The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz...The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz vectors, which aredifferent from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In com-parison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylovsubspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi al-gorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.展开更多
Based on the critical unstable of both crystal and magnetic structure of Gd-intermetallic compound near the competition of two strongly independent subsystem （＂local 4f7＂ and ＂conduction electron concentration＂）...Based on the critical unstable of both crystal and magnetic structure of Gd-intermetallic compound near the competition of two strongly independent subsystem （＂local 4f7＂ and ＂conduction electron concentration＂）, a new QPT （quantum point transition） is predicted by calculation of： （1） The band structure and density of state by density functional theory where a strong narrowing fluidity of fermions around EF with shifted to negative value ＂-0.8 eV ＂is observable in the Gd-intermetalliccompound system while in the Y-case, it is not dominated. An antiferromagnetic state on the fluidity of conduction band can be investigated; （2） The internal magnetic field due to short range exchange interaction Jij between the nearest neighbor of local magnetic moment of stable s-state of Gd （L = 0） through the mean field approximation and of Eigenvalue-Eigenfunction ~.（k） are calculated. While a strong negative value of Jy is predicted, the eigenvalue L（k） of the system shows a strong antiferromagnetic phase in the reciprocal lattice direction 〈010〉, 〈001〉 in the correlation length 3.38 ~A. Although the antiferromagnetic state at Rc 〈_ 3.38 °A is a puzzle but it is completely dominated at Rc = 9 °A after passing through the competition between ）.λmin（O） and λmin（π） in the ranger of 3.2 °A 〈 Rc 〈 9 °A. Since both of the antiferromagnetic subsystems are sensitive to the predicted KF, the effect of decreasing of conduction electron is proposed to investigate, the change of the antiferromagnetic ordering state to the competition of ferromagnetic state （in direction 〈010〉） and antiferromagnetic state （in direction 〈001 〉 and 〈 100〉） resulted to paramagnetic state in the long range Rc = 9 °A.展开更多
文摘The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz vectors, which aredifferent from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In com-parison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylovsubspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi al-gorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.
文摘Based on the critical unstable of both crystal and magnetic structure of Gd-intermetallic compound near the competition of two strongly independent subsystem （＂local 4f7＂ and ＂conduction electron concentration＂）, a new QPT （quantum point transition） is predicted by calculation of： （1） The band structure and density of state by density functional theory where a strong narrowing fluidity of fermions around EF with shifted to negative value ＂-0.8 eV ＂is observable in the Gd-intermetalliccompound system while in the Y-case, it is not dominated. An antiferromagnetic state on the fluidity of conduction band can be investigated; （2） The internal magnetic field due to short range exchange interaction Jij between the nearest neighbor of local magnetic moment of stable s-state of Gd （L = 0） through the mean field approximation and of Eigenvalue-Eigenfunction ~.（k） are calculated. While a strong negative value of Jy is predicted, the eigenvalue L（k） of the system shows a strong antiferromagnetic phase in the reciprocal lattice direction 〈010〉, 〈001〉 in the correlation length 3.38 ~A. Although the antiferromagnetic state at Rc 〈_ 3.38 °A is a puzzle but it is completely dominated at Rc = 9 °A after passing through the competition between ）.λmin（O） and λmin（π） in the ranger of 3.2 °A 〈 Rc 〈 9 °A. Since both of the antiferromagnetic subsystems are sensitive to the predicted KF, the effect of decreasing of conduction electron is proposed to investigate, the change of the antiferromagnetic ordering state to the competition of ferromagnetic state （in direction 〈010〉） and antiferromagnetic state （in direction 〈001 〉 and 〈 100〉） resulted to paramagnetic state in the long range Rc = 9 °A.
