PC-based implementation of Lanczos algorithm using packed and column pointer data structures for solving linear problems (PCBLAN)

Abstract

In the very beginning this method was used to find the solutions of vibrational frequencies in many engineering problems. Later on it was found that it is a powerful tool (Ajiz, PhD thesis, Queen's University, Belfast, 1982) not only to find the vibrational frequencies but also the solution of linear systems of equations [Ajiz, M.A., PhD thesis, Queen's University Belfast, 1982; Ajiz, M.A. and Jennings, A., Int. J. Numer. Method. Engng. 20 (1984) 949-66], particularly when the data structures are matrices with a small number of nonzeros as compared to zero elements [Jennings, A., Matrix Computations for Engineers and Scientists. John Wiley, 1977; Ajiz, M.A., Int. J. Comput. Struct., 28(2) (1988) 185-94; 31(4) (1989) 643-6; 38(2) (1991) 227-37; Jennings, A. and Ajiz, M.A., SIAM J. Sci. Stat. Comput., 5(4) (Dec. 1984) 978-87]. An algorithm is presented in this paper and is implemented using the above data structures which declares itself more efficient than the algorithm mentioned in (Wirth, N., Algorithms and Data Structures. Prentice Hall, 1986; Aho, A. V. et al., Data Structures and Algorithms. Addison-Wesley, 1983). It converts the original matrix into a tridiagonal matrix by taking full advantage of sparsity patterns of the original matrix. This tridiagonal matrix is further decomposed using Choleski root-free decomposition (Wirth, N., Algorithms and Data Structures. Prentice Hall, 1986; Aho, A.V. et al., Data Structures and Algorithms. Addison-Wesley, 1983). More than one right hand side can be processed at one time.