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Numerische Mathematik

, Volume 47, Issue 4, pp 483–504 | Cite as

An algorithm to compute a sparse basis of the null space

  • M. W. Berry
  • M. T. Heath
  • I. Kaneko
  • M. Lawo
  • R. J. Plemmons
  • R. C. Ward
Article

Summary

LetA be a realm×n matrix with full row rankm. In many algorithms in engineering and science, such as the force method in structural analysis, the dual variable method for the Navier-Stokes equations or more generally null space methods in quadratic programming, it is necessary to compute a basis matrixB for the null space ofA. HereB isn×r, r=n−m, of rankr, withAB=0. In many instancesA is large and sparse and often banded. The purpose of this paper is to describe and test a variation of a method originally suggested by Topcu and called the turnback algorithm for computing a banded basis matrixB. Two implementations of the algorithm are given, one using Gaussian elimination and the other using orthogonal factorization by Givens rotations. The FORTRAN software was executed on an IBM 3081 computer with an FPS-164 attached array processor at the Triangle Universities Computing Center and on a CYBER 205 vector computer. Test results on a variety of structural analysis problems including two- and three-dimensional frames, plane stress, plate bending and mixed finite element problems are discussed. These results indicate that both implementations of the algorithm yielded a well-conditioned, banded, basis matrixB whenA is well-conditioned. However, the orthogonal implementation yielded a better conditionedB for large, illconditioned problems.

Subject Classifications

AMS(MOS) 65F30 CR: G1.3 

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References

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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • M. W. Berry
    • 1
  • M. T. Heath
    • 2
  • I. Kaneko
    • 3
  • M. Lawo
    • 4
  • R. J. Plemmons
    • 1
  • R. C. Ward
    • 2
  1. 1.Departments of Mathematics and Computer ScienceNorth Carolina State UniversityRaleighUSA
  2. 2.Engineering Physics and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  3. 3.Department of CommerceHitotsubashi UniversityKunitachi, TokyoJapan
  4. 4.Department of Civil EngineeringUniversität Gesamthochschule, EssenEssen 1Federal Republic of Germany

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