Symbolic linear algebra, symbolic-numeric computation and applications

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For users symbolic computation is often viewed as having two main components. In the first instance, symbolic computation is a field for mathematical computation where algorithms work over exact (or inexact) arithmetic and where symbols or parameters are first class objects. Second is the equally important ability to transform symbolic expressions. This proposal involves elements of both.***Specifically  I propose research in three distinct subareas of symbolic computation:  symbolic linear algebra, sparse polynomial computation in numerical environments and applications of computer algebra.******In the area of symbolic linear algebra I propose to study algorithms for exact arithmetic of matrices of differential operators.  Our algorithms will focus on those domains  where growth of coefficient size becomes a fundamental concern. One goal will be to find fast algorithms for computation of matrix normal forms  for matrices of differential operators.  These normal forms are useful in a variety of applications  of mathematical, scientific and  engineering applications.  ***We are interested in sparse problems in multivariate polynomial interpolation along with the sparse decomposition of a signal built as a linear mixture of complex exponentials. This problem is closely connected to the shape from moments problem in the plane. In these cases efficient algorithms need to depend on the number of terms in the sparse representation rather than the possible number of terms found in a dense representation. Sparse representations are common in applications. Examples of of problems which will be addressed in this proposal include the numeric conditioning of  multi-dimensional sparse polynomial interpolation problems, generalizations to higher dimensional exponential interpolation and representations in terms of alternate bases. Applications include inverse problems such as the higher dimensional shape from moments problem and determination of sparse data in exponential, frequency data.  ******Our third research area focuses less on arithmetic but more on core  applications of computer algebra. Here we mention two directions for research : algorithms for manipulation and reduction of  higher order linear differential systems and algorithms for symmetry reduction of polynomial and dynamical systems  invariant under finite group actions or which require parameter or variable reduction.  In the first case  such algorithms should provide a new way for dealing with higher order systems of linear differential systems. In the second case the algorithms will impact the ability of solve dynamical systems and multivariate systems of polynomial equations.*** Software is part of our approach, as this enables us to verify that our theory agrees with actual practice. The long term goal is to ensure that symbolic computation and symbolic algorithms become more important problem solving tools for scientists and engineers.*** **