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

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4. Why Simple Zeros:
   

   Example:

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