The Use of Algebraic Algorithms in Scientific Computing.


Various computational tasks in Scientific Computing have a great deal of mathematical structure which is not utilized in present algorithms for these tasks. An obvious example is the solution of polynomial systems of equations in many variables, as they arise in Chemistry, Mechanics,etc. Algorithms exploiting the mathematical structure often exist within Computer Algebra, but they rely heavily on exact rational computation.

In the presentation, it will be shown how the implementation of such algorithms in an environment with data of limited accuracy requires a fundamental change in some of the underlying paradigms. After the algorithms have been redesigned in the appropriate manner, they may be executed in floating-point arithmetic and yet produce more meaningful results than in their original form. They can thus become part of packages for various problem areas in Scientific Computing. Examples will clarify the general considerations.