In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.
In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.
This paper presents a fabrication technology of enhancement-mode AlGaN/GaN high electron mobility transistors (HEMTs) using standard fluorine ion implantation. An 80 nm silicon nitride layer was deposited on the AlGaN as an energy-absorbing layer that slows down the high energy (similar to 25 keV) fluorine ions so that majority of the fluorine ions are incorporated in the AlGaN barrier. The threshold voltage was successfully shifted from -1.9 to +1.8 V, converting depletion mode HEMTs to enhancement-mode ones. The fluorine ion distribution profile was confirmed by Secondary Ion Mass Spectrometry (SIMS). (C) 2011 The Electrochemical Society. [DOI: 10.1149/1.3562273] All rights reserved.
A more accurate emission inventory of Black Carbon (BC) from China in 2000 was established based on county-level statistical data and recently published emission factors (EFs) from local measurements, which were further gridded at 0.5° × 0.5°. A comprehensive database for BC emission factors was compiled for main anthropogenic sources. BC emissions from China in 2000 were estimated to be 1228.52 Gg under normal operating conditions, and would increase to 2136.53 Gg if failures in control devices and combustion were considered. Spatial distribution of national BC emissions and emissions from different sources were determined; districts with extraordinarily high emissions cover 18.0% of China’s territory but generated 69.14% of the total emissions. Separate EFs were developed for each vehicle type fueled with gasoline or diesel; both the absolute value and relative share of BC emissions from vehicles in this work were higher than those in previous reports, suggesting that previous studies which did not differentiate vehicle types may have underestimated vehicle emissions.
