Applied Numerical Methods Using MATLAB, 2nd Edition

4Nonlinear Equations

Chapter Outline

• 4.1 Iterative Method toward Fixed Point
• 4.2 Bisection Method
• 4.3 False Position or Regula Falsi Method
• 4.4 Newton(‐Raphson) Method
• 4.5 Secant Method
• 4.6 Newton Method for a System of Nonlinear Equations
• 4.7 Bairstow's Method for a Polynomial Equation
• 4.8 Symbolic Solution for Equations
• 4.9 Real‐World Problems
• Problems

4.1 Iterative Method toward Fixed Point

Let us see the following theorem.

Fixed‐Point Theorem – Contraction Theorem [K-3, Section 5.1] Suppose a function g(x) is defined and its first derivative g′(x) exists continuously on some interval I = [x^o − r, x^o + r] around the fixed point x^o of g(x) such that

(4.1.1)

Then, if the absolute value of g′(x) is less than or equal to a positive number α which is strictly less than 1, that is,

(4.1.2)

The fixed‐point iteration starting from any point x₀ ∈ I

(4.1.3)

converges to the (unique) fixed point x^o of g(x).

Proof: The mean‐value theorem (MVT) (Appendix A) says that for any two points x₀ and x^o, there exists a point x between the two points such that

(1)

Taking the absolute ...