Number Cholesky decomposition

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. ... 

Although the number of degrees of freedom has been minimized, this approach is computationally intensive, and imposes severe limitations on the size of the system that can be studied. Since every particle interacts with every other particle, the calculation of the mobility matrix scales as 0 N ), where N is the number of Brownian particles. In addition, the covariance matrix for the random displacements requires a Cholesky decomposition of the mobility matrix, which scales as 0 N ) [27]. The computational costs of Brownian dynamics are so large that even today one cannot treat more than a few hundred Brownian particles [28]. 

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... 

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