Cholesky algorithm

In models of beads with full hydrodynamic interactions, for which the mobility tensor is represented by a dense matrix, the Cholesky decomposition of H requires 3N) /6 operations. Eor large N, this appears to be the most expensive operation in the entire algorithm. The only other unavoidable 0 N ) operation is the LU decomposition of the K x K matrix W that is required to solve for the K constraint forces, which requires /3 operations, or roughly... 

Cholesky s decomposition [56a,b] has been used to obtain S in a practical manner. The reason to choose Cholesky s algorithm is found essentially in the numerical stability of this procedure, but alternatively in the possible definition of a recursive pathway to evaluate S [56c], starting from a small number of functions up to any... 

A possible recursive algorithm, adapted to the already described Cholesky Proceduras 2 and 3, in order to build up the coefficients of the linear combination (5.2) can be described as in section 4.2 above. In this preliminary stage the variable m involved in Procedure 3 is taken as unity, that is only one function is added at each computational step. But, in general, m functions can be added at any time and in this manner has been the Cholesky algorithm described. 

Two center density expansions into separate centers have been employed successfully in order to overcome the many center integral problem. Here it is also proved how such a naive but elegant algorithm, based essentially on a recursive Cholesky decomposition, am be well adapted to modem computational hardware architectures. CETO functions appear in this manner as a plausible alternative to the present GTO quantum chemical computational flood, constituting the foundation of another step signaling the path towards STO integral calculation. 

Since the FCI corrections are based on different truncated virtual orbital spaces, it is very important to have an efficient algorithm for the transformation of two-electron integrals. A Cholesky decomposition of the two-electron matrix is then very convenient [31, 32], A two-electron integral [juv Ao] is related to the integral tables (obtained by the Cholesky decomposition) L t = 1,rs by the relation n... 

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. 

Many special linear systems Ax = b can be solved by special efficient algorithms [recall, for example, Choleski s method for symmetric positive-definite systems (Section in.G)]. Next, some typical special systems are considered (compare the end of Section II.D). 

PLDL factorization of an n x n symmetric matrix A (where P is a permutation matrix) can be computed, say, by Aasen s algorithm, using about /6 (rather than n /3) flops and 0 n ) comparisons for pivoting, even if A is not positive definite and has no Choleski s factorization. 

Usually, the Jacobian matrix of the system (7.38) is nonsymmetric. Thus, it is neither possible to solve the linear system by means of the Cholesky algorithm nor to halve memory allocation. The most efficient methods (Gauss or PLR variant) adopted for Jacobian factorization require twice as much time and memory allocation as the Cholesky algorithm. 

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. 

Special details of the numerical procedures are given in ref. [17,19]. The Cholesky code is not optimal, because the algorithm needs the values of the matrix recursively. This complicates vectorisation. The effective vector length is equal to the semi-bandwidth of the matrix A nd is greater than 200 only for large dimensions of A The iterative procedure can be vectorized much... 

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. ... 

R.WEINAR, W.PRZYBYJiO The BANACHIEWICZ-CHOLESKI algorithm for two level hypermatrices. V Conference on Computer Methods in Structural Mechanics of Polish Academy of Sciences, Karpacz 1981, Technological University of Wroclaw, Report No I-14/28/K9 Wrocjaw 1981, pp. 253-260. 

W.PRZYBYJiO, R.WEINAR Hypermatrix Block Frontal Solver for BANACHIEWICZ-CHOLESKI algorithm. Computational Mechanics 86.Theory and Applications. Proceedings of International Conference on Computational Mechanics, May 25-29, 1986, Tokyo. Editors G.YAGAWA, S.N. ATLURI. Springer-Verlag, Tokyo 1986, Volume 2, pp. 43-48. 

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