Abstract
This paper surveys software for the solution of sparse sets of linear equations. In particular we examine codes which can be used to solve equations arising in the solution of elliptic partial differential equations.
In this paper, we are not concerned with algorithms per se but rather their embodiment in robust, portable, well-documented code. They may, to some extent, be considered as alternatives and competitors to multigrid techniques.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ajiz, M.A. and Jennings, A. (1981). A robust ICCG algorithm. Report Civil Eng. Dept., Queen’s University, Belfast.
Buzbee, B.L., Golub, G.H. and Nielson, C.W. (1970). On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, pp.627–656.
Concus, P. and Golub, G.H. (1975). A generalized conjugate gradient method for nonsymmetric systems of linear equations. Comput. Sci. Dept. Stanford Report STAN-CS-75-535. Presented at IRIA Meeting, Paris, December 1975.
Detyna, E. (1981). Rapid elliptic solvers. In Sparse Matrices and their Uses. I.S. Duff (Ed.), Academic Press pp.245–264.
Dongarra, J.J., Bunch, J.R., Moler, C.B. and Stewart, G.W. (1979). LINPACK Users’ Guide. SIAM Press.
Duff, I.S. (1977a). A survey of sparse matrix research. Proc. IEEE 65, pp.500–535.
Duff, I.S. (1977b). MA28 — a set of Fortran subroutines for sparse unsymmetric linear equations. AERE Report R.8730, HMSO, London.
Duff, I.S. (1979). Practical comparisons of codes for the solution of sparse linear systems. In Sparse Matrix Proceedings 1978. I.S. Duff and G.W. Stewart (Eds.). SIAM Press pp.107–134.
Duff, I.S. (1981a). MA32 — A package for solving sparse unsymmetric systems using the frontal method. AERE Report R.10079, HMSO, London.
Duff, I.S. (1981b). ME28 — A sparse unsymmetric linear equation solver for complex equations. ACM Trans. Math. Softw. 7, pp.505–511.
Duff, I.S. (1982). A survey of sparse matrix software. AERE Report CSS 121. To appear in Sources and Development of Mathematical Software. W.R. Cowell (Ed.). Prentice-Hall.
Duff, I.S. and Reid, J.K. (1979). Performance evaluation of codes for sparse matrix problems. In Performance Evaluation of Numerical Software. L. Fosdick (Ed.), North Holland, pp.121–135.
Duff, I.S. and Reid, J.K. (1982). The multi-frontal solution of indefinite sparse symmetric linear systems. AERE Report CSS 122.
Eisenstat, C.S., Gursky, M.C., Schultz, M.H. and Sherman, A.H. (1977). Yale sparse matrix package I. The symmetric codes. II. The nonsymmetric codes. Reports 112 and 114. Dept. Computer Science, Yale University.
Felippa, C.A. (1975). Solution of linear equations with skyline-stored symmetric matrix. Computers and Structures 5, pp.13–29.
Fletcher, R. (1976). Conjugate gradient methods for indefinite systems. In Numerical Analysis, Dundee (1975). Lecture note 506. G.A. Watson (Ed.), pp.73–89.
Foerster, H. and Witsch, K. (1982). Multigrid software for the solution of elliptic problems on rectangular domains: MGOO (Release 1). These proceedings.
George, A. (1981). Direct solution of sparse positive definite systems: some basic ideas and open problems. In Sparse Matrices and their Uses. I.S. Duff (Ed.). Academic Press, pp.283–306.
George, A. and Liu, J.W.-H. (1981). Computer Solution of Large Sparse Positive Definite Systems. Prentice Hall.
George, A., Liu, J. and Ng, E. (1980). User guide for SPARSPAK: Waterloo Sparse Linear Equations Package. Research Report CS-78-30 (revised Jan.1980). Department of Computer Science. University of Waterloo.
Golub, G. and Kahan, W. (1965). Calculating the singular values and pseudoinverse of a matrix. J. SIAM Numer. Anal. 2, pp.205–224.
Grimes, R.G., Kincaid, D.R. and Young, D.M. (1980). ITPACK 2A: A Fortran implementation of adaptive accelerated iterative methods for solving large sparse linear systems. Report CNA-164. Center for Numerical Analysis. University of Texas at Austin.
Hageman, L.A. and Young, D.M. (1981). Applied Iterative Methods. Academic Press.
Hasbani, Y. and Engelman, M. (1979). Out-of-core solution of linear equations with non-symmetric coefficient matrix. Computers and Fluids 7, pp.13–31.
Hestenes, M.R. and Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49, pp.409–436.
Hood, P. (1976). Frontal solution program for unsymmetric matrices. Int. J. Numer. Meth. Engng. 10, pp.379–399.
IBM (1976). IBM System/360 and System/370 IBM 1130 and IBM 1800. Subroutine Library-Mathematics. User’s Guide. Program Product 5736-XM7. IBM Catalogue SH12-5300-1.
Irons, B.M. (1970). A frontal solution program for finite element analysis. Int. J. Numer. Meth. Engng. 2, pp.5–32.
Jacobs, D.A.H. (1980). A summary of subroutines and packages (employing the strongly implicit procedure) for solving elliptic and parabolic partial differential equations. Report RD/L/N 55/80, Central Electricity Research Laboratories.
