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Splitting dense columns in sparse linear systems. (English) Zbl 0727.65034
Let $$A=[S,D]$$ where S and D denote respectively the sparse and dense columns of a matrix A. The author gives an efficient and robust method for solving $$AA^ Tx=b.$$ The proposed method avoids the rank-deficiency problem that is common to the present algorithms and also makes an effective use of sparse matrix techniques. Applications to interior-point methods for linear programming and non-symmetric matrices are pointed out.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 65F50 Computational methods for sparse matrices 65K05 Numerical mathematical programming methods
ALPO
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##### References:
 [1] Gill, P.E.; Murray, W.; Saunders, M.A.; Tomlin, J.A.; Wright, M.H., On projected Newton methods for linear programming and an equivalence to Karmarkar’s projective method, Math. programming, 36, 183-209, (1986) · Zbl 0624.90062 [2] Adler, I.; Karmarkar, N.K.; Resende, M.G.C.; Veiga, G., An implementation of Karmarkar’s algorithm for linear programming, Math. programming, 44, 297-335, (1989) · Zbl 0682.90061 [3] Adler, I.; Karmarkar, N.K.; Resende, M.G.C.; Veiga, G., Data structures and programming techniques for the implementation of Karmarkar’s algorithm, ORSA J. comput., 1, 84-106, (1989) · Zbl 0752.90043 [4] Mehrotra, S., Implementations of affine scaling methods: approximate solutions of systems of linear equations using preconditioned conjugate gradient methods, () · Zbl 0782.90067 [5] Choi, I.C.; Monma, C.L.; Shanno, D.F., Further development of a primal-dual interior point method, () · Zbl 0757.90051 [6] Gill, P.E.; Murray, W.; Saunders, M.A., A single-phase dual barrier method for linear programming, () · Zbl 0815.65080 [7] Lustig, I.J.; Marsten, R.E.; Shanno, D.F., Computational experience with a primal-dual interior point method for linear programming, () · Zbl 0731.65049 [8] Marsten, R.E.; Saltzman, M.J.; Shanno, D.F.; Pierce, G.S.; Ballintijn, J.F., Implementation of a dual interior point algorithm for linear programming, () · Zbl 0752.90046 [9] McShane, K.A.; Monma, C.L.; Shanno, D.F., An implementation of a primal-dual interior point method for linear programming, ORSA J. comput., 1, 70-83, (1989) · Zbl 0752.90047 [10] Lustig, I.J.; Marsten, R.E.; Shanno, D.F., On implementing Mehrotra’s predictor-corrector interior point method for linear programming, () · Zbl 0771.90066 [11] Vanderbei, R.J., {\scalpo}: another linear program optimizer, () · Zbl 0777.90031 [12] Lustig, I.J.; Mulvey, J.M.; Carpenter, T.J., Formulating stochastic programs for interior point methods, () · Zbl 0739.90048 [13] Vanderbei, R.J., A brief description of {\scalpo}, () · Zbl 0446.90045 [14] Gay, D.M., Electronic mail distribution of linear programming test problems, Math. programming soc. COAL newslett., (1985)
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