathematica does that other pieces of software can do better, quicker, and much easier," says Stewart. "But the reason that Mathematica is in a class by itself [is that] it has solved the broadness question." Joe Mullich is a freelance writer based in Norristown, Pa.

Symbol-Crunching Programs: Mainframe Analytical Power Reaches The Desktop

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