Powering algorithm

As in the case of infrared, progress in computing and the development of powerful algorithms for Fourier transforms has made the development of pulse NMR possible. 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

In the power algorithm one first computes the matrix product of Cp with an initial vector of p random numbers v, yielding the vector w ... 

The normalization step prevents the elements in v from becoming either too large or too small during the numerical computation. The two operations above define the cycle of the powering algorithm, which can be iterated until convergence of the elements in the vector v within a predefined tolerance. It can be easily shown that after n iterations the resulting vector w can be expressed as ... 

A key operation in the power algorithm is the calculation of the deflated cross-product matrix which is independent of the contribution by the first eigenvector. This is achieved by means of the instmction ... 

Both MATLAB and ASCL include powerful algorithms for non-linear optimisation, which can also be applied for parameter estimation. Optimisation and parameter estimation are also discussed in greater detail in Sections 2.4.1 and 2.4.2. 

The Singular Value Decomposition, SVD, has superseded earlier algorithms that perform Factor Analysis, e.g. the NIPALS or vector iteration algorithms. SVD is one of the most stable, robust and powerful algorithms existing in the world of numerical computing. It is clearly the only algorithm that should be used for any calculation in the realm of Factor Analysis. 

Steady-state process simulation or process flowsheeting has become a routine activity for process analysis and design. Such systems allow the development of comprehensive, detailed, and complex process models with relatively little effort. Embedded within these simulators are rigorous unit operations models often derived from first principles, extensive physical property models for the accurate description of a wide variety of chemical systems, and powerful algorithms for the solution of large, nonlinear systems of equations. 

Powerful algorithms cut the data collection time to a few days. Using crystal samples cryocooled to 77 K sharpens the diffraction intensities by typical factors of 5 or so, and it hastens data collection to about 1 day. 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerful algorithm for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Substructure searching is probably the most basic approach to identifying compounds of interest. It is widely used for all kinds of problems. Most chemists take substructure searching for granted, testament to the decades of effort that has gone into the development of extremely powerful algorithms and database systems. 

In this example, we have inverted a 2 X 2 matrix. Perhaps an inversion by head could also be performed in the case of a 3 X 3 matrix. For larger matrices, however, a computer algorithm is necessary. In addition, matrix inversion is a very sensitive procedure, so that powerful algorithms, such as singular value decomposition (cf. Section 5.2), are to be applied. 

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