To confirm the correctness of an algorithm, we can use various techniques such as dry running, test cases, and formal proofs. Let's explore these techniques with an example:

Consider the following algorithm to calculate the factorial of a positive integer:

Step 1: Start

Step 2: Input a positive integer n

Step 3: Set a variable "factorial" to 1

Step 4: Set a variable "i" to 1

Step 5: Repeat the following steps while i <= n:

Multiply "factorial" by "i" and store the result in "factorial"

Increment "i" by 1

Step 6: Display the value of "factorial"

Step 7: Stop

Now, let's confirm the correctness of this algorithm using different techniques:

1. Dry Running: Dry running involves manually executing the algorithm step by step with sample inputs to check if it produces the expected results. For example, if we input n = 5 into the algorithm, we can dry run it as follows:

Step 2: Input n = 5

Step 3: Set factorial = 1

Step 4: Set i = 1

Step 5: Repeat the following steps while i <= 5:

i = 1, factorial = 1

i = 2, factorial = 2

i = 3, factorial = 6

i = 4, factorial = 24

i = 5, factorial = 120

Step 6: Display factorial = 120

By dry running the algorithm, we can see that it correctly calculates the factorial of 5 as 120.

2. Test Cases: Test cases involve running the algorithm with different inputs and comparing the output with the expected results. For the factorial algorithm, we can test it with various positive integers like 0, 1, 5, and 10 to ensure it produces the correct factorial values.

3. Formal Proofs: Formal proofs involve using mathematical techniques to demonstrate that an algorithm is correct. This typically involves using mathematical induction or loop invariants to prove the algorithm's correctness. However, formal proofs can be complex and require a deep understanding of mathematical concepts.

By using these techniques, we can confidently confirm the correctness of an algorithm. It's important to thoroughly test and validate an algorithm to ensure it produces the expected results for all possible inputs and edge cases.