Introduction

For several years, we have been involved with the development and use of research-oriented programs using iterative algorithms for solving large sparse linear systems

Au = b

with positive diagonal elements. One solves for the N component unknown vector u given the ${\bf N \times N}$ nonsingular coefficient matrix A and the N component right-hand side vector b. The current ITPACK software package of subroutines, version 2C, provides for the use of seven alternative iterative procedures. While these subroutines are not designed as production software, they should successfully handle industrial problems of moderate size, that is, ones that fit in high-speed memory. This package is written in standard FORTRAN-66 code. It has been tested over a wide variety of computer systems using various FORTRAN compilers, including one which is FORTRAN-77 compatible (see Acknowledgements). The seven iterative solution modules are based on several basic iterative procedures, such as the Jacobi method, the Successive Overrelaxation (SOR) method, the Symmetric SOR (SSOR) method, and the RS method for the reduced system. With the exception of SOR, the convergence of these basic methods are accelerated by Chebyshev (Semi-Iteration, SI) or Conjugate Gradient (CG) acceleration. All methods are available with adaptive parameter estimation and automatic stopping tests. When using the RS method it is required that the linear system be reordered into a ``red-black"⁴ system [6,12]. A switch to compute, if possible, the red-black indexing, permute the linear system, and permute associated vectors is provided. The successful convergence of iterative methods may be dependent on conditions that are difficult to determine in advance. For example, determining whether the coefficient matrix is positive definite can be as costly to check as solving the system. On the other hand, some conditions affecting convergence, such as positive diagonal elements, diagonal dominance, and symmetry are relatively easy to verify. For some applications, the theory may not exist to guarantee the convergence of an iterative method. The algorithms in ITPACK have been tested most extensively for linear systems arising from elliptic partial differential equations. The routines can be applied, formally, to any linear system which fits in high-speed memory. However, rapid convergence, and indeed convergence itself cannot be guaranteed unless the matrix of the system is symmetric and positive definite. Success can be expected, though not guaranteed, for mildly nonsymmetric systems. In other words, iterative methods may not converge when applied to systems with coefficient matrices which are completely general with no special properties. This article discusses the usage of ITPACK and gives a few test results. The description of the iterative methods is given in [4]. The underlying theory on which the iterative algorithms are based is described in [6]. A survey of the iterative methods in ITPACK is presented in [11]. Throughout this paper, we adopt notation such as SOR() when referring to a subroutine and A(*) for a single-dimensioned array. The residual vector is b-Au⁽ⁿ⁾ for the linear system Au=b and the pseudo-residual vector is Gu⁽ⁿ⁾+k-u⁽ⁿ⁾ for a basic iterative method of the form u⁽ⁿ⁺¹⁾=Gu⁽ⁿ⁾+k. The smallest and largest eigenvalues of the iteration matrix G are denoted m(G) and M(G), respectively.