Cholesky method

Chaise-current four vector, 545 Chebyshev approximation, 96 Chebychev inequality, 124 Chemoft, H., 102,151 Cholesky method, 67 Circuit, 256 matrix, 262... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

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. 

If B has been factorized with the modified Cholesky method B = LiDiLf... 

The Cholesky method (Buzzi-Ferraris and Manenti, 2010a), which requires a symmetric positive definite matrix. 

The generating random process we used is based on a rather subtle mathematical technique that we cannot describe here. Basically, we start from a symmetric, positive definite, correlation matrix A from which we deduce an accessory matrix B using the Cholesky method. The required vector U whose the components are the correlated velocity fluctuations is then equal to the matrix B multiplied by a vector whose components are uncorrelated, centered, normal variables of variances unity. The procedure first designed for an lD formulation has been extended to 2D-problems. Mean turbulence inhomogeneities can be accounted for in the process. Details can be found in Desjonqu res, 1987, Berlemont, 1987, Gouesbet et al, 1987, Berlemont et al, 1987, Desjonqu res et al, 1987. 

Using the Cholesky method, solve the following equations ... 

One approach to solve the linear Equations 5.12 is the method of Gill and Murray that uses the Cholesky factorization of H as in the following (Gill and Murray, 1974 Scales, 1985) ... 

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. 

The Gill-Murray modified Newton s method uses a Cholesky factorization of the Hessian matrix (Gill and Murray, 1974). The method is described in detail by Scales (1985). 

Discussion This is a general purpose subroutine for matrix inversion with accompanying solution of linear equations, using Choleski s method for a symmetric matrix. It has three options, indicated by the dummy variable M. In the chemical equilibrium program only option M = —1 is used. 

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. 

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. 

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