Power method iteration

One way to define a stochastic process is to rewrite the power method iteration Eq. (2.1) as... 

The nonlinear iterative partial least-squares (NIPALS) algorithm, also called power method, has been popular especially in the early time of PCA applications in chemistry an extended version is used in PLS regression. The algorithm is efficient if only a few PCA components are required because the components are calculated step-by-step. 

The Hamiltonian gradually filters the ground-state wavefunction from the trial wavefunction. To understand this filtering process, we expand the initial trial wavefunction in the exact wavefunctions of the Hamiltonian, ). With n iterations of the power method, we have... 

The two updates differ only by a factor of one-half before the first-order change from A and the second-order change. Unlike the wavefunction power method, the A -particle density matrices from each iteration in Eq. (Ill) are not exactly positive semidehnite until convergence. 

The 2-RDM is automatically antisymmetric, but it may require an adjustment of the trace to correct the normalization. The functionals in Table I from cumulant theory allow us to approximate the 3- and the 4-RDMs from the 2-RDM and, hence, to iterate with the contracted power method. Because of the approximate reconstruction the contracted power method does not yield energies that are strictly above the exact energy. As in the full power method the updated 2-RDM in Eq. (116) moves toward the eigenstate whose eigenvalue has the largest magnitude. 

Here we synthesize the concepts of the last four sections, (i) CSE, (ii) reconstruction, (iii) purification, and (iv) a contracted power method, to obtain an iterative algorithm for the direct calculation of the 2-RDM. 

An iterative approach has the advantage of allowing control of spurious fluctuations by interaction with the solution as it evolves. This may be done either automatically in the algorithm or by the exercise of human judgment. Although we reserve discussion of the most-powerful methods of interaction for the next chapter, it is appropriate here to set the stage by developing a family of iterative methods that have seen much use. 

This is of course a specific instance of the iterative process we associate with the scientific method. The postulated mechanisms surviving this process can be considered consistent with the experimental data. Kinetics provides a powerful method for eliminating putative reaction mechanisms, but kinetic methods alone can never establish a mechanism unambiguously. Other chemical and physical methods can be of help in this regard, but it must be acknowledged that all our models, at some level, are tentative and subject to revision. In practice, one must accept a certain amount of ambiguity, but for many, if not most, applications this is not crucial. 

In an admirably compact note in 1965 Nesbet used a very simple idea to generate a powerful method for the calculation of the individual eigenvalues and eigenvectors of very large matrices, which is particularly well-behaved if the lowest root is required. The method is iterative and uses the following technique ... 

The simplest method for PCA used in analytics is the iterative nonlinear iterative partial least squares (NIPALS) algorithm explained in Example 5.1. More powerful methods are based on matrix diagonalization, such as SVD, or bidiagonalization, such as the partial least squares (PLS) method. 

The power method. The power method may be used to determine the largest eigenvalue and its corresponding eigenvector. The method uses the iterative scheme Zk = Zk- = 1,2,3,... where A is the nxn matrix and zo... 

The first assertion of the above theorem is a direct consequence of Perron s theorem. The iterative method of (3.10) is commonly referred to as the power method of iteration, and the convergence of this iterative procedure is guaranteed by the fact that A exceeds in modulus all other eigenvalues of T. 

The main algorithms used for eigenvectors/eigenvalues computation differ in two aspects the matrix to work on, either X X (eigenvalue decomposition (EVD) and the POWER method) or X (singular value decomposition (SVD) and non-linear iterative partial least squares (NIPALS)). However SVD may work as well on X X (giving the same results as eigenvalue decomposition). Another difference is whether PCs are obtained simultaneously (EVD and SVD) or sequentially (POWER and NIPALS) for details and comparison of efficiency see Wu et al. [38]. In all the cases for which rows dimension I is much smaller than columns dimension /, one can operate on XX instead (EVD, POWER, SVD), and on X (NIPALS). 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

The DIIS convergence accelerator is available for all the SCF semiempirical methods. This accelerator may be helpful in curing convergence problems. It often reduces the number of iteration cycles required to reach convergence. However, it may be slower because it requires time to form a linear combination of the Fock matrices during the SCF calculation. The performance of the DIIS accelerator depends, in part, on the power of your computer. 

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