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An algorithm to compute a sparse basis of the null space (1985)

Berry, M. W. ; Heath, M. T. ; Kaneko, I. ; [et al.]

Springer

Numerische Mathematik 47 (1985), S. 483-504

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ISSN: 0945-3245

Keywords: AMS(MOS) ; 65F30 ; CR: G1.3

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01389453

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On computational procedures for the force method (1982)

Kaneko, I. ; Lawo, M. ; Thierauf, G.

Chichester [u.a.] : Wiley-Blackwell

International Journal for Numerical Methods in Engineering 18 (1982), S. 1469-1495

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ISSN: 0029-5981

Keywords: Engineering ; Engineering General

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mathematics , Technology

Notes: It is known that the matrix force method has certain advantages over the displacement method for a class of structural problems. It is also known that the force method, when carried out by the conventional Gauss-Jordan procedure, tends to fill in the problem data, making the method unattractive for large size, sparse problems. This poor fill-in property, however, is not necessarily inherent to the method, and the sparsity may be maintained if one uses what we call the Turn-Back LU Procedure. The purpose of this paper is two-fold. First, it is shown that there exist some close relationships between the force method and the least squares problem, and that many existing algebraic procedures to perform the force method can be regarded as applications/extensions of certain well-known matrix factorization schemes for the least squares problem. Secondly, it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint. Included in this unification is the Turn-Back LU Procedure, which was originally proposed by Topçu in his thesis.8 It is explained why this procedure tends to produce sparse and banded ‘self-stress’ and flexibility matrices with small band width. Some computational results are presented to demonstrate the superiority of the Turn-Back LU Procedure over the other schemes considered in this paper.

Additional Material: 11 Ill.