ERCIM News No.22 - July 1995

Dr. Bruno Codenotti, Director of the Institute for Computational Mathematics of the CNR, Italy: Computational Mathematics provides a sort of bridge between Computer Science and Mathematics


In the second half of this century, Computer Science has effected a new and significant change both on science and everyday life. What is primarily astonishing is the speed at which this development has taken place. First of all a few facts:

Computational Mathematics plays a central role in all of this. The examples mentioned above are mathematically posed problems, and fall within the domain of Computational Mathematics. The subject deals with the design and analysis of efficient algorithmic methods. One of the issues in Computational Mathematics is numerical stability - investigations make it possible to evaluate the effect of the necessary approximations and of the accumulation of errors.

As distinct from pure Mathematics, Computational Mathematics has a central interest in the complexity of constructive methods. In other words, embedding mathematics in the realm of computations makes one concerned with the amount of computational resources necessary and sufficient to solve the problems. Thus, the constructive aspects must be traded off against efficiency (space and time) and reliability (accuracy) requirements. From this line of argument it becomes clear that Computational Mathematics provides a sort of bridge between Computer Science and Mathematics. Its importance is also increasingly evident nowadays because new problems arising in areas such as life science and economics have to be tackled using the tools of computational mathematics. Thus, computing science has taken on the status of an autonomous scientific field, which is crucial to the practice of many other branches of science.

In my opinion Computational Mathematics will be highly influenced by parallel computing, by advances in pure mathematics and by the development of solid theoretical foundations in application domains such as life sciences. We still have not completely understood the full potential of parallelism and we still do not know if the dream of massively parallel computation is feasible; we do not even know yet which of the competing parallel machine architectures will win out in the long term. The intersection between computational mathematics and parallel computing is so considerable that any uncertainty about the future of parallel computing becomes an uncertainty as to how and where computational mathematics will evolve. This uncertainty poses severe constraints on the exploitation of parallelism itself: How cost-effective is it to develop software tools and algorithmic methodologies for classes of machines whose destiny is uncertain? Furthermore, the diversity in architectures of the current state-of-the-art parallel machines has created a new phenomenon almost absent in the case of sequential computations: parallel algorithms become machine-dependent in the sense that their efficiency can be radically different on different machines.

Let me conclude by saying that the scientists working in this field are living in very dynamic and exiting times, and are looking forward to the coming years.

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