Fixed-point iteration

Systems of nonlinear equations are solved by iterative methods. One of the simplest iterative methods is the fixed point or functional iteration which is often the basis of more sophisticated approaches. A fixed point x of a map ip is defined by the condition x — p x ). The problem of finding a solution of F x) = 0 is equivalent to finding a fixed point x of the map 

For stating convergence conditions for this sequence we need the follow- 

L p) is called a Lipschitz constant ofp. If L p) 1 then p is called a contraction. 

For contractive functions the existence of a fixed point can be shown  

We will apply this theorem now to construct Newton s method by an appropriate choice of B. Later, when dealing with discretized differential equations we will also apply the fixed point iteration directly to solve nonlinear systems. 

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In Figure 6.23 we plot the typical scenario for this fixed-point iteration to illustrate our algorithm. 

Fixed point iteration scheme for the simulation problem Figure 6.23... 

Moreover, there exist two constants Cq > 0 and h g> 0 such that M < C(, and h < h (, imply that the solution of Problem (28) is unique and is the limit of the fixed point iteration scheme (we omit the subscript h) ... 

Problem (26) can be viewed as a transport equation in r for given v, and a Stokes system in (v,p) for given t. The fixed point iteration scheme described in Theorem 6.1 does not use this fact. A more natural iterative scheme, which uncouples the r and the v equations, reads as follows ... 

Let be the estimate for the tear stream variables at the h iteration, Let F(X ) be the calculated result for that tear stream s values. The simplest iteration scheme is successive substitution, or fixed-point iteration ... 

Again, the multivariate form of fixed-point iteration is so unstable that it generally can be assumed that it will not work. Weighted iteration is also significantly more difficult. 

For a fixed point iteration, we generate the first three terms in the sequence a , that is, a , a, and U2. Next, use the A Aitken method to generate 8q. At t stage, we can assume that the newly generated Sq is a better approximation to a than U2, and then apply the fixed point iteration to Sq to generate the next sequence of S , dj and 82- The Aitken method is now applied to the sequence d ra = 0, 1, 2 to generate y , which is a better approximation to a, and the process continues. 

The All-in-One (A-i-O) method, also referred to as multidisciplinary feasibility (MDF), is the most common way of approaching the solution of MDO problems. In this method, the vector of DV x is provided to the coupled system of analysis disciplines and a complete multidisciplinary analysis (MDA) is performed via a fixed-point iteration with that value of x to obtain the system MDA output variable j(a ) that is then used in evaluating the objective fix, y(x)) and the constraints c(x, y (a )). The optimization problem is ... 

For the solution of the entire system (2.1), (2.3), (2.4) it suggests itself to use the Newton iteration. But due to the structure of (2.4) the resulting Jacobian matrix would be non-sparse The oxygen concentration in the upper part of the reactor depends on the values of both the carbon concentration and the temperature in the layers lying underneath. Thus (2.1), (2.3) have been discretized by means of the finite element method and afterwards been solved each individually. The latter was realized through two fixed point iterations for (2.1), (2.4) with fixed temperature T and for (2.3) with fixed concentrations Cc, Coaj respectively. For this (2.1) (including boundary conditions) is written as... 

Inserting (2.4) into (3.3) finally produces the fixed point iterations (in Cc and T)... 

The above considerations have also been used to construct a simple method for the iterative solution of the linear block system (2.9). Let A denote the Jacobian approximation in (2.9) and A the associated matrix with Qy replaced by 0. Then the system Ax = h can be solved easily because of its nearly upper triangular block structure. On this basis, we constructed a fixed point iteration which is known to converge with contraction rate not greater than the spectral radius p(I — A A), Obviously, since y = 0 in the starting point xo, we will have p < 1 in some neighborhood, which can be monitored. 

Similar calculations for the other steady state species need to be performed. In the next step then these relations are used to calculate the remaining set of differential equations which govern the non-steady state species. The overall reaction rate for each remaining species is thus determined by means of an inner iteration loop, a simple fixed-point iteration procedure is often employed. The remaining set of differential equations can be solved in different ways, and several numerical solvers for stiff differential equations are freely available. 

In contrast to the fixed point iteration, this methods requires the evaluation of the Jacobian F x) and the solution of linear systems in every step. Both tasks can be very time consuming. In multibody dynamics it is mainly the evaluation of the Jacobian, which is very costly, see Sec. 3.5. ... 

The simulation is performed by using the explicit Euler method as predictor and the implicit Euler method as corrector (with fixed point iteration iterated %ntil convergence , see Sec. 4 1-1)- I 4- l he results for the rotation of the chassis obtained with h = 10 are plotted. The thickness of the band around this solution curve corresponds to the estimated global error increment magnified by 5 10. This figure reflects clearly the influence of the second derivative of the solution on the size of the error. 

Like in the case of fixed point iteration the predictor solution is taken as starting value = The method demands an high computational effort, which is... 

We also have studied briefly the effects of increasing the depths of loop nests on the slicing time. This is interesting since loops will necessitate conventional fixed-point iterations in the RD data dependency analysis used by PDG-based slicing, whereas our SLV-based method uses a different mechanism to handle loops. 

Chen, J.Y., Tham, Y.F. Speedy solution of quasi-steady-state species by combination of fixed-point iteration and matrix inversion. Combust Flame 153, 634—646 (2008)... 

In the vicinity of the true stationary point, the fixed-point iterations based on (12.3.2) will converge rapidly to the true optimizer with the characteristic second-order convergence rate of Newton s method discussed in Section 11.5.2. Further away from the optimizer, the Newton step may not necessarily lead us towards the tme optimizer of the function since the stationary point of the local surface may no longer resemble the true optimizer of the fimction. In such cases, we should not apply the Newton step (12.3.2) directly but instead determine our step based on some other strategy. Such a strategy is presented in Section 12.3.2. 

The method of successive substitutions (also called fixed point iteration) is perhaps the simplest method of obtaining a solution to a nonlinear equation. This technique begins by rearranging the basic F(x) = 0 equation so that the variable x is given as some new function of the same variable x. The original equation is thus converted into an equation of the form ... 

From a more general point of view, the SCF iterations are a particular example of fixed-point iterations [2], frequently encountered in numerical analysis, in which we determine a fixed point X of a function f(x)... 

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