Jacobi’s method

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

Unfortunately, Jacobi s method is computationally intensive, requiring of the order of 6n arithmetic operations to calculate the eigensystem. 

Eq. (9) may be solved by standard numerical methods, e.g. Jacobi s method, to give both X and A. 

As 2 = —1 + —1, we can expect Jacobi s method to converge only if this matrix is indeed irreducible. 

Such questions may seem abstract, but they are in fact very important in practice. Above, our proof of convergence of Jacobi s method is based upon the assumed existence of a complete eigenvector basis for — A). 

In a space charge free region, the potential at point j,i is then simply the average of the four neighboring potentials. This iterative approach is known as Jacobi s method and dates back to the last century. 

A comparison of the performance of the three algorithms for eigenvalue decomposition has been made on a PC (IBM AT) equipped with a mathematical coprocessor [38]. The results which are displayed in Fig. 31.14 show that the Householder-QR algorithm outperforms Jacobi s by a factor of about 4 and is superior to the power method by a factor of about 20. The time for diagonalization of a square symmetric value required by Householder-QR increases with the power 2.6 of the dimension of the matrix. 

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of local and global minimization steps. Let p be the local density at lattice site i in the A th local and the /th global minimization step. A local estimate for the corresponding minimum value of fi is obtained via Newton s method (see Eq. (D.6)] that... 

An alternative method for preparing vinyl bromides can be incorporated into a synthesis of alkynes when combined with an E2 reaction. In Jacobi s synthesis of (-)-stemoamide,l aldehyde 196 was treated with... 

The Ixx, -- are the moments of inertia and the hy, are called the products of inertia. The inertia matrix I is symmetric, and diagonalization of this matrix by standard techniques (such as Jacobi s rotation method). ... 

As a demonstration of the useMness of Gershgorin s theorem, we generate a convergence criterion for the Jacobi iterative method of solving Ax = b. This example is typical of the use of eigenvalues in numerical analysis, and also shows why the questions of eigenvector basis set existence raised in the next section are of such importance. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The eigenanalysis of the MIL tensor is run via Jacobi method to calculate the main characteristics values, that is, eigenvalues (eo-g), and characteristic directions, that is, eigenvectors (co-s)-... 

N. Black, S. Moore and E. W. Weisstein, Jacobi Method from MathWorld, A Wolfram Web Resource. http //mathworld.wolfram.com/JacobiMethod.html. 

S. L. S. Jacoby, K S. Kowalik, and J. T. Pizzo, Iterative Methods for Nonlinear Optimization Problems, Prentice-Hall, Englewood Cliffs, N.J., 1972. 

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