Picard’s method

Finally, let us stress that the obtained asymptotic feature is entirely due to quadratic convergence characteristic of Newton s method. Thus no process with a linear convergence, e.g., Picard s method, would generate an asymptotic sequence. 

To illustrate the method of successive substitution, also known as Picard s method, we first consider the scalar differential equation... 

Thus, Equations 15.62 and 15.63 represent the rough approximation of Ad>(y) and y,(v) and can be plotted as a function of y in Figure 15.4 for the same data as in Figure 15.1. Both A(y) and j,(y) dependences are expected but the latter shows a quasi-limiting behavior not predicted by the condition of quasi-primary current distribution. The variation in jt(y) has to be zero if we consider all the components given by the Picard s method of successive approximation. Thus, the plots in Figure 15.4 have to be taken with care. 

Using Picard s method, find the approximate value of 7 and z corresponding to V = 0.1 given that... 

The philosophy underlying the Newton-Picard method [4] is to combine the good properties of both Newton s method and the d3mamic simulation. The procedure followed in... 

The Newton-Picard method, like Broyden s method, is a quasi-Newton method. This means that it produces approximations to a fixed point of F using the iteration scheme Xj+i = x< -I- Axj where Axj is some approximation of... 

In our implementation for each iteration of the Newton-Picard method we need 2p + 4 cycle simulations (and if G(a j) is small enough only 4). For more details about our implementation of the Newton-Picard method, we refer to [7]. If p is small compared to N, one iteration of the Newton-Picard method will be much more efficient than an iteration of Newton s method. 

Since in the first iteration of the Newton-Picard method the Jacobian is computed, this method will need at least as much time as Newton s method, which, because of the linearity of the systems, needs to compute the Jacobian only once. Therefore the Newton-Picard method is not optimal for the linear Systems A and B and it was not applied to these systems. 

Figure 2. The error versus the CPU time in seconds for different methods for System C and D. The computations for System D are initiated from two different starting values an empty bed and a saturated bed — dynamic simulation, O Broyden s method, + Newton-Picard method, Newton s method.

Presently we have tested four methods for obtaining periodic states of cyclic processes, namely the dynamic simulation, Newton s method, Broyden s method and the Newton-Picard method. These are compared with respect to computational efficiency. To this end the four methods were applied to four PSA test systems. 

For nonlinear systems we have introduced the Newton-Picard method as an alternative to methods more commonly used in chemical engineering literature. It is found that for the weakly nonlinear System C, Broyden s method is fastest and that the Newton-Picard method needs a CPU time equal to the dynamic simulation. For the more strongly nonlinear System D, Broyden s method is again the fastest but here the Newton-Picard also reduces the CPU time as compared to the dynamic simulation by a factor two. The main drawback of the Newton-Picard method is the lengthy first iteration, which arises from the construction of the Jacobian in order to determine a basis for the slowly converging subspace. 

The existence of solution to Eq. (4.1.15) is generally established by a procedure that guarantees the convergence of the method of successive approximations (also called Picard s iteration). This method consists in substituting into the right-hand side of (4.1.15), the nth approximation for the population density denoted by in order to calculate the (n + l)st approximant. Thus we have... 

Battistini, B, Picard, S, Borgeat, P and Sirois, P (1998) Measurements of prostanoids, leukotrienes, and isoprostanes by enzyme immunoassays. Methods Mol Biol, 105, 201-207. 

Kamel S, Brazier M, Neri V, Picard C, Samson L, Desmet G, Sebert JL. Multiple molecular forms of pyridinoline cross-lmlcs excreted in human urine evaluated by chromatographic and immunoassay methods. J Bone Miner Res 1995 10 1385-92. 

Atkinson, I. J. Erikson, G. H. Daksis, J. I. Picard, P. Kits and methods for purification of nucleic acids using heteropolymeric capture probes and duplex, triplex or quadruplex hybridization in soln. utilizing fluorescent intercalating dyes. U.S. Pat. Appl. Publ. US 2003049673, 2003 Chem. Abstr. 2003, 138, 232955. 

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