Jacobi Method

The Jacobi iterative method is similar to the Gauss-Seidel method with the exception that the newly calculated variables are not replaced until the end of each iteration is reached. In this section, we develop the Jacobi method in matrix form. 

Example 2.3 Solution of Chemical Reaction and Material Balance Equations Using the Jacobi Iteration for Predominantly Diagonal Systems of Linear Algebraic Equations. 

A chemical reaction takes place in a series of four continuous stirred tank reactors arranged as shown in Fig. E2.3. 

The conditions of temperature in each reactor are such that Uie value of the rate constant is different in each reactor. Also, the volume of each reactor Vf is different. The values of k, and Vj are given in Table E2.3. The following assumptions can be made regarding this system  

The rate of disappearance of component A in each reactor is given by 

Divide the equation by a and add x,- to the LHS and RHS (a similar procedure is used in Appendix A for the successive substitution method for solving nonlinear equations) to yield the equation 

The iteration process starts with an initial guessing vector x and the iteration equation used to generate the next iterated vector is 

The iteration process will proceed until one of the criteria in Eq. B.71 has been achieved. 

The second term in the RHS of Eq. B.74 is called the residual, and the iteration process will converge when the residual is approaching zero for all value of i. 

---

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

Using the most recently obtained values for each unknown (as opposed to the fixed point or Jacobi method), then... 

The conditions required by the dominant-diagonal theorem are sufficient to assure convergence of the Jacobi method. In certain important applications it happens that also A2 0. In that event all three methods converge, Jacobi the most slowly. 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

The data from sensory evaluation and texture profile analysis of the jellies made with amidated pectin and sunflower pectin were subjected to Principal component analysis (PC) using the statistical software based on Jacobi method (Univac, 1973). The results of PC analysis are shown in figure 7. The plane of two principal components (F1,F2) explain 89,75 % of the variance contained in the original data. The attributes related with textural evaluation are highly correlated with the first principal component (Had.=0.95, Spr.=0.97, Che.=0.98, Gum.=0.95, Coe=0.98, HS=0.82 and SP=-0.93). As it could be expected, spreadability increases along the negative side of the axis unlike other textural parameters. 

Repeated application of this equation constitutes the point-simultaneous relaxation method, also known as the point Jacobi method or the method of simultaneous displacements. 

The eigenvalues of A can be find by solving the characteristic equation of (1.61). It is much more efficient to look for similarity transformations that will translate A into the diagonal form with the eigenvalues in the diagonal. The Jacobi method involves a sequence of orthonormal similarity transformations, 12,... such that A(<+1 = TTkAkTk. The matrix Tk differs from the identity... 

Jacobi method, 101 Jahn-Teller theorem, 313-314 Javan, A., 137 j-j coupling, 51 Jolly, W. L., 320 Jordan, P., 94... 

In the Jacobi method, a series of similarity transformations is carried out. It is easily proven that similar matrices have the same eigenvalues. Let A = P, BP. The eigenvalues of A satisfy the secular equation (2.38) ... 

The Jacobi method is generally slower than these other methods unless the matrix is nearly diagonal. In SCF calculations one is faced with the non-orthogonal eigenvalue equation... 

Usually on the first iteration of an SCF calculation W is computed by the Schmidt orthogonalization method but thereafter W is chosen to be the C matrix from the previous iteration. This produces an F matrix which is nearly diagonal so the Jacobi method becomes quite efficient after the first iteration. Further, in the Jacobi method, F is diagonalized by an iterative sequence of simple plane-rotation transformations... 

The Jacobi method is traditional (first presented 1846) and simple. More efficient general methods were not found until 1954 (W. Givens, Oak Ridge), and some special features make it the first choice for many applications still today. In order to describe the method in more detail, consider the problem of finding a solution A,V to the problem... 

The Jacobi method works by successively transforming the matrices A and v in place , in a way which ensures that the non-diagonality measure... 

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. 

Brillouin, L. (1960). Poincare and the sortcomings of the Hamilton-Jacobi method for classical or quantized mechanics. Arch. Rational Mech. Analysis 5(1), 76-94. 

This is a system of equations of the form Ax = B. There are several numeral algorithms to solve this equation including Gauss elimination, Gauss-Jacobi method, Cholesky method, and the LU decomposition method, which are direct methods to solve equations of this type. For a general matrix A, with no special properties such as symmetric, band diagonal, and the like, the LU decomposition is a well-established and frequently used algorithm. 

In the Jacobi method, we rearrange the system of equations to place the contribution due to Xj on the LHS of the fth equation and the other terms on the RHS, and we divide both sides of the equation by an. The iteration equation for the Jacobi method is written as ... 

Iterative methods are sometimes used due to ease of computer coding and lesser computational storage requirements. The Jacobi method is the simplest iterative method but has slower convergence in comparison with the Gauss-Seidel method. In the Gauss-Seidel method, the (A -tl)th iteration of the value of the unknown x, is given by... 

Another sufficient condition for the convergence of the Jacobi method of iteration follows. Let R be the set of all starting vectors X0 for which the largest Hilbert norm of any matrix B generated by the iterative process has the property that... 

Show that if the conditions given by Eq. (15-27) are satisfied, then convergence can be assured for the Jacobi method of iteration. 

Transform each pair of oif-diagonal elements to zero in turn, subsequent transformations may regenerate elements, the method is iterative the Jacobi method-... 

The processes involved in all these methods can be appreciated by a detailed look at the Jacobi method which is extremely simple in concept and implementation. 

This is a straightforward implementation of the old faithful Jacobi method. The eigenvalues and eigenvectors are generated in order of lowest eigenvalue first. 

---