Conditions for convergence

A similar result takes place in the frequency domain. Substituting the Green s function (r r a ) in equation (13.197) with its complex conjugate, (r lr w), which satisfies the radiation condition for convergent waves, we arrive at the following migration formula in the frequency domain ... 

Show that if the F, s are linear, a necessary and sufficient condition for convergence is that each eigenvalue of B is less than unity. 

Obviously, the new arrangement for the iteration equation is better than the original iteration equation (Eq. A.8). Before we discuss the conditions for convergence, let us practice this method on a set of coupled nonlinear algebraic equations. 

The condition for convergence is conservative-, that is, if the condition is satisfied, the iteration process will converge. However, nothing is said about when the condition is not met. In such a case, the iteration process may converge or diverge. 

The disadvantage of the iterative methods is that they may not provide a convergent solution. Diagonal dominance (Eqs. B.8 and B.9) is the sufficient condition for convergence. The stronger the diagonal dominance the fewer number of iterations required for the convergence. 

A necessary and sufficient condition for convergence is that <1. This relation is known as a contraction mapping. However, the speed of convergence depends on how close is to zero. The number of iterations to reach a given tolerance IIAx ll < 8 can be estimated from the following relation ... 

As this iteration uses an (1,2,4) generalized inverse we can apply Th. 3.4.1 which states the conditions for converging to x with... 

A sufficient condition for convergence of Eq. 1.29) to the root x is dial g fjc) < I for all X in the search interval. Fig. 1,2b shows the case when this condition is not valid and the method diverges. This analytical test is often difficult in practice. In a computer program it is easier to determine whether x - jTjI < jjCj - jc, and, therefore, the. successive x values converge. The advantage of this method is that it can be started with only a single point, without the need for calculating the derivative of the function. 

The successive substitution method has successfully calculated flie two real roots of flic fourfli order polynomial in this case. However, flic convergence is slow, and follows what is known as linear convergence. This means that as the true solution is approached, the error at each iteration is some linear fraction of the error at flic previous iteration. Let s now study the convergence rate and the condition for convergence of the successive substitution method. Let s let X, =, where x is the true solution value and Sx =x - x is the error in... 

Fig. 9.14. The condition for convergence of the bipolar multipole expansioiL The condition a — b < R implies that the two systems are enclosed in spheres that do not intersect.

---