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Manuscript Title: ILUBCG2: a preconditioned biconjugate gradient routine for the solution of linear asymmetric matrix equations arising from 9-point discretizations.
Authors: A.E. Koniges, D.V. Anderson
Program title: ILUBCG2
Catalogue identifier: AALX_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 43(1987)297
Programming language: Fortran.
Computer: CRAY-1, CRAY X-MP.
Operating system: CTSS.
Word size: 64
Keywords: General purpose, Matrix, Partial differential eqs, Elliptic, Parabolic, Two-dimensional, Plasma physics, Implicit, Preconditioned conjugate, Gradient, Biconjugate gradient, Sparse asymmetric matrix, Mhd.
Classification: 4.8, 19.10.

Revision history:
Type Tit le Reference
adaptation 0001ILUBCG2-11 See below

Nature of problem:
Certain elliptic and parabolic partial differential equations that arise in plasma physics and other applications are solved in two dimensions. The implicit solution techniques used for these equations give rise to a system of linear equations whose matrix operator is sparse - often 9-banded - and generally asymmetric. We provide a fast algorithm for their solution.

Solution method:
An earlier matrix solver package used an incomplete L U decomposition preconditioning with a conjugate gradient iteration (Comp. Phys. Commun. 30(1983)31). We use the same preconditioning but instead use the biconjugate gradient (BDG) method. Although the BCG scheme requires the storage of two additional vectors, it converges in a significantly lesser number of iterations (often half), while the essential number of calculations remains the same.

Restrictions:
The discretization of the two-dimensional PDE and its boundary conditions must result in a spatial 9-point operator stencil which gives rise to a 9-banded matrix. The matrix must possess a reasonable amount of diagonal dominance for the preconditioning technique to be effective.

Unusual features:
The algorithm is arranged to produce a code similar to ILUCG2, and thus vectorization and optimization features for the Cray computers are retained.

Running time:
These are problem dependent because ill-conditioned matrices require more iterations than well-conditioned ones.

ADAPTATION SUMMARY
Manuscript Title: ILUBCG2-11: solution of 11-banded nonsymmetric linear equation systems by a preconditioned biconjugate gradient routine.
Authors: Y.-M. Chen, A.E. Koniges, D.V. Anderson
Program title: 0001ILUBCG2-11
Catalogue identifier: AALX_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 55(1989)359
Programming language: Fortran.
Computer: CRAY-1, CRAY X-MP.
Operating system: CTSS.
Word size: 64
Classification: 4.8, 19.10.

Nature of problem:
Certain elliptic and parabolic partial differential equations (PDE) that arise in fluid mechanics and other applications are solved in two dimensions. The implicit solution techniques used for these equations give rise to a system of linear equations whose matrix operator is sparse-often 9-banded-and generally nonsymmetric. For problems with periodic boundary conditions in one of the computational directions, the resulting matrix operator becomes 11-banded. We extend the ILUBCG2 code for solving linear equations having 9-banded matrices to those having 11-bands.

Solution method:
An earlier matrix solver package used an incomplete L-U decomposition preconditioning with a conjugate gradient iteration. We use the same preconditioning but instead use the biconjugate gradient (BCG) method. Although the BCG scheme requires the storage of two additional vectors, it converges in a significantly lesser number of iterations (often half), while the essential number of calculations per iteration remains the same.

Restrictions:
The discretization of the two-dimensional PDE and its boundary conditions must result in a spatial 9-point operator stencil which gives rise to an 11-banded matrix. The matrix must be sufficiently diagonally dominant for the preconditioning technique to be effective.

Unusual features:
The algorithm is arranged to produce a code similar to ILUBCG2, and thus vectorization and optimization features for the Cray computers are retained.

Running time:
These are problem dependent because ill-conditioned matrices require more iterations than well-conditioned ones.