Cholesky decomposition method

The optimal choice of preconditioner will ultimately depend on the computer architecture, in as much as some are more readily vectorizable or parallelizable. For example, the initial incomplete Cholesky decomposition methods work well on serial machines, but the forward and backward substitutions are not vectorizable. Simpler decompositions, such as diagonal scaling, run faster on machines like the Cray YMP. More complicated, vectorizable variations of the incomplete Cholesky decompositions have been developed (see, e.g., ref. 24) and are currently under investigation for their applicability to problems in biomolecular electrostatics. Studies of multigridding techniques are also very exciting. 

The example in the previous section is repeated using the Cholesky Decomposition Method and the matrix L is given by ... 

This is a system of equations of the form Ax = B. There are several numeral algorithms to solve this equation including Gauss elimination, Gauss-Jacobi method, Cholesky method, and the LU decomposition method, which are direct methods to solve equations of this type. For a general matrix A, with no special properties such as symmetric, band diagonal, and the like, the LU decomposition is a well-established and frequently used algorithm. 

This is specifically for the FE method applied to elliptic problems. Such problems yielda matrix which is symmetric and positive definite. The Choleski decomposition only exists for symmetric, positive definite matrices. 

A similar 0(N ) method, presented by Angeles and Ma in [2], uses the concept of an orthogonal complement to construct the joint space inertia matrix. The Cholesky decomposition of this matrix is used in solving the appropriate linear system for the joint accelerations. The computational complexity of this algoithm is slightly better than that in [42], but the algorithm is still not the most efficient It, too, is restricted to configurations of simple revolute and prismatic joints. 

In recent years there has been a growing interest in numerical techniques which can speed up quantum chemical computations. Various methods are available to approximate the four-index electron repulsion integrals as products of three-index intermediates. These methods are called density fitting (DF) or resolution of the identity (Rl), and Cholesky decomposition (CD) techniques. A general comparison... 

Weigend F, Kattannek M, Ahlrichs R (2009) Approximated electron repulsion integrals Cholesky decomposition versus resolution of the identity methods. J Chem Phys 130 164106... 

A rather efficient method to calculate the root of the hydrodynamic interaction tensor is Cholesky decomposition. The random displacements are then obtained via multiplying the root matrix with a vector of random numbers. The root is usually not unique, i.e., there are several matrices whose square is the diffusion tensor, but since any of these matrices yields random displacements which satisfy the condition eq. (3.22), this nonuniqueness averages out in the course of the simulation. These matrix operations become numerically rather intensive if the number of monomers becomes large (the number of operations is proportional to the third power of the number of monomers). The numerical algorithms for Langevin equations are well established, however, some details are still discussed today. ... 

This so-called Cholesky decomposition contains the symmetric orthogonalization as a special case, when V equals U. Unfortunately. there is no simple way to obtain information on the constraint for the charges in a semiempirical method, because the nonorthogonal basis is actually avoided and enters only through the cut-off expansion of S (2 T, is again limits our possibility for a clear-cut back transformation. It should be of the following form ... 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

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