Numerical Analysis, CS&E, SIAM: Then and Now

July 6, 2005

Opening a 1969 conference on mathematical and computer aids to design, George Forsythe thought back to 1946, when he was a young mathematician working at the Boeing Aircraft Company, to begin a survey of advances in computers, computing, and, especially, numerical analysis in the intervening years. Coming across a written version of the talk some thirty-six years later, Jim Crowley was impressed by Forsythe's prescience and by the relevance of much that he said to CS&E research today. Photo courtesy of Stanford University.

Talk of the Society
James Crowley

Anyone who ventures into my office knows that I am something of an archivist--I don't toss things out very readily. At the instigation of co-workers, though, I did take part in a recent office-wide clean-up. My reward was the discovery of an interesting item: the "Digest Record" of a 1969 conference, Mathematical and Computer Aids to Design (sponsored jointly by ACM, IEEE, and SIAM). One of the more interesting papers comes right at the beginning: the opening remarks, made by George Forsythe.

Forsythe's presentation, titled "Design--Then and Now," provides a remarkable perspective on numerical mathematics about two decades after the birth of the modern computer. By 1969 Forsythe had been at Stanford for more than a decade (he founded the university's computer science department in 1965). But at the 1969 conference he was looking back to his experiences as a young mathematician at Boeing, specifically to 1946, when computing meant rows of people manning (a poor choice of words: most operators were women) desk calculators.

At that time, Forsythe explained, Boeing engineers were analyzing theodolite pictures and "discovering missile velocities and acceleration by numerical differentiation and smoothing." A theodolite (also called a "transit" by American surveyors) is an instrument (essentially a telescope with angle readings) for measuring horizontal and vertical angles. After reading W.J. Eckert's "Punched Card Methods in Scientific Computation," Forsythe had boards of a punched-card tabulating machine wired to "apply simple linear formulas to a long table of data." Freeing up the desk calculator group from some of these tedious tasks, he wryly pointed out, enabled the group to "spend more of their time reading those fascinating theodolite pictures." This was the first digital computer at Boeing.

Only twenty-three years had passed when Forsythe gave the talk that distracted me from my office cleaning. But during those years, card-programmable digital computers had become ubiquitous---Forsythe reported that by 1969 Boeing had $100,000,000 worth of digital computers---and numerical analysis had undergone tremendous development and growth.
Forsythe also looked back over the past two decades of numerical analysis, where "the action is much slower" than in computer technology. He began his discussion of the state of the art with programming languages, listing first among them "general-purpose languages for the expression of algorithms: Cobol, PL/1, Algol, Fortran." Thirty-six years later, some of these languages still sound familiar to those who work in scientific computing. (I understand that Ed Block has also done some cleaning; information he un-earthed appears in his note on page 6 about the role of Grace Hopper in the creation of Cobol.)

Forsythe continued by identifying a tremendous set of discoveries and ad-vances in numerical methods. Among the examples he cited are linear and nonlinear programming, "a subject that did not exist in 1946" (and that of course owes much to the late George Dantzig), and optimization in general.
Other important advances in numerical analysis---all achieved in the decade before 1969---include the QR method for computing eigenvalues, Romberg integration, methods for integrating stiff sets of ordinary differential equations, and early methods for solving partial differential equations. Also mentioned are splines, discovered by Schoenberg in 1945 but becoming computationally useful only with new developments in the 1960s.

Forsythe also had much to say about interdisciplinary research--which to him meant mathematicians, computer scientists, and scientists/engineers working together in an application domain. This sounds a lot like what we now refer to as computational science and engineering (CS&E).

Discussing the team approach to solving large-scale scientific problems, for example, he noted that "each specialist will have to know more about what the other specialists can do." That is, as we often hear today, experts in each discipline involved in the solution of the problem need to be able to communicate with one another and work as a team to solve the problem.

"These ideas have implications about the organization of our universities," Forsythe pointed out. "Students are ever ready to specialize within a limited area" and "have a liking for the abstract. . . . Both tendencies interfere with their attention to problem solving, and with learning a broad enough base of methods to solve problems well." Many of these ideas are surfacing today in discussions of university programs in computational science and engineering or of multidisciplinary approaches to scientific discovery.

Forsythe's paper provides us with a "snapshot" of early developments in numerical analysis. Since that time, we have witnessed tremendous growth in numerical analysis and in scientific computing---reflected by the breadth of papers in some of our own journals, in particular SIAM Journal on Numerical Analysis and SIAM Journal on Scientific Computing.

Like Forsythe, who looked back twenty-three years to the first digital computer at Boeing, we can look back to 1969 and consider progress in the intervening years. In doing so, we see the roots of modern numerical analysis in work done in the late 1940s as well as in the science of numerical analysis circa 1969. We can also see in that 1969 meeting the origins of some very current SIAM developments.

The list of speakers, for instance, includes several people who are prominent and active members of SIAM today. I'm thinking in particular of two who followed each other in the program: Cleve Moler ("The Numerical Solution of Matrix Problems") and C.W. Gear ("The Numerical Solution of Large Systems of Ordinary Differential Equations").

Bill Gear, who was president of SIAM in 1987�88, regularly attends SIAM conferences (most recently the May 2005 SIAM Conference on Dyn-amical Systems in Snowbird) and continues to work to advance the state of the art in scientific computing. One of his former students, Linda Petzold, is also working to carry on the tradition in many ways, both scientifically and professionally. Most recently, Petzold agreed to be a candidate for SIAM president in our upcoming (fall 2005) election. Cleve Moler, whose talk at the 1969 conference predated MathWorks and who was a student of George Forsythe, has agreed to be the second candidate for SIAM president.

Both Petzold and Moler have long and distinguished records of research and have been active throughout their careers in SIAM activities. Further details about Moler and Petzold, and the complete slate of candidates running for office in 2005, will be posted on the SIAM Web site later this summer. Meanwhile, see the box on page 1 for a note about procedures for voting in that election, whether electronically or by traditional paper ballot. We encourage you to vote, by whatever method you choose.

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