Abstract
We propose a way to use the Markowitz pivot selection criterion for choosing the parameters of the extended ABS class of algorithms to present an effective algorithm for generating sparse null space bases. We explain in detail an efficient implementation of the algorithm, making use of the special MATLAB 7.0 functions for sparse matrix operations and the inherent efficiency of the ANSI C programming language. We then compare our proposed algorithm with an implementation of an efficient algorithm proposed by Coleman and Pothen with respect to the computing time and the accuracy and the sparsity of the generated null space bases. Our extensive numerical results, using coefficient matrices of linear programming problems from the NETLIB set of test problems show the competitiveness of our implemented algorithm.
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Abaffy, J., Broyden, C.G., Spedicato, E.: A class of direct methods for linear equations. Numer. Math. 45, 361–376 (1984)
Abaffy, J., Broyden, C.G., Spedicato, E.: ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Equations. Ellis Horwood, Chichester (1989)
Berry, M.W., Heath, M.T., Kaneko, I., Lawo, M., Plemmons, R.J.: An algorithm to compute a sparse basis of the null space. Numer. Math. 47, 483–504 (1985)
Le Borne, S.: Block computation and representation of a sparse nullspace basis of a rectangular matrix. Linear Algebra Appl. 428, 2455–2467 (2008)
Chen, Z., Deng, N.Y., Xue, Y.: A general algorithm for underdetermined linear systems. In: The Proceedings of the First International Conference on ABS Algorithms, pp. 1–13 (1992)
Coleman, T.S., More, J.J.: Estimation of sparse Jacobian matrices and graph coloring problem. SIAM J. Numer. Anal. 20, 187–209 (1983)
Coleman, T.S., More, J.J.: Estimation of sparse Hessian matrices and graph coloring problem. Math. Program. 28, 243–270 (1984)
Coleman, T.F., Pothen, A.: The null space problem I. complexity. SIAM J. Algebr. Discrete Methods 7(4), 527–537 (1986)
Coleman, T.F., Pothen, A.: The null space problem II. algorithms. SIAM J. Algebr. Discrete Methods 8, 544–563 (1987)
Davis, T.A.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004)
Davis, T.A.: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia (2006)
Gay, D.M.: Electronic mail distribution of linear programming test problems. Mathematical Programming Society COAL Newsleter 13, 10–2 (1985)
Gilbert, J.R., Heath, M.T.: Computing a sparse basis for the null space. SIAM J. Algebr. Discrete Methods 8(3), 446–459 (1987)
Gilbert, R., Ng, E.G., Peyton, B.W.: Separators and structure prediction in sparse orthogonal factorization. Linear Algebra Appl. 262, 83–97 (1997)
Gill, P.E., Murray, M., Wright, M.H.: Practical Optimization. Academic Press, New york (1981)
Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H.: Maintaining LU factors of a general sparse matrix. Linear Algebra Appl. 88/89, 239–270 (1987)
Hall, C.: Numerical solution of Navier–Stokes problems by the dual variable method. SIAM J. Algebr. Discrete Methods 6, 220–236 (1985)
Heath, M.T., Plemmons, R.J., Ward, R.C.: Sparse orthogonal schemes for structural optimization using the force method. SIAM J. Sci. Statist. Comput. 5, 514–532 (1984)
Kim, K., Nazareth, J.L.: A primal null-space affine-scaling method. ACM Trans. Math. Softw. 20, 373–392 (1994)
Kaneko, I., Lawo, M., Thierauf, G.: On computational procedures for the force method. Int. J. Numer. Methods Eng. 19, 1469–1495 (1982)
Lu, S.M., Barlow, J.L.: Multifrontal computation with the orthogonal factors of sparse matrices. SIAM J. Matrix Anal. Appl. 17, 658–679 (2006)
Markowitz, H.M.: The elimination form of the inverse and its application to linear programming. Manage. Sci. 3, 255–269 (1957)
Matstoms, P.: Sparse QR factorization in MATLAB. ACM Trans. Math. Softw. 20, 136–159 (1994)
Plemmons, R., White, R.: Substructuring methods for computing the nullspace of equilibrium matrices. SIAM J. Matrix Anal. Appl. 11, 1–22 (1990)
Reid, J.K.: A sparsity exploiting variant of Bartels–Golub decomposition for linear programming bases. Math. Program. 24, 55–69 (1982)
Spedicato, E., Bodon, E., Xia, E., Mahdavi-Amiri, N.: ABS method for continuous and integer linear equations and optimization. Cent. Eur. J. Oper. Res 18(1), 73–95 (2010)
Spedicato, E., Bodon, E., Popolo, A., Mahdavi-Amiri, N.: ABS methods and ABSPACK for linear systems and optimization: a review. 4OR 1(1), 51–66 (2003)
Stern, J.M., Vavasis, S.A.: Nested dissection for sparse nullspace bases. SIAM J. Matrix Anal. Appl. 14, 766–775 (1993)
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 (1979, in German)
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Khorramizadeh, M., Mahdavi-Amiri, N. An efficient algorithm for sparse null space basis problem using ABS methods. Numer Algor 62, 469–485 (2013). https://doi.org/10.1007/s11075-012-9599-1
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DOI: https://doi.org/10.1007/s11075-012-9599-1