Iterative method direct

The filtered backprojection can be applied to 3D image reconstruction with some manipulations. The 3D data sinograms are considered to consist of a set of 2D parallel projections, and the FBP is applied to these projections by the Fourier method. The iteration methods also can be generally applied to the 3D data. However, the complexity, large volume, and incomplete sampling of the data due to the finite axial length of the scanner are some of the factors that limit the use of the FBP and iterative methods directly in 3D reconstruction. To circumvent these difficulties, a modified method of handling 3D data is commonly used, which is described below. 

For large Cl calculations, the frill matrix is not fonned and stored in the computer s memory or on disk rather, direct CF methods [ ] identify and compute non-zero and inunediately add up contributions to the sum jCj. Iterative methods [, in which approximate values for the Cj coefficients are refined tlirough sequential application of to the preceding estimate of the vector, are employed to solve... 

When three-point interpolation fails to yield a convergent calculation, you can request a second accelerator for any SCFcalculation via the Semi-empirical Options dialog box and the Ab Initio Options dialog box. This alternative method. Direct Inversion in the Iterative Subspace (DIIS), was developed by Peter Pulay [P. Pulay, Chem. Phys. Lett., 73, 393 (1980) J. Comp. Chem., 3, 556(1982)]. DIIS relies on the fact that the eigenvectors of the density and Fock matrices are identical at self-consistency. At SCF convergence, the following condition exists... 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

The CPHF equations are linear and can be determined by standard matrix operations. The size of the U matrix is the number of occupied orbitals times the number of virtual orbitals, which in general is quite large, and the CPHF equations are normally solved by iterative methods. Furthermore, as illustrated above, the CPHF equations may be formulated either in an atomic orbital or molecular orbital basis. Although the latter has computational advantages in certain cases, the former is more suitable for use in connection with direct methods (where the atomic integrals are calculated as required), as discussed in Section 3.8.5. 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

In this chapter economical direct and iterative methods are designed for numerical solution of difference elliptic equations. 

Direct and iterative methods. Recall that the final results of the difference approximation of boundary-value problems associated with elliptic equations from Chapter 4 were various systems of linear algebraic equations (difference or grid equations). The sizes of the appropriate matrices are extra large and equal the total number N of the grid nodes. For... 

In the two-dimensional case the iterative alternating direction method or the direct decomposition method turns out to be more economical, but for the multidimensional Dirichlet problem ATM is the most economical one among other available methods. This advantage is stipulated by the special structure of the operator A (see Chapter 4, Section 5) ... 

Gibrid (combined) methods. In mastering the difficulties involved in solving difference elliptic equations, some consensus of opinion is to bring together direct and iterative methods in some or other aspects as well as to combine iterative methods of various types (two-step methods). All the tricks and turns will be clarified for the iteration scheme... 

Solution of difference equations by direct or iterative methods selected on the basis of the economy criteria for the corresponding computational algorithms. 

Since the modified iterative method is completely numerical, data can be used directly from the monodisperse chromatograms to characterize the axial dispersion, eliminating the need for a specific axial dispersion function. The monodisperse standards were used to represent the spreading behavior for particle ranges as given in reference (27). 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

Although the evaluation of partial derivatives is not usually an insurmountable obstacle in networks involving one-phase flow in pipes, several investigators (C3, L2) have explored alternative iterative methods which do not require direct evaluation of partial derivatives. These methods are generally based on linearized approximations using secants rather than tangents. ... 

Sparse matrices are ones in which the majority of the elements are zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Duff, I. S., J. K. Reid, and A. M. Erisman (eds.), Direct Methods for Sparse Matrices, Clarendon Press, Oxford (1986) Saad, Y., Iterative Methods for Sparse Linear Systems, 2d ed., Society for Industrial and Applied Mathematics, Philadelphia (2003). The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n x n matrix. 

This equation can also be solved by an iterative method (128), however, the reliability of this method is in doubt (84) because it is the compressibility, not the specific volume or density that dominates (equation 23). This is also the reason why the precision of the sound speeds can only be compared directly with the precision of compressibility (not successive iterations of density). 

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