Symbolic linear algebra, symbolic-numeric computation and applications

对于用户来说，符号计算通常被视为有两个主要组成部分。在第一种情况下，符号计算是一个数学计算领域，其中算法研究精确(或不精确)算术，其中符号或参数是第一类对象。其次是同样重要的转换符号表达的能力。这一建议涉及两者的元素。*具体地说，我建议研究符号计算的三个不同的子领域：符号线性代数、数值环境中的稀疏多项式计算和计算机代数的应用。*在符号线性代数领域，我建议研究微分算子矩阵的精确算术算法。我们的算法将专注于那些系数大小增长成为基本问题的领域。一个目标是找到用于计算微分算子矩阵的矩阵标准形的快速算法。这些标准形在数学、科学和工程应用的各种应用中都是有用的。*我们感兴趣的是多元多项式插值中的稀疏问题，以及作为复指数的线性混合构建的信号的稀疏分解。这一问题与平面上的力矩形状问题密切相关。在这些情况下，有效的算法需要依赖于稀疏表示中的项的数量，而不是在密集表示中找到的项的可能数量。稀疏表示在应用程序中很常见。这一建议中将解决的问题的例子包括多维稀疏多项式插值问题的数值条件、对高维指数插值的推广以及用交替基来表示。应用包括逆问题，如从矩的高维形状问题和确定指数、频率数据中的稀疏数据。*我们的第三个研究领域较少关注算术，而更多地关注计算机代数的核心应用。这里我们提到了两个研究方向：高阶线性微分系统的处理和约化算法，以及多项式和动力系统在有限群作用下不变或需要参数或变量约化的对称性约简算法。在第一种情况下，这样的算法应该为处理高阶线性微分系统的系统提供一种新的方法。在第二种情况下，算法将影响求解动力系统和多元多项式方程系统的能力。*软件是我们方法的一部分，因为这使我们能够验证我们的理论与实际相符。长期目标是确保符号计算和符号算法成为科学家和工程师更重要的问题解决工具。

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.*** **