Research On The Methods Of Solving Several Special Linear Equations

Sovling linear equations occupies a very important position in computationalscience, applied mathematics, and engineering fields, but also the center in scientificcomputing. Special matrix has a wide range of applications in optimization theory,digital signal processing, automatic control, system identification, engineeringcalculations, and many other fields. The concept of generalized inverse matrixoriginates the linear equations. The saddle-point problem is a special linear algebraicequations. Therefore, it’s of important theoretical value and practical significance forthe research topic of getting the stable and rapid algorithm of the linear equation withspecial coefficient matrix by taking advantage of the special structure of these specialmatrix itself, getting the generalized inverse matrix algorithm when we study the linearequations, seeking fast and efficient iterative solution of saddle-point problems. Basedon the above-mentioned research purposes, our specific studies are as follows:By constructing special block matrix for the m×nCauchy-Type matrix C withfull rank, and then studying its triangular decomposition of the inverse or directtriangular decomposition, we get the corresponding fast algorithms for incompatibleequations which Cas coefficient matrix of the minimal norm least squares solution inthis paper. The three new fast methods reduce the calculation of a level of complexityrelativing to the general methods, such as solving normal equations and the orthogonalmethod. Numerical experiments show that the new methods of computing are moreefficient.We propose a fast algorithm of Moore-Penrose inverse for an m×nCauchy-Typematrix Cwith full column rank by forming a special block matrix and researching itsinverse. This method reduces the computational complexity than conventional methods.Numerical experiments show that the new method is more efficient. We get unilateralinversion formulas of the left inverse and the right inverse of m×nCauchy-Typematrix Cbased on whether equation has solution.We propose several Generalized Uzawa iterative methods for solving saddle-pointproblems and generalized saddle-point problems based on the positive definite andskew-Hermitian splitting (PSS) iterative method, and analyze the convergence of thesemethods. Numerical experiments illustrates the effectiveness of the algorithms.