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We introduce the two‐sided Rayleigh quotient shift to the QR algorithm for non‐Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two‐sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double‐shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two‐sided 2D Grassmann–Rayleigh quotient double‐shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single‐shift and the Francis double‐shift. [ABSTRACT FROM AUTHOR]

We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift. [ABSTRACT FROM AUTHOR]

A new method is proposed for analyzing the nonlinear dynamics and stability in lane changes on highways for tractor- semitrailer under rainy weather. Unlike most of the literature associated with a simulated linear dynamic model for tractor- semitrailers steady steering on dry road, a verified 5DOF mechanical model with nonlinear tire based on vehicle test was used in the lane change simulation on low adhesion coefficient road. According to Jacobian matrix eigenvalues of the vehicle model, bifurcations of steady steering and sinusoidal steering on highways under rainy weather were investigated using a numerical method. Furthermore, based on feedback linearization theory, taking the tractor yaw rate and joint angle as control objects, a feedback linearization controller combined with AFS and DYC was established. The numerical simulation results reveal that Hopf bifurcations are identified in steady and sinusoidal steering conditions, which translate into an oscillatory behavior leading to instability. And simulations of urgent step and single-lane change in high velocity show that the designed controller has good effects on eliminating bifurcations and improving lateral stability of tractor-semitrailer, during lane changing on highway under rainy weather. It is a valuable reference for safety design of tractor-semitrailers to improve the traffic safety with driver-vehicle-road closed-loop system., 1. Introduction Tractor-semitrailer, as an articulated vehicle system, has complex nonlinear dynamic characteristics. Vehicle dynamics bifurcation mechanisms and driving stability have basically been confirmed based on nonlinear theory and tire [...]

A two‐sided Rayleigh quotient iteration is proposed for the periodic eigenvalue problem of a formal matrix product. Local convergence analysis is presented and the cubic local convergence rate is proved. Numerical examples are provided for testing the proposed method. [ABSTRACT FROM AUTHOR]

Flotation separation of scheelite from calcite is difficult because of the existence of the same cation in the minerals and similar physicochemical characteristics. Many studies have been performed to determine the selective depressants for calcite, but the separation of scheelite from calcite is still a big problem. In this article, the flotation separation of scheelite from calcite has been studied by flotation tests. The results show that benzohydroxamic acid has weak collecting ability to both scheelite and calcite. Lead ions can activate the flotation of scheelite and calcite, but the flotation separation of scheelite from calcite can’t be realized. The depressant saline sodium silicate (mixture of sodium silicate and lead ions) has selective depression effect on calcite and the optimum ratio of sodium silicate to lead ions is 3:1. The use of saline sodium silicate as a depressant can achieve the flotation separation of scheelite from calcite. The zeta potential results and adsorption studies showed that the pre-adsorption of saline sodium silicate only interferes with the adsorption of benzohydroxamic acid on calcite surface; this is the main reason for the selective depression effect of saline sodium silicate on calcite. [ABSTRACT FROM PUBLISHER]

In order to address the difficulty of arranging large calibration objects and the low measurement accuracy of small calibration objects in traffic accident scene photogrammetry, a photogrammetric method based on a composite of small calibration objects is proposed. Several small calibration objects are placed around the traffic accident scene, and the coordinate system of the composite calibration object is given based on one of them. By maintaining the relative position and coplanar relationship of the small calibration objects, the local coordinate system of each small calibration object is transformed into the coordinate system of the composite calibration object. The two-dimensional direct linear transformation method is improved based on minimizing the reprojection error of the calibration points of all objects. A rectified image is obtained using the nonlinear optimization method. The increased accuracy of traffic accident scene photogrammetry using a composite small calibration object is demonstrated through the analysis of field experiments and case studies. [ABSTRACT FROM AUTHOR]