Numerical Analysis Chapter 4 Summary of Knowledge Points-Finding Roots of Nonlinear Equations

Finding roots of nonlinear equations

One, dichotomy

1. The root interval is [a, b], pay attention to the selection of a0, b0.
2. Error estimation and the number of iterations needed to find.
   
3. Frequent exam problem solving ideas: Know the error, find the number of iterations; know the number of iterations, find the absolute error limit and the final root value

2. Simple iteration method (successive approximation method)

1. General format: iteration function, iteration format
   
2. Convergence conditions: φ(x) is derivable on the interval [a, b] and satisfies the following two conditions. (Often used to prove the convergence of the iterative format)
   
3. Error estimate
   
4. Convergence order: P-order convergence proof methods are generally as follows:
   
5. Aitken acceleration method
   Nw
6. Frequent exam problem solving ideas: construct an iterative equation, write an iterative format, prove the convergence of the iterative format, find the order of convergence

Three, Newton iteration method

1. Iterative format: According to the first-order Taylor expansion of f(x) at x0
   
2. Convergence of the square ( will prove )
   
3. Precautions for convergence
   (1) Local convergence
   
   (2) Strict selection of initial values
   

Fourth, the deformation of Newton iteration method

1. Simplified Newton iteration method
   
2. Secant method: 1.618 order convergence
   
3. Newton iteration method with parameters
   
4. Newton Iteration Method for Directly Finding Multiple Roots of Equation
   

5. Summary of question types and solution ideas

1. Explain that the equation has a unique root in a certain interval.

• Find the derivative of the equation and prove the monotonicity. According to the zero theorem, there must be a unique root

2. Prove that the iterative scheme converges

• According to the two properties of function convergence, namely function range and derivative range, it is proved that the iterative scheme converges

3. Find the convergence order of the iterative scheme for convergence

• According to two methods for judging the order of convergence: p-th order derivative is not zero, constructing p-th order infinitesimal equation

4. Limit the absolute error limit and find the number of iterations

• Solve according to the error estimation formula of each method

5. Solve the roots of a multiple root equation

• Solve according to two root finding methods. One is to give the number of multiple roots, the other is to find the second derivative, which is not affected by the number of multiple roots