Mathematical Sciences: Matrix Inequalities and Matrix Perturbation Theory

研究者继续研究矩阵不等式及其应用。本项目的目标是(a)扩展矩阵函数的摄动理论，并开发有效的算法来计算这些函数的条件数，(b)推导矩阵分解的良好摄动界，(c)扩展矩阵范数理论，特别是幺正相似不变范数，以及Hadamard乘子的诱导范数。项目(a)和(b)在数值线性代数中有应用。(c)的问题本身就很有趣，并且与获得(a)和(b)中的结果相关。在数值计算中有一些不可避免的错误。例如，无论何时用计算机进行计算，都有可能产生小的舍入误差——例如，1/3 = 0.333333……可以近似为0.3333。另一个误差来源是输入的数据可能是不精确的测量结果。这些小错误可能会导致最终答案出现大错误——大到毫无用处。研究者研究了数值线性代数计算中小误差的影响。这将有助于确定一种特定的计算方法是否产生可靠的结果，并事先知道在测量中需要多大的精度才能确保最终答案的给定精度。

The investigator continues research in matrix inequalities and their applications. The objectives of this project are to (a) extend the perturbation theory for functions of matrices, and develop efficient algorithms to compute the condition number for these functions, (b) derive good perturbation bounds for matrix factorizations, (c) extend the theory of matrix norms, in particular unitary similarity invariant norms, and the induced norms of Hadamard multipliers. Projects (a) and (b) have applications in numerical linear algebra. The problems of (c) are of interest in their own right and are relevant to obtaining the results in (a) and (b). There are some unavoidable errors in numerical calculations. For example, whenever calculations are done by computer there is the possibility that small roundoff errors will be incurred -- for example, 1/3 = 0.333333... may approximated by 0.3333. Another source of errors is that the input data may be the result of measurements that are not exact. It is possible that these small errors can cause a large error in the final answer -- so large that it is useless. The investigator studies the effects of small errors in numerical linear algebraic calculations. This will help one to decide whether a particular method of calculation produces reliable results, and to know in advance what accuracy is needed in measurements to ensure a given accuracy in the final answer.