Non-Recursive Algorithm:

A non-recursive algorithm is a type of algorithm that does not use recursion to solve a problem. It typically involves iteration or loops to perform repetitive tasks.

To perform a mathematical analysis of a non-recursive algorithm, you would typically consider the time complexity and space complexity of the algorithm.

Time complexity refers to the amount of time it takes for an algorithm to run as a function of the input size. Common notations used to express time complexity include O(n), Ω(n), and Θ(n), where "n" represents the input size.

Space complexity refers to the amount of memory space required by an algorithm as a function of the input size. Similarly, it is often expressed using the big O notation, such as O(n), Ω(n), or Θ(n).

To analyze a non-recursive algorithm, you would analyze the number of iterations performed by the loops and the amount of memory used by variables or data structures. By determining how these quantities grow as the input size increases, you can derive the time and space complexity of the algorithm.

Recursive Algorithm:

A recursive algorithm is a type of algorithm that calls itself to solve a problem by breaking it down into smaller subproblems. It involves dividing the problem into simpler instances of the same problem until a base case is reached.

To perform a mathematical analysis of a recursive algorithm, you would typically consider the recurrence relation and the base case.

The recurrence relation describes the relationship between the input size and the number of recursive calls made by the algorithm. It helps in understanding how the problem size reduces with each recursive call.

The base case defines the terminating condition for the recursion. It specifies the smallest instance of the problem that can be solved directly without further recursion.

To analyze a recursive algorithm, you would express the time complexity and space complexity using the recurrence relation. You would determine how the problem size reduces with each recursive call and combine it with the time and space complexity of the base case.

By solving the recurrence relation and considering the base case, you can determine the overall time and space complexity of the recursive algorithm.

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