Jacobs, D.A.H. (1981a). The exploitation of sparsity by iterative methods. In Sparse Matrices and their Uses. I.S. Duff (Ed.). Academic Press, pp.191–222.
Jacobs, D.A.H. (1981b). Preconditioned conjugate gradient methods for solving systems of algebraic equations. Report RD/L/N 193/80. Central Electricity Research Laboratories.
Jennings, A. (1966). A compact storage scheme for the solution of symmetric linear simultaneous equations. Comput. J. 9, pp.281–285.
Jennings, A. (1977). Matrix Computation for Engineers and Scientists. Wiley.
Kuo-Petravic, G. and Petravic, M. (1979). A program generator for the incomplete LU decomposition-conjugate gradient (ILUCG) method. Computer Physics Comm. 18, pp.13–25.
Kuo-Petravic, G. and Petravic, M. (1981). A program generator for the Incomplete Cholesky Conjugate Gradient (ICCG) method with a symmetrizing preprocessor. Comput. Phys. Comm. 22, pp.33–48.
Manteuffel, T. (1977). The Tchebychev iteration for nonsymmetric linear systems. Numer. Math. 28, pp.307–327.
Markowitz, H.M. (1957). The elimination form of the inverse and its application to linear programming. Management Science 3, pp.255–269.
Meijerink, J.A. and van der Vorst, H.A. (1977). An iterative solution method for linear systems of which the coefficient matrix is a symmetrix M-matrix. Math. Comp. 31, pp.148–162.
Munksgaard, N. (1980). Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients. TOMS 6, pp.206–219.
O’Leary, D.P. and Widlund, O. (1981). Algorithm 572: Solution of the Helmholtz equation for the Dirichlet problem on general bounded three-dimensional regions. TOMS 7, pp.239–246.
Paige, C.C. and Saunders, M.A. (1975). Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, pp.617–629.
Paige, C.C. and Saunders, M.A. (1982). LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. To appear.
Parlett, B.N. (1978). A new look at the Lanczos algorithm for solving symmetric systems of linear equations. Lin. Alg. and its Applics. 29, pp.323–346.
Rice, J.R. (1978). ELLPACK 77. User’s Guide. Report CSD-TR-289. Computer Science Department, Purdue University.
Schumann, U. (Ed.) (1978). Computers, Fast Elliptic Solvers, and Applications. Advance Publications.
Sherman, A.H. (1978). Algorithm 533. NSPIV, a Fortran subroutine for sparse Gaussian elimination with partial pivoting. TOMS 4, pp.391–398.
Stone, H.L. (1968). Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. 5, pp.530–558.
Swarztrauber, P.N. (1977). The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle. SIAM Rev. 19, pp.490–501.
Swarztrauber, P.N. (1981). Vectorizing the FFT’s. To appear in Methods in Computational Physics. Volume on Parallel Algorithms. Lawrence Livermore Series.
Swarztrauber, P.N. and Sweet, R.A. (1979). Algorithm 541. Efficient Fortran subprograms for the solution of separable elliptic partial differential equations. TOMS 5, pp.352–364.
Thompson, E. and Shimazaki, Y. (1980). A frontal procedure using skyline storage. Int. J. Num. Meth. Engng. 15, pp.889–910.
Thomsen, et al (1977). SIRSM: Fortran package for the solution of sparse systems by iterative refinement. Report NI-77.13. Numerisk Institut, Lyngby, Denmark.
van der Vorst, H.A. and van Kats, J.M. (1979). Manteuffel’s algorithm with preconditioning for the iterative solution of certain sparse linear systems with a non-symmetric matrix. Report TR.11, ACCU, Utrecht.
van Kats, J.M. and van der Vorst, H.A. (1979). Software for the discretization and solution of second order self-adjoint elliptic partial differential equations in two dimensions. Technical Report TR10, ACCU, Utrecht.
van Kats, J.M., Rusman, C.J. and van der Vorst, H.A. (1980). ACCULIB documentation. Minimanual. Report No.9, ACCU, Utrecht.
Vinsome, P.K.W. (1976). Orthomin, an iterative method for solving sparse sets of simultaneous linear equations. Paper number SPE 5729. 4th SPE Symposium on Numerical Simulation of Reservoir Performance.
Young, D.M. and Kincaid, D.R. (1980). The ITPACK package for large sparse linear systems. Report CNA.160. Center for Numerical Analysis. University of Texas at Austin.
Zlatev, Z., Barker, V.A. and Thomsen, P.G. (1978). SSLEST: A Fortran IV subroutine for solving sparse systems of linear equations. User’s Guide. Report NI-78-01, Numerisk Inst., Lyngby, Denmark.
Zlatev, Z. and Wasniewski, J. (1978). Package Y12M — Solution of large and sparse systems of linear algebraic equations. Report No.24. Mathematisk Institut, Copenhagen, Denmark.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Duff, I.S. (1982). Sparse matrix software for elliptic PDE’s. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lecture Notes in Mathematics, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069936
Download citation
DOI: https://doi.org/10.1007/BFb0069936
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11955-5
Online ISBN: 978-3-540-39544-7
eBook Packages: Springer Book Archive