Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

Thermosolutal LTNE Porous Mixed Convection Under the Influence of the Soret Effect

The effects of horizontal pressure gradient and Soret coefficient on the onset of double-diffusive convection in a fluid-saturated porous layer under the influence of local thermal nonequilibrium (LTNE) temperatures are analyzed. Darcy's law with local acceleration term, which involves the two-field temperature model describing the fluid and solid phases separately and the approximation of Oberbeck-Boussinesq, is used. The dynamics of small-amplitude perturbations on the basic mixed convection flow is studied numerically. Using the Galerkin method along with the QZ-algorithm, the eighth order eigenvalue differential equation obtained by employing linear stability analysis is solved. The solution provides the neutral stability curves and determines the threshold of linear instability, and the critical values of thermal Darcy-Rayleigh number, wave number, and the frequency at the onset of instability are determined for various values of control parameters. It is found that, rather than the stationary motion, the instability is found to be via oscillatory motion. Besides, the contribution to each parameter on stability characteristics is explored in detail, and some relevant findings have been described that have not been reported hitherto in the literature.

Stability Analysis of MHD Fluid Flow over a Moving Plate with Pressure Gradient Using the Chebyshev Spectral Method

The purpose of this paper is to investigate the magnetohydrodynamic stability external flow of a viscous, incompressible and electrically conducting fluid over a moving flat plate using temporal linear stability analysis. Using a similarity variables based on the Skan-Falkner transformation, the governing differential equations of mean flow are transformed into a nonlinear ordinary differential equation, which is then solved numerically by the Runge-Kutta method. The MHD stability equation is solved numerically by using the Chebyshev spectral collocation method, which is based on the eigenfunction expansion in terms of Chebyshev polynomials, collocation points, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ algorithm. The influence of the pressure gradient, magnetic field and wall velocity on the dimensionless mean velocity are presented graphically and discussed. For the disturbance flow in the presence of magnetic field and moving wall, the critical Reynolds number increases with increasing the magnetic field parameter and wall velocity, indicating that these parameters stabilize the flow.