An Upper Bound on the Error at Each Iteration When Finding a Root or Fixed Point
Generally, the exact value of the rootof a function(or the exact value of a fixed point) is unknown, so it is impossible to evaluate the error of iterative methods,but
we can find an upper bound on the error using the difference between two iterates.
Definition: A sequenceis called contractive if there exists a positive constantsuch that
Theorem
Ifis contractive with constantthen,
i. is convergent with limit
ii.is bounded above and
Hence, forgiven,provides an upper bound onthe error at
iteration
Example:The first fiveare given below.
0.0000  1.0000  2.0000  3.0000  4.0000  5.0000  
1.0000  1.5000  1.3710  1.4297  1.4077  1.4169  
 0.5000  0.1290  0.0587  0.0220  0.0092  

 0.2580  0.4550  0.3750  0.4180  
0.4142  0.0858  0.0392  0.0155  0.0065  0.0027 
K can be hard to find. It can be shown that we can takefrom the above table so