Tridiagonal matrix algorithm

Abstract: An optimized parallel algorithm is proposed to solve the problem occurred in the process of complicated backward substitution of cyclic reduction during solving tridiagonal linear systems. Adopting a hybrid parallel model, this algorithm combines the cyclic reduction method and the partition method. This hybrid algorithm has simple backward substitution on parallel computers comparing with the cyclic reduction method. In this paper, the operation count and execution time are obtained to evaluate and make comparison for these methods. On the basis of results of these measured parameters, the hybrid algorithm using the hybrid approach with a multi-threading implementation achieves better efficiency than the other parallel methods, i.e., the cyclic reduction and the partition methods. Among them, the cyclic reduction method is previously found to be the fastest algorithm in many ways for solutions. In particular, the approach involved in this paper has the least scalar operation count and the shortest execution time on multi-core computer when the size of an equation is large enough. The hybrid parallel algorithm improves the performance of the cyclic reduction and partition methods by 30% and 20% respectively.

Abstract: While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the eigenstates expansion of H(+) are precisely the kernel polynomials of those associated with H. This formalism is here applied to the case of the Morse oscillator which may have a finite discrete energy spectrum in addition to a continuous one. This completes the treatment of tridiagonal Hamiltonians with pure continuous energy spectrum, a pure discrete one, or a spectrum of mixed discrete and continous parts.

Abstract: We propose a system for the real-time computation of optical flow along contours of significant intensity change. Hildreth 1] formulated the energy functional for this problem and presented a conjugate gradient method to find the global minimum of the quadratic energy functional. For a contour withNpoints, the conjugate gradient method requiresO(N) iterations (i.e.,O(N2) operations) to converge to a solution. The direct analytical methods we present here require onlyO(N) operations. Using current desktop computing power (a Sun SPARCstation 10), the direct methods make it possible to compute the optical flow in real time.In the finite difference formulation of the problem, the structure of the coefficient matrix for open contours is block tridiagonal, and that for closed contours is cyclic block tridiagonal 2,3]. Therefore, it is natural to consider block extensions of the tridiagonal matrix solvers abundant in mathematics literature. This approach is graceful in that the properties of the tridiagonal matrix solvers carry over to the corresponding block tridiagonal solvers. Some of these properties are low computational complexityO(N) operations), high numerical stability, and parallelism. Based on these guiding principles, we propose block tridiagonal Gaussian elimination for open contours and the generalized Ahlberg?Nilson?Walsh method for closed contours. Assuming that the computation of Laplacian of Gaussian of the images in a sequence, and the detection of the image contours, can be done in real time using parallel hardware, the computation of optical flow using the two methods can be done in real time with common desktop hardware (we report results using a 70 MHz Sun SPARCstation 10). Both of these methods can be further speeded up by implementation on parallel hardware using a block generalization of Wang's partition method.

Abstract: In this paper we present an implementation of multisection and parallel bisection method on a transputer network for finding the eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix which lie in a specified interval ( a , b ). Although several similar studies in the literature have been reported significant speedups over the sequential versions of the algorithms, it remains to be determined which multiprocessor configuration is the most advantageous for these problems.