Matrix 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... 

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 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... 

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 ... 

Since the matrix J J is symmetric, it is reasonable to think the system might be solved by either the Cholesky factorization or one of its variants. 

The computation of the matrices appearing in the formulae above is not necessary. An efficient computation avoids both the matrix inversion and matrix multiplications, replacing them with the solution of systems of equations and matrix-vector multiplications. Apart from the calculation of inertia terms and applied forces, the bulk of the computational effort in all formulations is constituted by the computation of the Cholesky factor of the matrix G. Once the factor is computed, formulations 1 and 3 require the solution of a single equation with matrix G, the others require two. For some systems it may be advisable to organise the computations in a different manner, solving instead equations with the indefinite system matrix... 

The special structure of a positive-definite matrix allows us to perform Cholesky factorization more quickly than LU decomposition. We start by writing A = Llfi exphcitly. 

For a positive-definite matrix A, the existence of the Cholesky factorization A = R R, allows us to use a preconditioner A mJMi, such that the transformed system... 

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. 

Suppose we factorize Gj using Cholesky (possible only if the matrix is positive definite) ... 

The preconditioner M is a matrix obtained during a partial factorization of the matrix A by Cholesky by limiting as possible the filling-up problem (Nocedal and Wright, 2000). 

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. 

If the matrix J was factorized QR, the UJ factorization, which is required to factorize the matrix J J with Cholesky, is already available. 

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