奇异线性代数方程组的算法与理论研究

We have fulfilled the plan and published 49 papers, 14 of them have been cited in SCI journals, 11 of them have been cited by EI journals. We attended 7 International Conferences and presented the talk. We developed the Krylov subspace methods for solving the singular linear.equations, which was independent of the Isreal professor A. Sidi and was one of the earliest papers in this field and has arised the attension by the foreign researchers. We investigated incomplete semi-iterative method, successive matrix squaring method, index-splitting method and analyzed the initial guess and its convergence. We gave the perturbation.analysis of the singular linear equation with index 1 and parallel Cramer rule for computing the minimum T-norm, S-least squares solution. The perturbation expression for the Tikhonov regularization and weighted pseudoinverse were expressed under the weakest condition. We also studied the perturbation bound of the constrained and weighted linear least.squares problem. For the Drazin inverse, we developed the perturbation theory, solved the hard problem posed by S. Campbell and C. Meyer in 1975 and improved the classical result due to M. Drazin in 1958. In Hilbert space, we gave the perturbation analysis of the least squares problem and the perturbation bound of the Drazin inverse was presented in Banach.space. We also investigated the perturbation, splitting, the representation and approximation theorem of the generalized inverse AT.S (2) ,.which unified the results of the common generalized inverses.

研究具有实际应用背景的奇异线性代数方程组的数值算法及其理论研究.包括:奇异线性代数方程组的通解结构;奇异线性代数方程组的Krylov子空间迭代法和不完全半迭代算法;同时我们将这两类方法运用于经济学中的投入-铲除分析的Leontief闭模型,并建立这类问题的计算解和准确解的误差估计.本课题具有重要的理论意义和广泛的应用前景.