Published in *The Mathematica Journal* (Spring 1995), Vol. 5, issue 2.

Jerry Keiper, leader of the numerics research and development group at Wolfram Research, was killed in a bicycle accident on January 18, 1995 at the age of 41.

Keiper’s life was a rare and wonderful mixture of brilliance and achievement with modesty and humanity. He was driven by a profound desire to do good in the world, while not burdening it with any of his own personal needs.

Keiper was born in Medina, Ohio on October 20, 1953, the second of eight children. He spent his early years on the family farm. Then, after graduating from high school, he enrolled in a technical school, planning to become an electronic technician. But he excelled in mathematics, and even though none of his family had ever gone to college before, he decided to enroll at Ohio State University. He received a bachelor’s degree in mathematics from there in 1974, and a master’s degree a year later. His master’s thesis showed that the Riemann zeta function could be expressed as a fractional derivative of the gamma function—the first of many results he was to obtain about special functions.

Throughout his life, Keiper was deeply influenced by religion. He was raised as an Apostolic Christian, but in his college years joined the Mennonite church—a branch of protestantism with a prohibition against military service and a tradition of humanitarian activity. Keiper’s religious views initially made him decide not to pursue a career in mathematics, and instead to become a high-school teacher. He spent a brief time in 1977 as a teacher in the Michigan public school system, but found that, in his own words, "there was very little teaching involved in the work."

Having been disappointed by teaching, Keiper spent a year constructing a pipe organ—indulging his lifelong enthusiasm for things mechanical. But in 1979 he returned to mathematics, and in 1981 he earned a master’s degree in applied mathematics at the University of Toledo, Ohio. That same year he left the U.S. to work with the Mennonite church in Nigeria, and after various delays and adventures spent three years teaching at a university in central Nigeria.

Keiper returned to the U.S. in 1984, and enrolled as a graduate student in computer science at the University of Illinois. He specialized in numerical analysis, working particularly with the well-known numerical analyst Bill Gear on the solution of differential algebraic equations.

In the spring of 1987, Keiper heard about the early development of *Mathematica*, and approached me about working on the project. Knowing his interests in both special functions and numerical analysis, I suggested that Keiper might work on finding general methods for the numerical evaluation of special functions. Existing academic and other work had been concerned mostly with evaluating specific functions to a specific precision for specific ranges of parameters. But I wanted Keiper to make *Mathematica* be able to evaluate any of the functions found in standard books of tables, to any precision, anywhere in the complex plane. Many numerical analysts thought this was an absurdly ambitious project, but undaunted, Keiper set about doing it.

His crucial idea was to use the symbolic capabilities of *Mathematica* to automate the process of finding optimal approximation algorithms. In the past, such algorithms had mostly been worked out by hand, on a case-by-case basis. But Keiper wrote systematic *Mathematica* programs to find algorithms for any function. Sometimes it took a month of CPU time to generate a particular optimal algorithm. But once generated, the algorithm could be executed very rapidly. And the result was that for the first time it became possible to assemble reliable algorithms for evaluating hundreds of special functions to any degree of precision for any values of their parameters.

In addition to special function evaluation, Keiper also worked on other numerical features of *Mathematica*, particularly numerical quadrature and root finding. Initially he used mainly refinements on algorithms already in the literature, but increasingly he developed entirely new algorithms, typically based on integrating the numerical and symbolic capabilities of *Mathematica*.

After *Mathematica* was released in 1988, Keiper briefly returned to his Ph.D. thesis project concerning differential algebraic equations, and with the help of the capabilities he had put into *Mathematica*, he was rapidly able to complete his thesis, officially receiving his Ph.D. in 1989.

