Converging interval methods for the iterative solution of a non-linear

Chemical

Engineering Science,

1973, Vol. 28, pp. 2 187-2 193.

Converging

Pergamon

Press.

Printed in Great Britain

interval methods for the iterative solution of a non-linear equation

MORDECHAI SHACHAM and EPHRAIM KEHAT Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel (Received

7 August 1972)

Abstract-The convergence criterion (x~+~-1~1 < E is more reliable than p(xi) 1< E,particularly ifat successive iterations the values of xi are forced to alternate on both sides of the solution, by the iterative method. Two such methods are proposed. The first is a modification of the Interval Newton method and the second is a combination of the Newton and the Secant methods. These two methods require two starting points and numerical differentiation can be used with no loss of accuracy or rate of convergance. If numerical differentiation is not required, these methods can be started with one initial point and converge faster.

problems encountered by chemical engineers require a numerical solution of an algebraic non-linear equation. A large number of numerical iterative methods is available [ I-51 and the relative merits of such methods for the solution of specific chemical engineering problems are discussed in the literature[3,6,71. The efficiency of a method for the solution of a single non-linear equation is an interesting academic problem, whereas the efficiency of a method for the solution of a large number of non-linear equations (e.g. mass and energy balances for complex flowsheets, steady state and dynamic (staged separation processes)) is a more acute problem. However, an efficient new method for the solution of a single non-linear equation can lead to the development of an efficient method for the solution of a system of equations. Major problems in this area that remain to be solved are: 1. The determination of an initial starting point in the neighbourhood of the solution. 2. An efficient convergence criterion. 3. The determination of a step size for numerical differentiation, used in methods that require derivatives. This paper deals mainly with the last two problems and describes two new efficient methods for the solution of a non-linear equation. MANY

ITERATIVE METHODS FOR THE SOLUTION OF A NON-LINEAR EQUATION

A non-linear general as :

equation f(x)

can be described =o

in (1)

where f(x) is given and a value of x that satisfies Eq. (1) with a predetermined accuracy is desired. An iterative solution begins by guessing an initial value for x and computing another value of X, closer to the solution by an iterative function 4 (11): Xi+1= +(Xi, Xi-l, * * *Xi-n).

(2)

The iteration function for the classic Newton method is: Xi+1=

(3)

xi-f(Xi>lf’(X*)

and for the Secant method is; Xi+1

=

x*-f(X*)

(X*-Xi

-l)l(f(-G>

-f(x*

-1))

-

c4)

If x* is the exact solution of Eq. (l), the exact error at each interation step is: ef = IXi--X*1.

(5)

However, the value of x* is not known, and Eq. (5) cannot be used to check for convergence.

2187

MORDECHAI CONVERGENCE

The three generally criteria are ([8], p. 25):

SHACHAM

CRITERIA

used

convergence

(a)

If(xi) I < E

(6)

(b)

l~i+l--4

(7)

(c)

‘x”x:,x”