Iterative Solution

Definition

An iterative solution uses repeated steps to move toward an answer.

Basic form:

start with a value
apply an update rule
repeat

Iterative solution = answer approached through repeated updates.

---

Simple Example

Suppose we want to count down to zero.

Start:

x = 3

Rule:

x = x - 1

Steps:

3 -> 2 -> 1 -> 0

The result comes from repeated application of a rule.

Iteration turns one rule into a sequence of states.

---

Approximation Example

Some iterative methods approximate an answer.

Example goal:

find a number close to sqrt(2)

Instead of writing the exact value directly, a method can improve a guess step by step.

Example:

guess 1
better guess
better guess
better guess

Each step should move closer to the target.

Iteration is often used when direct solution is difficult.

---

Iteration Needs A Rule

An iterative solution needs an update rule.

Example:

current value -> next value

Without a rule, repeated steps are not controlled.

The rule decides how the next step is produced.

Iteration is structured repetition, not random repetition.

---

Stopping Condition

An iterative method also needs a stopping condition.

Examples:

stop when x = 0
stop after 100 steps
stop when change is very small
stop when error is below a limit

Without a stopping condition, the process may continue forever.

Iteration needs a way to decide when enough steps have been taken.

---

Final Summary

• An iterative solution uses repeated updates.
• It starts from an initial value.
• It applies an update rule.
• It may produce an exact result or an approximation.
• It needs a stopping condition.
• Iteration is step-based rather than direct.