Algorithms and Convergence

Algorithms and Convergence
An algorithm is a procedure that describes, in an unambiguous manner, a finite sequence of steps to be performed in a specified order. The object of the algorithm is to implement a procedure to solve a problem or approximate a solution to the problem.
MATLAB program is used to implement algorithms for the methods described in this course. We have to learn some important statements that needed to build programs. The structure of the algorithm will be as follows:
Input: input variables.
Output: results required.
Steps 1 … n: steps of the solution.
Results: type results.
Stop: end of the algorithm.

Looping techniques in the algorithms are either counter-controlled, such as,
For i = 1, 2, . . . , n
Set xi = a + i · h
or condition-controlled, such as

While i < N do Steps 3–6.

To allow for conditional execution, we use the standard

If . . . then or If . . . then . . . else
constructions.
Not all numerical procedures give satisfactory output for arbitrarily chosen input. As a consequence, a stopping technique independent of the numerical technique is incorporated into each algorithm to avoid infinite loops.