【作者】 陈金林；

【导师】 韩志斌；

【作者基本信息】 华中科技大学 ， 计算数学， 2007， 硕士

【摘要】 广义逆的理论和方法不仅是许多数学分支的基本工具,而且在经济学、统计学、测量学、最优化、信息处理、自动控制、工程技术和运筹学等应用学科中都有着广泛的应用。在研究最小二乘问题,长方、病态线性、非线性问题,无约束、约束规化问题,控制论和系统识别问题,网络问题等等中间,广义逆更是不可缺少的研究工具。本文,首先引入对值域空间、零域空间和一般广义逆的秩理论知识,对具有指定的值域与零域空间的广义逆的表示进行了具体的论述。主要通过广义逆矩阵的表示理论和矩阵分解去研究广义逆的计算表示方法,在此解决了如下的问题:第一:Gauss-Jordan消元法在对非奇异矩阵求逆过程中,使用初等行(或列)变换来得到非奇异矩阵的逆,而且这个变换也能判断出一个矩阵是否是非奇异矩阵。然而它并不能直接的运用到广义逆的计算过程中。这里对广义逆的表示进行了一些改进,从而使它可以利用Gauss-Jordan消元法去求解。第二:图像复原是一个病态问题,一般需要通过规整化技术得到一个合适的解。规整化的难点在于抑制噪声的同时保持图像的边缘等重要图像信息。在本文中,把新的广义逆的求解方法利用于图像复原的保持边缘规整化改进算法中,从而用与原有的方法比较,说明它的高效性。更多还原

【Abstract】 The theory and methods of generalized inverse matrices are important basis tools in all mathematical disciplines and have extensive applications in economics statistics, surveying, optimization techniques, information processing, automatic control, engineering techniques, operations research and so on. On the other hand, the study of algebra structure of associative ring is prevalent, and matrices generalized inverse over a ring is an important tool in revealing algebra structure of associative ring.In this paper, first introduced rang, null space and generalized inverse of the rank theoretical knowledge, designated rang and null space the generalized inverse, said the specific expositions. Through expression theory of generalized inverse and matrix factorization, mainly, study generalized inverse calculation. We will study and settle following several problems.The first, it is celebrated that Gauss-Jordan elimination method compute the inverse of a nonsingular matrix by executing elementary row (or column) operation. Moreover the operation can be used to determine whether or not a matrix is nonsingular. However, one cannot directly use this method on a generalized inverse of a rectangular matrix or a square singular matrix. In this paper, said here on expression of the generalized inverse of a number of improvements, Gauss-Jordan elimination method so that it can be used to solve.The second, image restoration is an ill posed problem and must be regularized. Usually, the difficulty of regularization lies in avoiding the smoothing of edges while suppressing the noise. The improver algorithm of edge- preserving regularization used the new solution of the generalized inverse, thus comparing to the original method, it shows its effectiveness.更多还原