A numerical algorithm for asymptotically approximating the solution to periodic tridiagonal Toeplitz linear systems

New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems

The generalized minimal residual (GMRES) method is widely used to solve a linear system . This paper establishes upper and lower bounds for GMRES residuals for solving an tridiagonal Toeplitz linear system. For normal matrix A, this problem has been studied previously by Li [Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47(3) (2007), 577–599.]. Also, Li and Zhang [The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer. Math. 112 (2009), pp. 267–293.] for non-symmetric matrix A, presented upper bound for GMRES residuals. In fact, our main goal in this paper is to find the upper and lower bounds for GMRES residuals on normal tridiagonal Toeplitz linear systems, and lower bounds for residuals of GMRES on solving non-normal tridiagonal Toeplitz linear systems.

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Solution of Single Tridiagonal Linear Systems and Vectorization of the ICCG Algorithm on the Cray-1

Iterative methods such as generalized minimal residual (GMRES) method are used to solve large sparse linear systems. This paper is considered the GMRES method for solving tridiagonal block Toeplitz linear systems with diagonal blocks, and establishes upper bounds for GMRES residuals. The coefficient matrix becomes an m-tridiagonal Toeplitz matrix, and tridiagonal toeplitz systems are subcases of these systems. Also, we show that the GMRES method on linear system computes the exact solution in at most N steps.

The rate of convergence of GMRES on a tridiagonal toeplitz linear system. II

The numerical algorithms used to solve the physics equation in codes which model laser fusion are examined, it is found that a large number of subroutines require the solution of tridiagonal linear systems of equations. One dimensional radiation transport, thermal and suprathermal electron transport, ion thermal conduction, charged particle and neutron transport, all require the solution of tridiagonal systems of equations. The standard algorithm that has been used in the past on CDC 7600's will not vectorize and so cannot take advantage of the large speed increases possible on the Cray-1 through vectorization. There is however, an alternate algorithm for solving tridiagonal systems, called cyclic reduction, which allows for vectorization, and which is optimal for the Cray-1. Software based on this algorithm is now being used in LASNEX to solve tridiagonal linear systems in the subroutines mentioned above. The new algorithm runs as much as five times faster than the standard algorithm on the Cray-1. The ICCG method is being used to solve the diffusion equation with a nine-point coupling scheme on the CDC 7600. In going from the CDC 7600 to the Cray-1, a large part of the algorithm consists of solving tridiagonal linear systems on each L line of the Lagrangian mesh in a manner which is not vectorizable. An alternate ICCG algorithm for the Cray-1 was developed which utilizes a block form of the cyclic reduction algorithm. This new algorithm allows full vectorization and runs as much as five times faster than the old algorithm on the Cray-1. It is now being used in Cray LASNEX to solve the two-dimensional diffusion equation in all the physics subroutines mentioned above.

Periodic tridiagonal linear systems of equations typi- cally arise from discretizing second order differential equations with periodic boundary conditions. In this paper a parallel-vector algorithm is introduced to solve such systems. Implementation of the new algorithm is carried out on an Intel iPSC/2 hypercube with vector processor boards attached to each node processor. It is to be noted that t his algorithm can be extended to solve other periodic banded linear systems.

Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

In this paper, a new analysis technique in the study of dynamical systems with periodically varying parameters is presented. The method is based on the fact that all linear periodic systems can be replaced by similar linear time-invariant systems through a suitable periodic transformation known as the Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is suggested. In this procedure, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original differential problem to a set of linear algebraic equations from which the state transition matrix (STM) can be constructed over the period in closed form. Application of Floquet theory and eigen analysis to the resulting STM yields the L-F transformation matrix in a form suitable for algebraic manipulations. The utility of the L-F transformation in obtaining solutions of both linear and nonlinear dynamical systems with periodic coefficients is demonstrated. It is shown that the application of L-F transformation to free and harmonically forced linear periodic system directly provides the conditions for internal and combination resonances and external resonances, respectively. The application of L-F transformation to quasilinear periodic systems provides a dynamically similar quasilinear systems whose linear parts are time-invariant and the solutions of such systems can be obtained through an application of the time-dependent normal form theory. These solutions can be transformed back to the original coordinates using the inverse L-F transformation. Two dynamical systems, namely, a commutative system and a Mathieu type equation are considered to demonstrate the effectiveness of the method. It is shown that the present technique is virtually free from the small parameter restriction unlike averaging and perturbation procedures and can be used even for those systems for which the generating solutions do not exist in the classical sense. The results obtained from the proposed technique are compared with those obtained using the perturbation method and numerical solutions computed via a Runge-Kutta type algorithm. The technique is found to be very powerful in the analysis of linear and nonlinear periodic dynamical systems.

In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax=b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for residuals of CG, MINRES, and GMRES on solving a tridiagonal Toeplitz linear system, where A is Hermitian or just normal. These expressions and bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first or second kind.

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛ X −1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over A’s spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both A’s spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This paper will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind.

Solving diagonally dominant tridiagonal linear systems is a common problem in scientific high-performance computing (HPC). Furthermore, it is becoming more commonplace for HPC platforms to utilise a heterogeneous combination of computing devices. Whilst it is desirable to design faster implementations of parallel linear system solvers, power consumption concerns are increasing in priority. This work presents the oclspkt routine. The oclspkt routine is a heterogeneous OpenCL implementation of the truncated SPIKE algorithm that can use FPGAs, GPUs, and CPUs to concurrently accelerate the solving of diagonally dominant tridiagonal linear systems. The routine is designed to solve tridiagonal systems of any size and can dynamically allocate optimised workloads to each accelerator in a heterogeneous environment depending on the accelerator’s compute performance. The truncated SPIKE FPGA solver is developed first for optimising OpenCL device kernel performance, global memory bandwidth, and interleaved host to device memory transactions. The FPGA OpenCL kernel code is then refactored and optimised to best exploit the underlying architecture of the CPU and GPU. An optimised TDMA OpenCL kernel is also developed to act as a serial baseline performance comparison for the parallel truncated SPIKE kernel since no FPGA tridiagonal solver capable of solving large tridiagonal systems was available at the time of development. The individual GPU, CPU, and FPGA solvers of the oclspkt routine are 110%, 150%, and 170% faster, respectively, than comparable device-optimised third-party solvers and applicable baselines. Assessing heterogeneous combinations of compute devices, the GPU + FPGA combination is found to have the best compute performance and the FPGA-only configuration is found to have the best overall estimated energy efficiency.

A parallel algorithm solving a tridiagonal Toeplitz linear system