Main Idea

1. Algorithm:

   An Algorithm is a procedure that describes, in an unambiguous manner, a finite sequence of steps to be performed in a specified order

Characterizing Algorithms

1. Stability:
   
   • Stable Algorithm: an algorithm where small changes in the initial data produce correspondingly small changes in the final results.
   • Conditionally Stable Algorithm: an algorithm that is stable only for certain choices of initial data.
2. Error Growth:
   
3. Stability and Error-Growth:
   
   • Stable Algorithm: an algorithm that exhibits linear growth of error.
   • UnStable Algorithm: an algorithm that exhibits exponential error growth.

Rates of Convergence

1. Rate of Convergence:
   

   \(\beta_n \ = \ \dfrac{1}{n^p}, \ \ \ \\) for the largest number \(p > 0\).

2. Big-Oh Notation:
   

3. Example [determining rate of convergence]:
   Show Derivation