Practical preconditioners

Unfortunately, there is no criterion regarding the selection of the best preconditioner for a generic matrix. 

Very often, however, the physical problem that we are solving provides the most appropriate answer. It is, therefore, fundamental that the user exploits all the features of its problem. 

1) The preconditioner M is a band matrix obtained from the Hessian. 

2) The preconditioner M is a matrix obtained during a partial factorization of the matrix A by Cholesky by limiting as possible the filling-up problem (Nocedal and Wright, 2000). 

If several processors are available and if the compiler is predisposed to use the openMP directives, this will speed up the calculations. 

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The 9x9 matrix = o.l + 0.9 S,j has one eigenvalue close to 2. Use Davidson s method to find the eigenvalue and the corresponding eigenvector. Note In this case, the preconditioner (3.2.11) will be singular. Common practice is that whenever the preconditioning formula will require a divide by 0, it is replaced by 1. 

The Hessians of potential energy functions, for example, separate naturally into local terms (among atom pairs involved in bonds, bond angles, and dihedral angles) and nonlocal terms (among nonbonded atom pairs). The number of local terms increases linearly with n, whereas the nonlocal terms increase as n2. Thus, a preconditioner from the local terms is a good choice that has performed well in practice.23-82... 

The TN code in CHARMM uses a preconditioner from the local chemical interactions (bond length, bond angle, and dihedral-angle terms). This sparse matrix is rapid to compute and was found to be effective in practice. Other possibilities of preconditioners in general contexts have also been developed, such as a matrix derived from the BFGS update (defined in Section 6.1). ... 

This result is the motivation behind the use of a preconditioner, a transformation of the linear system into a related one whose condition number is closer to 1. This reduces the number of iterations necessary for iterative methods to converge to a solution, and is common practice in the numerical solution of BVPs. Let us choose some nonsingular upper-triangular matrix Mi defining a coordinate transformation,... 

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