Cholesky factorization

This technique (also known as the Grout reduction or Cholesky factorization) is based on the transfonnation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as... 

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). 

A simpler procedure that may result in a suitable value of / is to apply a modified Cholesky factorization as follows ... 

A set of nonlinear parameters Aj, in general case, is unique for each function To satisfy the requirement of square integrability of the wave function, each matrix must be positively defined. It imposes certain restrictions on the values that the elements of matrix A may take. To ensure the positive definiteness and to simplify some calclations, it is very convenient to represent matrix A in a Cholesky factored form. 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

The preconditioner is problem dependent and should be chosen in large-scale applications as a sparse approximation to H that can be factored rapidly. A Cholesky factorization of a positive-definite matrix M produces... 

When the preconditioner is constructed from a natural separability of the problem into terms of differing complexity, it may not necessarily be positive-definite, as required for straightforward implementation of PCG. A very useful technique for optimization has been a replacement of the standard Cholesky factorization of positive-definite systems by a modified Cholesky (MC) al-gorithm.5 137139 The MC process detects indefiniteness during the factorization itself and produces a decomposition for... 

We assume that the function value and gradient are evaluated together in an operations (additions and multiplications), where n is the problem size and a is a problem-dependent number. The Hessian can then be computed in (a/2)n(n + 1) operations. When a sparse preconditioner M is used, we denote its number of nonzeros by m and the number of nonzeros in its Cholesky factor, L, by /. (We assume here that M either is positive-definite or is factored by a modified Cholesky factorization.) Thus M can be computed in about (a/2)nm operations. As discussed in the previous section, it is advantageous to reorder the variables a priori to minimize the fill-in for M. Alternatively, a precon-... 

E. Eskow and R. B. Schnabel, Software for a New Modified Cholesky Factorization. Computer Science Department Technical Report CU-CS-443-89, University of Colorado, Boulder, 1989. 

T. Schlick, SMM ]. Sci. Statist. Comput., in press. Modified Cholesky Factorizations for Sparse Preconditioners. 

Here r is a 3n x 1 vector of Cartesian coordinates for the n particles, Lk is an n x n lower triangular matrix of rank n and I3 is the 3x3 identity matrix, k would range from 1 to A where N is the number of basis functions. The Kronecker product with I3 is used to insure rotational invariance of the basis functions. Also, integrals involving the functions k are well defined only if the exponent matrix is positive definite symmetric this is assured by using the Cholesky factorization LkL k. The following simplifications will help keep the notation more compact ... 

This is achieved through Cholesky factorization, which is a method to simulate multivariate normal returns, based on the assumption that the covariance matrix is symmetric and positive-definite. It is used to ensure the simulated series have a certain desired correlation. 

In view of the fact that the matrix iV is a product of the matrix A and its transpose, it follows that matrix N is symmetric and positive definite, so in the following only triangular decomposition of a symmetric and positive-definite matrix, called Cholesky factorization, will be discussed ... 

Miller" (p. 6), using an argument based on the Cholesky factorization of X X, shows that... 

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