Methods for Numerical Solution of Nonlinear Equations
- PDF / 451,634 Bytes
- 20 Pages / 441 x 666 pts Page_size
- 108 Downloads / 252 Views
Methods for Numerical Solution of Nonlinear Equations
In this chapter, we consider the systems of n arbitrary equations: F1 (x1 , x2 , . . . , xn ) ≡ F1 (x) = 0 F2 (x1 , x2 , . . . , xn ) ≡ F2 (x) = 0 .. .
(3.1)
Fn (x1 , x2 , . . . , xn ) ≡ Fn (x) = 0 with n unknowns, creating the vector x = [x1 , x2 , . . . , xn ]. In the case when at least one of the functions Fi (x) of the system, where i = 1, 2, 3, . . . , n, is nonlinear with respect to at least one unknown (variable) x j , where j = 1, 2, 3, . . . , n, the system is nonlinear. The nonlinearity of the function Fi (x) with respect to the variable x j ⊂ x should be understood in the following way. Value changes of these functions and related changes of the corresponding variables are not related by means of constant coefficients, independently of the value of variables. Contrary to the linear case, the systems of nonlinear equations cannot be solved by means of direct (simple) methods, because such methods are not elaborated up to now. In consequence, in case of the systems of nonlinear equations, the iterative numerical methods are chiefly used and the most popular among them are: – – – –
Method of direct iterations; Iterative parameter perturbation procedure; Newton iterative method and Equivalent minimization strategies.
3.1 The Method of Direct Iterations The algorithm of the method of direct iterations is very similar to that of the Gauss– Seidel method used for solving systems of linear equations, see Sect. 1.2.2. During the first stage of the procedure, the system of equations (3.1) is transformed to the following equivalent form: S. Rosłoniec, Fundamental Numerical Methods for Electrical Engineering, C Springer-Verlag Berlin Heidelberg 2008
49
50
3 Methods for Numerical Solution of Nonlinear Equations
x1 = f 1 (x2 , x3 , . . . , xn ) x2 = f 2 (x1 , x3 , . . . , xn ) .. .
(3.2)
xn = f n (x1 , x2 , . . . , xn−1 ) We begin our calculation by taking the approximate initial solution, i.e.: x 1 = a1 , x 2 = a2 , x 3 = a3 , . . . , x n = an Then the expressions which permit to find successive, more accurate, approximations of the solution can be written in the following form: x1 = f 1 (a2 , a3 , . . . , an−1 , an ) x2 = f 2 (x1 , a3 , . . . , an−1 , an ) .. . xi = f i (x1 , x2 , . . . , xi−1 , ai+1 , . . . , an ) .. .
(3.3)
xn = f n (x1 , x2 , . . . , xn−2 , xn−1 ) The calculating process, performed according to the formula (3.3) has iterative form, and it means that the approximate solution obtained in the current iteration constitutes the initial approximation (starting point) for the next iteration. These calculations are continued, until the difference
R=
n
|xi − ai |
(3.4)
i=1
obtained from the two consecutive solutions (found in the previous and the current iteration) would become sufficiently small. In the limit case (for an infinite number of iterations) the difference R should attain the value equal to zero. The applicability condition for this method is simply identical to the convergence of the vector x, see (3.3), towards a ce
Data Loading...
