Other iterative methods

Other iterative methods apply equally well to problem (2). Among them the method with the recurrence relation... 

Three-layer iteration schemes. So far we have considered two-layer iteration schemes available for solving operator equations of the form Au = / with a self-adjoint operator A under the assumption that the spectral bounds and for the operator A are known in advance either in a space H or in a space Hb, where B = B > 0 is some stabilizator. Other iterative methods find a wide range of applications in some or other aspects. 

Of course, the roots of Eq. (46a) may easily be found. The unknown quantities fi and 2m may be found by substituting hm from Eq. (46a) into the imaging equations (18) and normalization equation (3). Because of the nonlinear nature of these equations, a Newton-Raphson (or other iterative) method of solution would be necessary. 

Other iterative methods have been proposed that are less well justified. For example, one approach [30] employs equations (48) and (49) directly, with the nth approximation for g(T) substituted into the integrals to obtain the (n + l)st approximations for A and g(z). A variational approach also has been developed [25], based on the introduction of / = as a new variable, so that equation (46) can be written in a standard form, d ildrj = — Aco/r], None of these approaches circumvent the cold-boundary difficulty unless is modified suitably, for example, by the introduction of 0. The approximations to be discussed next bypass the cold-boundary difficulty in a natural way. 

When using the Conjugate Gradient method (or other iterative methods) to solve the electronic part of the problem, it has to be remembered that errors in the Hellmann-Feynman forces on the atoms are first order in errors in the wavefunclions, whereas the error in E is second order. This means that the electron system should be somewhat relaxed towards its ground stale before calculating the Hellmann-Feynman forces and letting the atoms respond — especially near structural equilibrium where the forces are small (Payne et al., 1992). 

In recent years the solution of problems of large amplitude motions (LAM s) has usually been based on grid representations, such as DVR,[11, 12] of the Hamiltonians coupled with solution by sequential diagonalization and truncation (SDT[13, 9]) of the basis or by Lanczos[2] or other iterative methods[14]. More recently, filter diagonalization (FD)[5, 4] and spectral transforms of the iterative operator[15] have also been used. There has usually been a trade-off between the use of a compact basis with a dense Hamiltonian matrix, or a. sinij)le but very large DVR with a s])arse H and a fast matrix-vector product. 

Since tends to the zero vector 0 if and only if M is convergent [25], it follows that the iterative procedure of (4.6) is necessarily convergent. However, other iterative methods based on (4.5) may be more rapidly convergent. Consider the iterative method defined by... 

A -> 0, and a gain of a factor V2 in the asymptotic rate of convergence is achieved. While other iterative methods exist [43] which are based on the direct inversion of larger submatrix equations, the iterative method SLOR shows generally how these are applied, and what type improvements in rate of convergence can be obtained. 

An approach that has met with success is the Adamson and Ling iterative method." This method converges to an acceptable solution much faster than any other iterative method although it is not a general iterative solution of the Fredholm equation since the initial approximation to /(y) is obtained by using the condensation approximation. Thus the method utilizes information concerning the physical form of the local isotherm functions (see Section 4 for details and applications). 

We note that while the Newton method is the most robust and most widely used in nonlinear finite element software, it is also computationally expensive primarily due to the necessity to solve a system of linear equations. It also imposes considerable computer memory requirements since a global system matrix is used. This method also is not as easily parallelized as some other iterative methods. In order to achieve the optimal performance of the Newton method, it is crucial to calculate the tangent stiffness matrix that is indeed tangent or, in other words, is the derivative with respect to unknowns that are calculated very accurately. 

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