NEWTON'S METHOD OF SOLVING A SYSTEM OF NONLINEAR EQUATIONS

Аннотация

The concept of a system of nonlinear equations, the stages of solving the problem, the geometric interpretation of the solution of the equation and the concept of iterative processes are given and their application is shown in the examples. The problem of numerical solution of a number of practical problems consisting of a system of nonlinear equations is considered. There are a number of approximate computational methods for solving systems of nonlinear equations, including Newton's method. Using these methods, a number of specific practical problems were solved, a computational algorithm and a block diagram were developed. An approximate method of finding the true roots of a system of nonlinear equations is given, based on examples, graphs are used in the form of results, and appropriate conclusions are drawn. Keywords: System of nonlinear equations, Maple, Newton's method, algorithm, graph, block diagram, approximate method, geometric interpretation, iterative process, numerical solution. Many practical problems lead to the solution of a system of nonlinear equations. In general, a system of n unknown nonlinear algebraic or transcendental equations is written as follows: 𝑓(x ,x ,…,x ) = 0 1 2 n 𝑓 (x ,x ,…,x ) = 0 { 3 1 2 n (1) 𝑓 (x ,x ,…,x ) = 0 n 1 2 n This (1) system can be written in vector form as follows: 𝑓 ( x ) = 0. (2) here x = (x ,x ,…,x )𝑇 – vector column of arguments; 1 2 n (𝑓 ,𝑓 ,…,𝑓 )𝑇 – vector column of functions; ( … )𝑇 – sign of transponder 1 2 n operation. Search for a solution of a system of nonlinear equations – this is a much more complex problem than solving a single nonlinear equation. Generalizing the methods used to solve a single equation to solve a system of nonlinear equations requires a lot of calculations or cannot be applied in practice. In particular, this applies to the method of dividing the interval into two equal parts. Nevertheless, a number of iterative methods of solving nonlinear equations can be generalized to the solution of a system of nonlinear equations. 174 Scientific research results in pandemic conditions (COVID-19) (2) We use the series approximation method to solve the system of equations. Suppose that vector (2) is isolated from equation 𝑥 = (x ,x ,…,x ) which is one of the roots k – approach 1 2 n 𝑥𝑘 = ( 𝑥𝑘,𝑥𝑘,… 𝑥𝑘 ) 1 2 𝑛 be found. In this case (2) is the exact root of the vector equation x = 𝑥𝑘 + 𝜀𝑘 , (3) can be expressed in the form, here 𝜀𝑘 = ( 𝜀𝑘,𝜀𝑘,… 𝜀𝑘 ) – error 1 2 𝑛 correction limit (root error). Substituting (3) into (2), we obtain the following equation: 𝑓 ( 𝑥(𝑘) + 𝜀(𝑘) ) = 0. (4) Suppose, 𝑓 ( x ) – this x and 𝑥(𝑘) Let any bubble containing D be a continuously differentiable function in the field. (4) to the right of equation side 𝜀(𝑘) - we spread the line by the levels of the small vectors and are limited to the linear terms of this line: 𝑓( 𝑥(𝑘) + 𝜀(𝑘) ) = 𝑓( 𝑥(𝑘))+ 𝑓′( 𝑥(𝑘))𝜀(𝑘) = 0 (5) It follows from formula (5) that 𝑓′( 𝑥 ) as a product x ,x ,…,x - in 1 2 n relation to variables 𝑓 ,𝑓 ,…,𝑓 The following Jacob matrix of the system of 1 2 n functions is understood: 𝜕 𝑓1 𝜕 𝑓1 𝜕 𝑓1 𝜕 𝑥1 𝜕 𝑥1 𝜕 𝑥n 𝑓′( 𝑥 ) = 𝑊 (𝑥) = |𝜕 𝑓1 𝜕 𝑓1 𝜕 𝑓1 | , |… 𝜕 𝑥 …2 𝜕 … 𝑥2… … 𝜕 𝑥 …n| 𝜕 𝑓n 𝜕 𝑓n 𝜕 𝑓n 𝜕 𝑥1 𝜕 𝑥2 𝜕 𝑥n or if we write it in the form of a short vector, 𝑓′( 𝑥 ) = 𝑊 (𝑥) = | 𝜕 𝑓 i|, i,j = 1,n. 𝜕 𝑥 j (𝑘) (5) the system is the limit that corrects this error 𝜀 ( i = 1,n ) ) is a 𝑖 linear system with matrix W (x). From this formula (5) can be written as follows: 𝑓( 𝑥(𝑘))+ W( 𝑥(𝑘))𝜀(𝑘) = 0. From here, W( 𝑥(𝑘)) - assuming a non-specific matrix, we have: 𝜀(𝑘) = −𝑊−1( 𝑥(𝑘))𝑓( 𝑥(𝑘)) The result is this 𝑥(𝑘+1) = 𝑥(𝑘) −𝑊−1( 𝑥(𝑘))𝑓( 𝑥(𝑘)), k = 0,1,2, … (6) We come to the formula of the Newtonian method, in this 𝑥(0) - as a zero approximation can be obtained the rough value of the root sought. In practice (2), the calculations for solving a system of nonlinear equations by this method are continued according to formula (6) until the following condition is satisfied: 175 Scientific research results in pandemic conditions (COVID-19) |𝑥(𝑘+1) + 𝑥(𝑘+1) | < 𝜀 . (7) Based on the above, Newton We write the algorithm of the method as follows: 1. 𝑥(0) - initial approximation determined. 2. The value of the root is given by formula (6) determined by 3. If condition (7) is met, then the issue will be resolved and 𝑥(𝑘+1) – (2) of the vector equation is taken as the root, otherwise and go to step 2.In calculations (2) of a system of nonlinear equations f (x) functions and their derivatives the matrix W (x) is clearly given, in which case it is a block diagram of the solution of the system As shown in Figure 1. Figure 1. System of nonlinear equations algorithm of Newton's method for solving 𝑓(𝑥) the vector-function x is continuously differentiable up to twice around the root and the Jacob matrix 𝑊(𝑥) the non-specific, multidimensional Newtonian method has a quadratic approximation: |𝑥(𝑘+1) −𝑥| < 𝐶 |𝑥(𝑘) − x |2 We emphasize that the successful selection of the initial approximation is imp