For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such a...For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.展开更多
By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made in...By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made into a general program in Maple, which enables the complete discrimination system and complete root classification of a polynomial to be automatically generated by computer, without any human intervention. Besides, by using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.展开更多
The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of rea...The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.展开更多
Design approaches were proposed for both continuous and discrete LQI (linear quadratic with integral) controllers, if exist, to stabilize the inner loops in addition to stabilizing the closed loops and minimizing the ...Design approaches were proposed for both continuous and discrete LQI (linear quadratic with integral) controllers, if exist, to stabilize the inner loops in addition to stabilizing the closed loops and minimizing the quadratic cost functionals. Derived from the algebraic Riccati equations involved in continuous and discrete LQI control, the design approaches were straightforward obtained by setting the quadratic cost functionals to decoupled forms with positive definite weighting matrices. Examples were provided to verify the effectiveness of the approaches.展开更多
文摘For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.
基金Project supported by the "Scale" Plan and the "863" Plan of China.
文摘By establishing a complete discrimination system for polynomials, the problem of complete root classification for polynomials with complex coefficients is utterly solved, furthermore, the algorithm obtained is made into a general program in Maple, which enables the complete discrimination system and complete root classification of a polynomial to be automatically generated by computer, without any human intervention. Besides, by using the automatic generation of root classification, a method to determine the positive definiteness of a polynomial in one or two indeterminates is automatically presented.
基金Supported by the National Natural Science Foundation of China(11361074)the Foundation of Science and Technology Department of Guizhou Province([2015]7206)+1 种基金the Natural Science Programs of Education Department of Guizhou Province([2015]420)the Research Foundation of Guizhou Minzu University(15XRY004)
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371109, 11426075）, the Natural Science Foundation of Heilongjiang Province （No. QC2014C001）, and the Fundamental Research Funds for the Central Universities.
文摘The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.
基金The National High Technology Research and Development Program ( 863 ) of China(2006A-A09Z233)
文摘Design approaches were proposed for both continuous and discrete LQI (linear quadratic with integral) controllers, if exist, to stabilize the inner loops in addition to stabilizing the closed loops and minimizing the quadratic cost functionals. Derived from the algebraic Riccati equations involved in continuous and discrete LQI control, the design approaches were straightforward obtained by setting the quadratic cost functionals to decoupled forms with positive definite weighting matrices. Examples were provided to verify the effectiveness of the approaches.
