# An algorithm to compute a sparse basis of the null space

- 381 Downloads
- 39 Citations

## Summary

Let*A* be a real*m×n* matrix with full row rank*m*. 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 matrix*B* for the null space of*A*. Here*B* is*n×r, r=n−m*, of rank*r*, with*AB*=0. In many instances*A* 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 matrix*B*. 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 matrix*B* when*A* is well-conditioned. However, the orthogonal implementation yielded a better conditioned*B* for large, illconditioned problems.

## Subject Classifications

AMS(MOS) 65F30 CR: G1.3## Preview

Unable to display preview. Download preview PDF.

## References

- Berry, M., Plemmons, R.: Computing a banded basis of the null space on the Denelcor HEP multiprocessor. Proc. AMS/SIAM Summer Conference on Linear Algebra in Systems Theory. AMS Series on Contemp. Math. (to appear, 1985)Google Scholar
- Berry, M., Health, M., Plemmons, R., Ward, R.: Comparison of some orthogonal schemes for structural optimization. Transactions of the Army Research Office Conference on Applied Math. and Comp., pp. 477–485 (1983)Google Scholar
- George, A., Heath, M.: Solution of linear least squares problems using Givens rotations. Linear Algebra Appl.
**34**, 69–83 (1980)Google Scholar - George, A., Liu, J.: Computer solution of large sparse positive definite systems. Englewood Cliffs, N.J.: Prentice-Hall 1981Google Scholar
- George, A., Heath, M., Plemmons, R.: Solution of large-scale least squares problems using auxiliary storage. SIAM J. Sci. Stat. Comput.
**2**, 416–429 (1981)Google Scholar - Gill, P., Murray, W., Saunders, M., Wright, M.: Null space methods for large-scale quadratic programming. Paper presented at the Sparse Matrix Symposium 82, Fairfield Glade, Tenn (1982)Google Scholar
- Golub, G., Plemmons, R.: Large-scale least squares adjustment by dissection and orthogonal decomposition. Linear Algebra Appl.
**34**, 3–28 (1980)Google Scholar - Hall, C.: Numerical solution of Navier-Stokes problems by the dual variable method. SIAM J. Algebraic Discret. Methods
**6**, 220–236 (1985)Google Scholar - Heath, M.T., Plemmons, R.J., Ward, R.C.: Sparse orthogonal schemes for structural optimization using the force method. SIAM J. Sci. Stat. Comput.
**5**, 514–532 (1984)Google Scholar - Kaneko, I.: Block angular decomposition for structuring. Unpublished lecture notes, Dept. of Civil Eng., University of Essen, Federal Republic of Germany (1981)Google Scholar
- Kaneko, I., Plemmons, R.: Minimum norm solutions to linear elastic analysis problems. Int. J. Numer. Meth. Eng.
**20**, 983–998 (1984)Google Scholar - Kaneko, I., Lawo, M., Thierauf, G.: On computational procedures for the force method. Int. J. Numer. Meth. Eng.
**19**, 1469–1495 (1982)Google Scholar - Lawson, C., Hanson, R.: Solving Least Squares Problems. Englewood Cliffs, N.J.: Prentice-Hall 1974Google Scholar
- Rice, J.: PARVEC workshop on very large least squares problems and supercomputers. CSD-TR-464. Purdue University, Lafayette, Ind. (1983)Google Scholar
- Topcu, A.: A contribution to the systematic analysis of finite element structures using the force method. Doctoral Dissertation, University of Essen, Essen, Federal Republic of Germany (in German)Google Scholar
- Ursic, S.: The ellipsoid algorithm for linear inequalities in exact arithmetic. Paper presented at the Lin Alg. and Applic. Conf., Raleigh, N.C. (1982)Google Scholar