On Solving Pentadiagonal Linear Systems via Transformations

Many authors have studied numerical algorithms for solving the linear systems of pentadiagonal type. The well-known fast pentadiagonal system solver algorithm is an example of such algorithms. The current paper describes new numerical and symbolic algorithms for solving pentadiagonal linear systems via transformations. The proposed algorithms generalize the algorithms presented in El-Mikkawy and Atlan, 2014. Our symbolic algorithms remove the cases where the numerical algorithms fail. The computational cost of our algorithms is better than those algorithms in literature. Some examples are given in order to illustrate the effectiveness of the proposed algorithms. All experiments are carried out on a computer with the aid of programs written in MATLAB

Generalized scans and tridiagonal systems

AbstractMotivated by the analysis of known parallel techniques for the solution of linear tridiagonal system, we introduce generalized scans, a class of recursively defined length-preserving, sequence-to-sequence transformations that generalize the well-known prefix computations (scans). Generalized scan functions are described in terms of three algorithmic phases, the reduction phase that saves data for the third or expansion phase and prepares data for the second phase which is a recursive invocation of the same function on one fewer variable. Both the reduction and expansion phases operate on bounded number of variables, a key feature for their parallelization. Generalized scans enjoy a property, called here protoassociativity, that gives rise to ordinary associativity when generalized scans are specialized to ordinary scans. We show that the solution of positive-definite block tridiagonal linear systems can be cast as a generalized scan, thereby shedding light on the underlying structure enabling known parallelization schemes for this problem. We also describe a variety of parallel algorithms including some that are well known for tridiagonal systems and some that are much better suited to distributed computation

Load-balanced parallel banded-system solvers

AbstractSolving banded systems is important in the applications of science and engineering. This paper presents a load-balancing strategy for solving banded systems in parallel when the number of processors used is small. An optimization-based load-balancing analysis is given to determine how many loads should be assigned to each processor in order to minimize the time requirement. Some experimentations are carried out on the nCUBE 2E multiprocessor to demonstrate the speedup advantage of the proposed load-balancing strategy. The speedup improvement ratio ranges from 47% to 66% (from 12% to 24%) when using 4 (8) processors

Design and evaluation of tridiagonal solvers for vector and parallel computers

Postprint (published version