Jacobi iterative algorithm

Figure 8.10 illustrates the mechanism of iteration for the Jacobi iterative scheme (Algorithm 1). 

The main difference from the Jacobi iteration is that as soon as a new value is found, it is used immediately. The steps for this algorithm are described below ... 

Calculation of eigenvectors requires an iterative procedure. The traditional method for the calculation of eigenvectors is Jacobi rotation (Section 3.6.2). Another method—easy to program—is the NIPALS algorithm (Section 3.6.4). In most software products, singular value decomposition (SVD), see Sections A.2.7 and 3.6.3, is applied. The example in Figure A.2.7 can be performed in R as follows ... 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

The FSCC equation (2.7) is solved iteratively, usually by the Jacobi algorithm. As in other CC approaches, denominators of the form Eq —E ) appear, originating in the left-hand side of the equation. The well-known intruder state problem, appearing when some Q states are close to and strongly interacting with P states, may lead to divergence of the CC iterations. The intermediate Hamiltonian method avoids this problem in many cases and allows much larger and more flexible P spaces. 

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