IoIIIIIIIIIII EEEEEEEEEEEEEE EEEEEEEE

We describe a set of procedure7 for computing and updating an LU factorization of a sparse matrix A, where A may be square (possibly singular) or rectangular. The procedures include a Markowitz factorization and a Bartels-Golub update, similar to those of Reid (1976, 1982). The updates provided are addition, deletion or replacement of a row or column of A, and rank-one modification. (Previously, column replacement has been the only update available.) Various design features of the implementation (LUSOL) are described, and computational comparisons are made with the LA05 and MA28 packages of Reid (1976) and Duff (1977).

[1]  Michael A. Saunders,et al.  On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method , 1986, Math. Program..

[2]  P. Gill,et al.  A Note on Nonlinear Approaches to Linear Programming. , 1986 .

[3]  L. Foster.  Rank and null space calculations using matrix decomposition without column interchanges , 1986 .

[4]  Jack J. Dongarra,et al.  Distribution of mathematical software via electronic mail , 1985, SGNM.

[5]  William G. Poole,et al.  A Structurally Stable Modification of Hellerman–Rarick’s ${\text{P}}^4 $ Algorithm for Reordering Unsymmetric Sparse Matrices , 1985 .

[6]  Danny C. Sorensen,et al.  Analysis of Pairwise Pivoting in Gaussian Elimination , 1985, IEEE Transactions on Computers.

[7]  P. Gill,et al.  Sparse Matrix Methods in Optimization , 1984 .

[8]  John K. Reid,et al.  A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases , 1982, Math. Program..

[9]  Andrew H. Sherman,et al.  Algorithms for sparse Gaussian elimination with partial pivoting , 1978, TOMS.

[10]  Eli Hellerman,et al.  Reinversion with the preassigned pivot procedure , 1971, Math. Program..

[11]  John K. Reid,et al.  The Solution of Large Sparse Unsymmetric Systems of Linear Equations , 1971, IFIP Congress.

[12]  R. Bartels.  A stabilization of the simplex method , 1971 .

[13]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[14]  F. Ficken,et al.  The Simplex Method of Linear Programming. , 1961 .

[15]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

[16]  John G. Lewis,et al.  Proposed sparse extensions to the Basic Linear Algebra Subprograms , 1985, SGNM.

[17]  Jack J. Dongarra,et al.  A proposal for an extended set of Fortran Basic Linear Algebra Subprograms , 1985, SGNM.

[18]  Iain S. Duff,et al.  MA28 --- A set of Fortran subroutines for sparse unsymmetric linear equations , 1980 .

[19]  Eli Hellerman,et al.  The Partitioned Preassigned Pivot Procedure (P4) , 1972 .

[20]  J. Stoer,et al.  A Realization of the Simplex Method Based on Triangular Decompositions , 1971 .