Following his deeply-held personal and religious beliefs, Keiper lived in a very simple manner. He wore simple clothes, ate simple food, and used a bicycle as his primary means of transportation. He also felt that to be consistent in not supporting the military, he should avoid paying taxes to the government. For a while, this meant that he would accept almost no salary. But in the end he worked out a scheme for donating all but a small percentage of his salary to charity. In addition, Keiper set up a foundation, which he named the Michael and Margarethe Sattler Foundation, after two early Mennonite martyrs. As part of Keiper’s compensation, Wolfram Research then made donations to this foundation. The foundation solicited proposals, and in turn supported various colleges, giving them both funds and copies of *Mathematica*.

As the popularity of *Mathematica* grew, Keiper was very happy to see his work used so widely. But in 1990 he felt a need to contribute more directly to education, and so he decided to apply for teaching positions at a number of colleges. Assured of financial support from Wolfram Research, he sent out a resume with the line "salary goal: not an issue," and planned to ask for no salary for his teaching. The reaction he got from the academic establishment was less than appreciative, and as a result he decided to pursue his educational interests in other ways.

For about a year he moved to Kansas and helped set up an educational lab based on *Mathematica*, while continuing his work on the development of numerical algorithms for *Mathematica*. During this time, he also began writing a textbook of numerical analysis based on *Mathematica*, in collaboration with Bob Skeel, a numerical analyst at the University of Illinois. The book was published by McGraw-Hill in 1993 under the title *Elementary Numerical Computing with Mathematica*, and is now a standard text in numerical analysis courses.

Since the mid-1970s, Keiper had maintained a keen interest in analytic number theory and its investigation by computer. In early 1988, Keiper used a prototype of *Mathematica* to explore various relations between zeros of the Riemann zeta function. He hesitantly wrote to D. H. Lehmer, a pioneer of computational number theory, describing his results, and Lehmer replied warmly, encouraging him to publish what he had discovered.

In the course of the next several years, Keiper began to overcome his shyness, and to publish some of his mathematical work. He was particularly interested in finding formulations of the Riemann Hypothesis that would make it more amenable to investigation by numerical methods. He did many large computer experiments both on the ordinary Riemann zeta function and on generalizations and related functions such as the Ramanujan tau functions. A few months before he died, Keiper told me he felt he had made considerable progress. And when he died there were several of his programs found on computers at Wolfram Research that had been running for more than 2000 CPU hours—generating results intended for Keiper to interpret.

Although Keiper did his work on the zeta function mainly to investigate basic questions in number theory, he always made sure that relevant pieces were integrated into *Mathematica*. And in 1990 it was his work that made possible the six-foot-long poster of the Riemann zeta function that Wolfram Research produced for the International Congress of Mathematicians in Kyoto. This poster is now to be found displayed in most mathematics departments around the world. (A special new memorial edition of the poster is being produced.)

In the past few years, Keiper became interested in the fundamentals of computer arithmetic, and forthcoming versions of *Mathematica* will include some major innovations that he made in the basic handling of numbers on a computer.

Keiper attended *Mathematica* conferences around the world, speaking about the numerical capabilities of *Mathematica*. He was also a frequent participant in discussions on computer network newsgroups. He was always extremely patient, although in private he would often express his frustration at those who chose to attack *Mathematica* without understanding it or giving it the thought that it deserved.

Keiper was outstandingly modest about his own abilities. But in his quiet and unassuming way, he over and over again managed to far surpass what others had done. His published papers provide hints of his ability, but his greatest professional achievements are embodied in the internal operation of the numerical functions of *Mathematica*. And although only specialists may be concerned with exactly how these functions work, a million people around the world make use of them, executing over and over again the code and algorithms that Jerry Keiper created.

Keiper is survived by his former wife of fifteen years, Susan Diehl, as well as by his parents, five brothers, and two sisters. Wolfram Research is planning to establish a Keiper Memorial Fund which will be used to support educational programs of the type in which Jerry Keiper was interested.

[Since this article was written, Wolfram Research, Inc. has established the Jerry Keiper Memorial Center for Teacher Development in conjunction with the Illinois Council of Teachers of Mathematics.]