Chapter 3 NONLINEAR EQUATIONS

1. Chapter 3NONLINEAR EQUATIONS

2. 3.1 Numerical Solution in Brief3.1.1 Direct methods and uninterrupted approximation • Root finding problem: f(x) = 0 • a: f(a) = 0 – root or zero of the function f(x) • A method that would narrow the searching interval in case when we know the dimension and that the root exists for sure, and finally converge to the root. • bisection method • inverse linear interpolation • Can be applied for the majority of non-linear problems • However: slow speed of convergence

3. Continuous approximation method: we start with the initial approximation x1 and then build the sequence of approximation values following a repetition formula. • For example, given n-th approximation value xn, the next value is found by xn+1 = f(xn) (n = 1,2,…) • The sequence of approximation values x1,x2… • : Newton-Raphson method • : Bailey method

4. Table 3.1 Numerical methods for solving non-linear equations

5. Rolle’s theorem: Suppose f(x) is continuously differentiable function on [a,b]. If f(a) = f(b) = 0, then a number c in (a,b) exists with f’(c) = 0 • Mean-value theorem: If f belongs to [a,b] and f is differentiable on (a,b), then a number θ in (a,b) exists such that:

6. 3.2 Direct Search Methods3.2.1 Bisection Method • Root-finding problem f(x) = 0 • Assume: • f – continuous function defined on interval [x1,x2] • f(x1) and f(x2) are of opposite sign (f(x1)f(x2)<0) • the root in the interval is unique • The method calls for a repeated halving of subintervals of [x1,x2] and, at each step, locating the half containing x (3.8) • If f(x) = 0, then x is the solution • If not, then • if f(x)f(x1)>0 (same sign)  solution belongs to (x, x2), we set x1 = x and x2 is the same • if f(x)f(x2)>0 (same sign)  solution belongs to (x1, x), we set x2 = x and x1 is the same

7. Figure 3.3 Bisection method convergence graph

8. Figure 3.4 Bisection method algorithm

9. Bisection method calculation procedure

10. Some stopping procedures that can be applied on step 2.3 of bisection method algorithm: • select the tolerance ε>0 and generate x1,…xN until one of the following conditions is met: • If follow the first stopping condition, then the total number of mathematical operations is

11. 3.3 Continuous Approximation Methods3.3.1 Newton-Raphson method • One of the most powerful and well-known numerical methods for solving a root-finding problem f(x) = 0 Algorithm: • For finding the root of the problem we need • If we expand the function f(x) in the neighborhood of xn in Taylor series • If we neglect everything except first three terms • This sets the stage for the method, which starts with an initial approximation x0 and generates the sequence {xn+1} (3.13)

12. Figure 3.6 Newton-Raphson method

13. Figure 3.7 Newton-Raphson algorithm

14. Newton-Raphson calculation procedure

15. 3.3.2 Bailey method • The extension of Newton-Raphson method Algorithm: • If we expand the function f(x) in the neighborhood of xn in Taylor series • If we neglect everything except first two terms of the remainder, with x = xn+1 and f(xn+1) = 0 • Substituting Eq.(3.13) from Newton-Raphson method, (3.18)

16. 3.4 Non-linear Simultaneous Equations • Consider m non-linear simultaneous equations with m unknowns: • In brief vector form • The initial estimation is x(0), which neighborhood is x(0)+δx, where δ = (δx1, δx2, …, δxm)T

17. Writing the Taylor series for the vector function f(x(0)+δx) • from where (3.20)

18. J – Jacobi matrix (Jacobian) • Then, saying x(0) + δx = a, and f(a) = 0, • and the approximation is which is the recursive formula for Newton’s method