The Research On The Relative Problems Of Solving Special Linear Equations

This dissertation has two parts, the first part is mainly about solving linear equations, whose coefficient matrices are tridiagonal,pentadiagonal or block period tridiagonal such as sparse matrix. We present some simple and practical algorithms to directly solve them. We also present the upper bound of rounding-off error concerning the algorithms for solving them. The second part is mainly based on the properties of H-matrix and monotone matrix; we give some results about the convergence of the generalized alternating iterative method and the parallel alternating iterative methods for H-splittings of a nonsingular H-matrix and so.Based on the above two parts of this paper commenced, the main results are as follows:1) Suppose x? is the solution of Ax=b by Gaussian Elimination with partial pivoting, backward rounding-off error analysis shows (?) is the solution of (?),εis the precision of arithmetic operation. If A∈Rn×n is tridiagonal, we use Thomas algorithm to solve Ax = f , then we present a backward rounding-off error estimation. The upper bound forδA∞is irrelevant to n obtained by the usual backward error analysis where round-off errors are attributed totally to the system matrix A. We also present a modification of the Thomas algorithm for solving weakly irreducible diagonally dominant tridiagonal linear system of equations and a backward rounding-off error estimation for it.2) If A∈Rn×nis pentadiagonal, we also can use forward elimination and backward substitution algorithm to solve Ax=b, we present a backward rounding-off error estimation for it. The upper bound forδA∞is irrelevant to n obtained by the usual backward error analysis where round-off errors are attributed totally to the system matrix A.3) The forward elimination and backward substitution algorithm is good for solving a block diagonally dominant block-tridiagonal linear system of equations. In this paper, we propose a bidirectional two-parametric method for solving a block-tridiagonal linear system whose coefficient matrix is not block diagonally dominant. The new algorithm require about the same amount of work as the two-parametric method for solving a block-tridiagonal system. But the new algorithm can concurrently compute the solution on both directions. If the coefficient matrix is centrosymmetric, our method is better than the two-parametric method.4)The forward elimination and backward substitution algorithm is also good for solving a diagonally dominant periodic tridiagonal linear system of equations. In this paper, we propose a bidirectional three-parametric method for solving a periodic block-tridiagonal linear system whose coefficient matrix is not block diagonally dominant. The new algorithm require about the same amount of work as the three-parametric method for solving a periodic block-tridiagonal system. But the new algorithm can concurrently compute the solution on both directions. The bidirectional three-parametric method is easily extended to solve the linear system whose coefficient matrix has the structure like "N". If the coefficient matrix is also centrosymmetric, our method can save nearly half computation load for solving it.5) J.-J. Climent and C. Perea [Appl. Math. Comput. 143 (2003) 1–14] introduced a generalized alternating iterative method, and established convergence results for weak nonnegative splittings of a monotone matrix, and for P-regular splittings of a symmetric positive definite matrix. In this paper, we establish convergence results for H-splittings of a nonsingular H-matrix and give an upper bound to the spectral radius of iterative matrix. Also, we establish some comparison theorems of spectral radius of the iterative matrix on a generalized alternating iterative method for weak nonnegative splittings of a monotone matrices linear system. Moreover, we give some numerical examples to show our results.6) J.-J. Climent, C. Perea et al.[Appl. Math. and Comp., Vol.148, (2004), pp.497-517] introduced two models of parallel multisplitting nonstationary iterations for solving the system of linear equations Ax = b. It is shown that when matrix A is monotone and the multisplittings are weak nonnegative of the first or second type, both models lead to convergent schemes. When matrix A is symmetric positive definite and the multisplittings are P-regular, the schemes are also convergent. We give convergence theorems for both models of parallel multisplitting nonstationary iterations if A is a nonsingular H-matrix and the multisplittings are H-splittings. Furthermore, we give one example to show our results.