Jacobi iteration

In another important class of cases, the matrix A is positive definite. When this is so, both the Gauss-Seidel iteration and block relaxation converge, but the Jacobi iteration may or may not. 

In the Jacobi iteration, one introduces arbitrary initial values (say jcj, y, zi) of jc, y, z in the right-hand sides of equations (6) this provides (left side) a new set of values (jc2, v , z2) which are reintroduced in the right sides and so on. In the Gauss-Seidel iteration the new value of a variable is reintroduced in the next equation as soon as it has been computed from xi, y i, Zi one calculates x2, fronts, y i, Zi, one calculates y2, and so on. 

Note that the synchronous treatment of a Boolean system is in fact a Jacobi (parallel) iteration. Our treatment may also be considered a kind of iteration, but it is neither a Jacobi iteration (in which all commutations are synchronous) nor a Gauss-Seidel iteration (in which the commutations take place one at a time but in a predetermined, arbitrary, order). We consider all the successions of states implicitly contained in the state table which one is followed depends on the values of the delays. 

Figure 8.10 Schematic representation of the Jacobi iterative scheme.

COROLLARY 1. Jacobi iteration converges if A is a diagonally dominant matrix Gauss-Seidel iteration converges if A is symmetric positive definite. 

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. 

Solution Co Example 2.3. This program solves a set of % linear algebraic equations by the Jacobi iterative % method, using the function JACOBI,M, to find the % concentrations of a seres of CSTRs. 

Solves a set of simultaneous linear algebraic equations that represent the material balances for a set of continuous stirred tank reactors using the Jacobi Iterative method (Jacobi.m). 

Jacobi Iterative method for solution of predominantly diagonal sets of simultaneous linear algebraic equations. 

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. 

The optimum over-relaxation parameter is known only for a small class of linear problems and for select boundary conditions. The iteration matrix has eigenvalues each one of which reflects the factor by which the amplitude of an eigenmode of undesired residual is suppressed for each iterative step. Obviously the modulus of all these modes must be less than 1. The modulus of the factor with the largest amplitude is called the spectral radius and determines the overall long term convergence of the procedure for many iterative steps. If Pj is the spectral radius of the Jacobi iteration then the optimum value of X is known to be ... 

The first method to be discussed is the Jacobi iteration. This is achieved according to the following scheme ... 

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 ... 

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