The Method of Successive Substitution

Equations B.1.16 and B.1.17 represent the system for which we already have the solution 

The method of successive substitution is useful for solving matrix differential equations in which the coefficients are functions of the independent variable t. 

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

Equation B.2.5 gives x in terms of an integral of itself. Approximate expressions for x 

Now we have the first two terms of a series solution to Eq. B.2.4. The third term on the right-hand side of Eq. B.2.6 involves x so, once again, we replace it using Eq. B.2.5 to give the next approximation 

The simplest one-point iterative root-finding technique can be developed by rearranging the function/(x) so that x is on the left-hand side of the equation 

The function g(x) is a formula to predict the root, In fact, the root is the intersection of the line y = x with the curve y = g(x). Starting with an initial value of x, as shown in Fig, 1,2a, we obtain the value of x  

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Method ofWegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure Xi + = F xC) is revised by setting x + i = F xi) and then taking Xi + = qxi -i- (1 — q)xi + i, where is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q s are ... 

These values of CA and CB are used for the rest of the calculations. It may be noted that the method of successive substitution used here is not the best or the most reliable method of iteration. In this particular example however, the convergence is rapid leading to final values of ... 

For systems involving recycle streams or intermediate feed locations, the method of successive substitution can be used [Mohan and Govind, 1988a]. Moreover, multiple reactions including side reactions and series, parallel or series-parallel reactions result in strongly coupled differential equations. They have been solved numerically using an implicit Euler method [Bernstein and Lund, 1993]. 

Once the tear streams are identified and the sequence of calculations specified, everything is in order for the solution of material and energy balances. All that has to be done is to calculate the correct values for the stream flow rates and their properties. To execute the calculations, many computer codes use the method of successive substitution, which is described in Appendix L. The output(s) of each module on interation k is expressed as an explicit function of the input(s) calculated fi om the previous iteration, A - 1. For example, in Fig. 5.16 for module 1,... 

Wegstein s method, which is used in many flowsheeting codes, accelerates the convergence of the method of successive substitutions on each iteration. In the secant method, the approximate slope is... 

The method of successive substitution can be a very effective way of computing the from Eqs. 8.3.24 when the mole fractions at both ends of the diffusion path y-g and y g, are known. In practice, we start from an initial guess of the fluxes and compute the rate factor matrix [< >]. The correction factor matrix [a] may be calculated from an application of Sylvester s expansion formula (Eq. A.5.20)... 

The solution to the matrix differential Eq. 10.4.6 can be found using the method of successive substitution (Appendix B.2). Here we follow closely the treatment by Taylor (1981b) (see, also Krishna, 1982). The solution to Eq. 10.4.6 can be written as... 

This equation admits two solutions. One solution occurs at the time less than and the other at time greater than When the method of successive substitution is applied to the equation... 

Nonlinear Volterra eqnations are solved by iterative methods. Two iterative methods are presented here. One is Newton s method, and the other is the method of successive substitution. 

The iterative solution can be carried out by one of various algorithms, for example Newton s approximation to find roots, steepest descent to find a minimum quadratic error, rough search, successive substitution. Newton s method in four dimensions works reasonably well, although instability can set in if the incremental changes are allowed to be too large. Hence some deceleration is required to stabilise the algorithm. The method of successive substitution is more efficient, but also... 

Wegstein s method can be employed to accelerate convergence when the method of successive substitutions requires a large number of iterations. As shown in Figure 4.11b, the previous two iterates of/ x and x are extrapolated linearly to obtain the next value of x as... 

Thus, weights q and 1 — q are applied, respectively, to x and/(jc ). Equation (4.10), withq defined by the slope, is usually employed when the slope is less than 1, such that q < 0. Typically, q is bounded between —20 and 0 to ensure stability and a reasonable rate of convergence. Wegstein s method reduces to the method of successive substitutions, x = / x ), when = 0. 

Alternatively, so-called secant methods can be used to approximate the Jacobian matrix with far less effort (Westerberg et al., 1979). These provide a superlinear rate of convergence that is, they reduce the errors less rapidly than the Newton-Raphson method, but more rapidly than the method of successive substitutions, which has a linear rate of convergence (i.e., the length of the error vector is reduced from 0.1, 0.01, 10 , 10 , 10 , ...). These methods are also referred to as quasi-Newton methods, with Broyden s method being the most popular. 

To compare the method of successive substitutions with the Newton-Raphson method, or the quasi-Newton methods, the former can be written ... 

In the dominant-eigenvalue method, the largest eigenvalue of the Jacobian matrix is estimated every third or fourth iteration and used in place of Sj in Eq. (4.20) to accelerate the method of successive substitutions, which is applied at the other iterations (Orbach and Crowe, 1971 Crowe andNishio, 1975). 

Other methods The method of successive substitution (SS), and the full Newton method for solving the system of nonlinear algebraic equations of flash have certain limitations. They have also certain desirable... 

Figure El.la Solution using the method of successive substitution.

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 ... 

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