2.1/

Bisection Technique

1. What?
2. Why?
3. Method:
   
4. Algorithm:
   
5. Drawbacks:
   
6. Stopping Criterions:
   

   The best criterion is:

7. Convergence:
   

   It:

8. Rate of Convergence \ Error Bound:
   
9. The problem of Percision:
   We use,
   
10. The Signum Function:
    We use,
    

2.2/

Fixed-Point Problems

1. Fixed Point:
2. Root-finding problems and Fixed-point problems:
   

   Root Finding and Fixed-point problems are

3. Why?:
   
4. Existence and Uniqueness of a Fixed Point.:
   

Fixed-Point Iteration

1. Approximating Fixed-Points:
   
2. Algorithm:
   
3. Convergence:
   • Fixed-Point Theorem:
     
   • Error bound in using \(p_n\) for \(p\):
     

     Notice:
     

4. Using Fixed-Points:
   

   Question: \(\ \ \ \ \\)

   Answer:
   

5. Newton’s Method as a Fixed-Point Problem:
   

2.3/

Newton’s Method

1. What?:
   • Newton’s (or the Newton-Raphson) method is:
   • 
     
   • 
     
2. Derivation:
   
3. Algorithm:
   
4. Stopping Criterions:
   

Convergence using Newton’s Method

1. Convergence Theorem:
   Theorem:
   

   The crucial assumption is

   
   

   Theorem 2.6 states that,
   (1)
   

   (2)

The Secant Method

1. What?
   In Newton’s Method
   We approximate \(f'( p_n−1)\) as:
   

   To produce:
   

2. Why?
   

   \(\ \ \ \ \ \ \ \ \\):

   Frequently,

   Note:

3. Algorithm:
   
4. Convergence Speed:
   

The Method of False Position

1. What?
   
2. Why?
   
3. Method:
   
4. Algorithm:
   

2.4/

Order of Convergence

1. Order of Convergence:
   
2. Important, Two cases of order:
   
3. An arbitrary technique that generates a convergent sequences does so only linearly:
   

   Theorem 2.8 implies

4. Conditions to ensure Quadratic Convergence:
   

5. Theorems 2.8 and 2.9 imply:
   (i)
   

   (ii)
   

6. Newtons’ Method Convergence Rate:
   

Multiple Roots

1. Problem:
   
2. Zeros and their Multiplicity:
   
3. Identifying Simple Zeros:
   
   • Theorem:
   • Generalization of Theorem 2.11:

     The result in Theorem 2.12 implies

       </xmp>
     

4. Why Simple Zeros:
   

   Example:

5. Handling the problem of multiple roots:
   
   • We \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • We define \(\ \ \ \ \ \ \ \ \ \\) as:
     
   • Derivation:
   • Properties:

2.5/

Aitken’s \(\Delta^2\) Method

1. What?
   
2. Why?
   
3. Derivation:
   
4. Del [Forward Difference]:
   
5. \(\hat{p}_n\) [Formula]:
   
6. Generating the Sequence [Formula]:
   

Steffensen’s Method

1. What?:
2. Zeros and their Multiplicity:
   
3. Difference from Aitken’s method:
   
   • Aitken’s method:
   • Steffensen’s method:

   Notice
   

4. Algorithm:
   
5. Convergance of Steffensen’s Method:
   

2.6/

Algebraic Polynomials

1. Fundamental Theorem of Algebra:
   
2. Existance of Roots:
   
3. Polynomial Equivalence:
   

   This result implies

Horner’s Method

1. What?
   
2. Why?
   
3. Horner’s Method:
   
4. Algorithm:
   
5. Horner’s Derivatives:
   
6. Deflation:
   
7. MatLab Implementation:
   

Complex Zeros: Müller’s Method

1. What?
   
   • It is a:
   • Müller’s method uses
2. Why?
   
   1. First:
      
      
   2. Second:
      
      

      If the initial approximation is a real number,

3. Complex Roots:
   
4. Algorithm:
   
5. Calculations and Evaluations:
   Müller’s method can: