For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly s...For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations, and they may show advantages on problems that the HSS method is ineffective.展开更多
The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, ...The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.展开更多
基金the National Basic Research Program(Grant No.2005CB321702)The ChinaOutstanding Young Scientist Foundation(Grant No.10525102)the National Natural Science Foundation of China(Grant No.10471146)
文摘For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations, and they may show advantages on problems that the HSS method is ineffective.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10471074, 10771116)the Doctoral Program of the Ministry of Education of China (Grant No. 20060003003)
文摘The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.
