FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination

Abstract

In the numerical solution of boundary-value problems for ordinary differential equations, hnear systems with a partmular block structure are encountered. In this paper two packages of FORTRAN subroutines, COLROW and ARCECO, which use alternate row and column elimination for solving such systems, as discussed by Varah, are described Varah's procedure is stable, and introduces no fillin, that is, reqmres no additional storage To improve its efficiency, use is made of the fact that after each sequence of eliminations a reducible matrix is obtained. In addition, the packages presented treat systems with a more general structure than those considered by Varah. The results of numerical experiments, which demonstrate the effectiveness of the new packages and their superiority over a prevmusly pubhshed code, on a suitably restrmted class of problems, are presented

A computational study of finite element methods for second order linear two-point boundary value problems

“…However, these systems can be solved more efficiently using packages developed by the authors [8], [29]. The packages COLROW and ROWCOL, described in [8] and [29], respectively, use a variant of the procedure of alternate row and column ehmination described by Varah [23], in which row elimination with row pivoting (the usual method) is alternated with column ehmination with column pivoting, switching from one to the other when fill-in would occur otherwise. This procedure is stable and, in contrast to the procedure implemented in SOLVEBLOK, generates no fill-in.…”

“…This procedure is stable and, in contrast to the procedure implemented in SOLVEBLOK, generates no fill-in. Detailed descriptions of our implementations of alternate row and column elimination are given in [8] and [29] along with listings of the FORTRAN routines. 5.…”

A Preconditioned Conjugate Gradient Method for Nonselfadjoint or Indefinite Orthogonal Spline Collocation Problems

“…Since A 2 and B 2 are almost block diagonal, so is S i . Hence a linear system with S i can also be solved by two calls to the routine COLROW [15,16].…”

Modular Analysis of Sequential Solution Methods for Almost Block Diagonal Systems of Equations

“…The famous COLROW algorithm [2], which is highly regarded for its performance, was incorporated in several applications [3][4][5][6][7]. It utilizes Lam's alternating column/row pivoting [8] and Varah's correspondingly alternating column/row scalar elimination [9].…”

“…This approach was proposed by Varah [9] in his LGU decomposition method. It was developed afterwards into a more efficient LU version-termed here Scalar Column/Scalar Row (SCSR) Elimination-that was adopted by the COLROW solver [2].…”