We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.

Supplementary data are available at Bioinformatics online.

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The exponents method for calculating the concentrations of species in multimetal-multiligand systems is introduced. This method uses the NewtonRaphson method with restricted step iteration, which guarantees a monotonically decreasing objective function. Variable transformation and scaling are performed to avoid underflows and overflows during the calculations. A special linear solver using the eigenvalues and eigenvectors of the Jacobian matrix is implemented for overcoming disastrous singularity of this matrix, and the singular value decomposition method is applied for setting the initial guess. In addition, polynomial extrapolation is used for improving the performance when simulating a diagram of concentrations of species. The method was tested with 14 systems of different sizes over the whole pH range and presented robust and efficient behavior. 0 1995 by John Wiley & Sons, Inc. each ligand, and the protonic concentration are known as well as the stability constant for each of the metal-ligand complex reactions, then the problem is to calculate the concentration of metals and ligands in the unbound state. Afterward, the concentrations of the various species of metal-ligand complex can be determined. Conversely, when the equilibrium constants are unknown, they may be found known concentrations using repeated solutions of the forward problem.

ABSTRACT. This article presents improvements to the hybrid preconditioner previously developed for the solution through the conjugate gradient method of the linear systems which arise from interior-point methods. The hybrid preconditioner consists of combining two preconditioners: controlled Cholesky factorization and the splitting preconditioner used in different phases of the optimization process. The first, with controlled fill-in, is more efficient at the initial iterations of the interior-point methods and it may be inefficient near a solution of the linear problem when the system is highly ill-conditioned; the second is specialized for such situation and has the opposite behavior. This approach works better than direct methods for some classes of large-scale problems. This work has proposed new heuristics for the integration of both preconditioners, identifying a new change of phases with computational results superior to the ones previously published. Moreover, the performance of the splitting preconditioner has been improved through new orderings of the constraint matrix columns allowing savings in the preconditioned conjugate gradient method iterations number. Experiments are performed with a set of large-scale problems and both approaches are compared with respect to the number of iterations and running time.

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