Iterative large linear system solution

Large Linear System Solution with Iterative Methods... 

Young, D. M. Iterative Solution for- Large Linear Systems. Academic, New York (1971). 

Young, D.M. Iterative solution of large linear systems. New York Academic Press 1971... 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

As for the solution of the linear system, the standard approach based on the inversion of D matrix (see equation (48)) becomes unmanageable for very large solutes due to both the computational time and the disk memory occupation it requires. To deal with these cases an iterative procedure has been developed, [112] which is able to solve equation (48) without defining and inverting the full D matrix. A specific two-step extrapolation technique proved very effective in the solution of this problem, especially for the PCM variant based on the normal... 

As the dimension of the CPKS linear system is too large to make direct methods of solution feasible, the CPKS equations are solved iteratively, and this is the most expensive phase of the second derivative calculation. Evaluation of the explicit second derivatives of the XC energy (the third term in Eq. (46)) is not insignificant and warrants careful implementation, as discussed in Ref. [68] however, as this step is not dominant, we will not discuss it here. Likewise, the matrix Q may be evaluated in terms of known quantities and needs to be done only once, and so for a discussion of its implementation the reader is directed to Ref. [68]. 

The computational requirements of the time stepping algorithm are concentrated in the solution of linear systems with the matrices and. For large-scale and accurate models, the dimension of these matrices will be very large. Therefore, it will be advantageous to solve them iteratively. Then we have to answer the following questions... 

Linearization and iteration The nonlinear system of equations, Eq. 57, is linearized and solved for a first estimate solution of [7], as discussed in connection with Eq. [39]. The solution is then inserted in the retained quadratic terms, and the linear system is solved for an improved estimate of the I7). This iterative procedure is repeated until the 7 converge within a desired tolerance. For the bond-stretch constraint, there is just one nonlinear (quadratic) term in its Taylor expansion (see later, Eq. [95]), and the linearization and iteration procedure is a fairly good approximation, justified even for relatively large corrections. For the bond-angle and torsional constraints, with infinite series Taylor representations, tighter limits are imposed on the allowable constraint... 

The FDA transforms a differential equation into a system of linear algebraic equations in which the unknowns are the function values at the grid points. Although the number of grid points and thus the number of equations required for acceptable accuracy can become very large, especially in 3D, the linear system is sparse. That is, most of the coefficients in any equation are zero since the FDA formulas involve function values at only a small number of neighboring points. Furthermore, the coefficient matrix has a banded structure of very simple form, except possibly near a boundary, which makes iterative methods a common choice for the solution of the linear equations. 

With these basic definitions in hand, we now begin to consider the solution of the linear system Ax = b, in which x, b and is an A x A/ real matrix. We consider here elimination methods in which we convert the linear system into an equivalent one that is easier to solve. These methods are straightforward to implement and work generally for any linear system that has a unique solution however, they can be quite costly (perhaps prohibitively so) for large systems. Later, we consider iterative methods that are more effective for certain classes of large systems. 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

It may appear as if this is no great improvement, since finding a solution to a linear equation system with direct methods requires about n3 operations, about half as many as the inversion. However, the solution of the linear equation system can be accomplished by iterative methods where, in each step, some product jv is formed. Superficially, this cuts down, the number of operations, but still requires the Jacobian to be computed and stored. However, for a very large class of important problems, such a product can be efficiently computed without the need of precalculating or storing the Jacobian. 

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