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
Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use t...Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.展开更多
Based on the NCEP/NCAR reanalysis daily mean temperature data from 1948 to 2005 and random time series of the same size,temperature correlation matrixes（TCMs） and random correlation matrixes（RCMs） are constructed ...Based on the NCEP/NCAR reanalysis daily mean temperature data from 1948 to 2005 and random time series of the same size,temperature correlation matrixes（TCMs） and random correlation matrixes（RCMs） are constructed and compared.The results show that there are meaningful true correlations as well as correlation＂noises＂in the TCMs.The true correlations contain short range correlations（SRCs） among temperature series of neighboring grid points as well as long range correlations（LRCs） among temperature series of different regions,such as the El Nino area and the warm pool areas of the Pacific,the Indian Ocean,the Atlantic,etc.At different time scales,these two kinds of correlations show different features：at 1-10-day scale,SRCs are more important than LRCs;while at 15-day-or-more scale,the importance of SRCs and LRCs decreases and increases respectively,compared with the case of 1-10-day scale.It is found from the analyses of eigenvalues and eigenvectors of TCMs and corresponding RCMs that most correlation information is contained in several eigenvectors of TCMs with relatively larger eigenvalues,and the projections of global temperature series onto these eigenvectors are able to reflect the overall characteristics of global temperature changes to some extent.Besides,the correlation coefficients（CCs） of grid point temperature series show significant temporal and spatial variations.The average CCs over 1950-1956,1972-1977,and 1996-2000 are significantly higher than average while that over the periods 1978-1982 and 1991-1996 are opposite,suggesting a distinctive oscillation of quasi-10-20 yr.Spatially,the CCs at 1-and 15-day scales both show band-like zonal distributions;the zonally averaged CCs at 1-day scale display a better latitudinal symmetry,while they are relatively worse at 15-day scale because of sea-land contrast of the Northern and Southern Hemisphere.However,the meridionally averaged CCs at 15-day scale display a longitudinal quasi-symmetry.展开更多
Let DD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(jj)|≥A_iA_j,i≠j,i,j∈N}.PD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(kk)|≥A_iA_jA_k,i≠j≠k,i,j,k∈N}. In this paper,we show DD_0(R)PD_0(R),and the conditions under which the nu...Let DD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(jj)|≥A_iA_j,i≠j,i,j∈N}.PD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(kk)|≥A_iA_jA_k,i≠j≠k,i,j,k∈N}. In this paper,we show DD_0(R)PD_0(R),and the conditions under which the numbers of eigen vance of A∈PD_0(R)\DD_0(R)are equal to the numbers of a_(ii),i∈N in positive and negative real part respectively.Some couter examples are given which present the condnions can not be omitted.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.
文摘Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.
基金Supported jointly by the National Natural Science Foundation of China under Grant Nos. 40930952, 40875040, and 40905034the National Basic Research Program of China under Grant No. 2006CB400503the National Science & Technology Support Program of China under Grant Nos. 2007BAC03A01 and 2007BAC29B01
文摘Based on the NCEP/NCAR reanalysis daily mean temperature data from 1948 to 2005 and random time series of the same size,temperature correlation matrixes（TCMs） and random correlation matrixes（RCMs） are constructed and compared.The results show that there are meaningful true correlations as well as correlation＂noises＂in the TCMs.The true correlations contain short range correlations（SRCs） among temperature series of neighboring grid points as well as long range correlations（LRCs） among temperature series of different regions,such as the El Nino area and the warm pool areas of the Pacific,the Indian Ocean,the Atlantic,etc.At different time scales,these two kinds of correlations show different features：at 1-10-day scale,SRCs are more important than LRCs;while at 15-day-or-more scale,the importance of SRCs and LRCs decreases and increases respectively,compared with the case of 1-10-day scale.It is found from the analyses of eigenvalues and eigenvectors of TCMs and corresponding RCMs that most correlation information is contained in several eigenvectors of TCMs with relatively larger eigenvalues,and the projections of global temperature series onto these eigenvectors are able to reflect the overall characteristics of global temperature changes to some extent.Besides,the correlation coefficients（CCs） of grid point temperature series show significant temporal and spatial variations.The average CCs over 1950-1956,1972-1977,and 1996-2000 are significantly higher than average while that over the periods 1978-1982 and 1991-1996 are opposite,suggesting a distinctive oscillation of quasi-10-20 yr.Spatially,the CCs at 1-and 15-day scales both show band-like zonal distributions;the zonally averaged CCs at 1-day scale display a better latitudinal symmetry,while they are relatively worse at 15-day scale because of sea-land contrast of the Northern and Southern Hemisphere.However,the meridionally averaged CCs at 15-day scale display a longitudinal quasi-symmetry.
文摘Let DD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(jj)|≥A_iA_j,i≠j,i,j∈N}.PD_0(R)={A∈C^(n×#)||Rea_(ii)Rea_(kk)|≥A_iA_jA_k,i≠j≠k,i,j,k∈N}. In this paper,we show DD_0(R)PD_0(R),and the conditions under which the numbers of eigen vance of A∈PD_0(R)\DD_0(R)are equal to the numbers of a_(ii),i∈N in positive and negative real part respectively.Some couter examples are given which present the condnions can not be omitted.
