THE APPLICATION OF THE PRECONDITIONED BICONJUGATE GRADIENT ALGORITHM TO NLTE RATE MATRIX EQUATIONS

This paper reports the success of the preconditioned biconjugate gradient (PBCG) and the conjugate gradient square (CGS) algorithms in solving the matrix equations resulting from the discretization of systems of population rate equations which arise in nonequilibrium kinetics modeling. The success of the PBCG and CGS can be attributed to two main ideas: First, the singularity of the rate matrix resulting from population conservation requirement was removed through a reduction of matrix order so as to improve the condition number of the matrix. Second, an efficient preconditioner was found to reduce the reduce the eigenvalue spread of the rate matrix. The preconditioning matrix was selected on the basis of retaining the largest few rates in each column of the well conditioned rate matrix. This preconditioner, along with the reduced rate matrix, enabled the algorithms to converge very rapidly so as to make them an attractive alternative to standard direct methods.