C++/Recursion Programs/Program

Power using Recursion

Calculate Power using Recursion in C++

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C++
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#include <iostream>
using namespace std;

// Recursive function to calculate power
double power(double base, int exponent) {
    // Base cases
    if (exponent == 0) {
        return 1;
    }
    if (exponent == 1) {
        return base;
    }
    
    // Handle negative exponent
    if (exponent < 0) {
        return 1.0 / power(base, -exponent);
    }
    
    // Optimized: Divide and conquer
    // If exponent is even: base^exp = (base^(exp/2))^2
    // If exponent is odd: base^exp = base * (base^(exp/2))^2
    if (exponent % 2 == 0) {
        double half = power(base, exponent / 2);
        return half * half;
    } else {
        double half = power(base, (exponent - 1) / 2);
        return base * half * half;
    }
}

int main() {
    double base;
    int exponent;
    
    cout << "Enter base: ";
    cin >> base;
    cout << "Enter exponent: ";
    cin >> exponent;
    
    double result = power(base, exponent);
    cout << base << "^" << exponent << " = " << result << endl;
    
    // Test various powers
    cout << "\nVarious powers:" << endl;
    cout << "2^10 = " << power(2, 10) << endl;
    cout << "3^5 = " << power(3, 5) << endl;
    cout << "5^-2 = " << power(5, -2) << endl;
    cout << "10^0 = " << power(10, 0) << endl;
    
    return 0;
}

Output

Enter base: 2
Enter exponent: 8
2^8 = 256

Various powers:
2^10 = 1024
3^5 = 243
5^-2 = 0.04
10^0 = 1

This program teaches you how to calculate Power using Recursion in C++. The recursive power calculation uses a divide-and-conquer approach to efficiently compute exponents. This optimized method reduces time complexity from O(n) to O(log n) by halving the problem at each step.

1. What This Program Does

The program demonstrates optimized recursive power calculation:

  • Divide-and-conquer approach
  • Handling even and odd exponents
  • Supporting negative exponents
  • Efficient O(log n) time complexity

Optimized recursion provides efficient power calculation.

2. Header Files Used

  1. #include <iostream>
    • Provides cout and cin for input/output operations.

3. Understanding Power Calculation

Power Definition:

  • base^exponent = base × base × ... × base (exponent times)
  • Example: 2^8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

Optimization Idea:

  • Instead of multiplying base exponent times
  • Use divide-and-conquer: base^exp = (base^(exp/2))^2
  • Reduces recursive calls significantly

4. Base Cases

Stopping Conditions:

if (exponent == 0) { return 1; // Any number to power 0 = 1 } if (exponent == 1) { return base; // Any number to power 1 = itself }

How it works:

  • Power 0: always returns 1
  • Power 1: returns base itself
  • Stops recursion
  • Essential base cases

5. Handling Negative Exponents

Negative Exponent:

if (exponent < 0) { return 1.0 / power(base, -exponent); }

How it works:

  • Negative exponent: base^(-exp) = 1 / base^exp
  • Converts to positive exponent
  • Returns reciprocal
  • Handles negative exponents

6. Optimized Recursive Case

Even Exponent:

if (exponent % 2 == 0) { double half = power(base, exponent / 2); return half * half; }

How it works:

  • base^even = (base^(even/2))^2
  • Example: 2^8 = (2^4)^2 = 16^2 = 256
  • Reduces problem by half
  • More efficient

Odd Exponent:

else { double half = power(base, (exponent - 1) / 2); return base * half * half; }

How it works:

  • base^odd = base × (base^((odd-1)/2))^2
  • Example: 2^9 = 2 × (2^4)^2 = 2 × 256 = 512
  • Handles odd exponents
  • Maintains efficiency

7. Time Complexity

Efficiency:

  • Naive: O(n) - n multiplications
  • Optimized: O(log n) - log n recursive calls
  • Much faster for large exponents
  • Divide-and-conquer benefit

Example:

  • 2^1000: naive needs 1000 operations
  • Optimized: only ~10 recursive calls
  • Significant improvement

8. When to Use This Approach

Best For:

  • Large exponents
  • Performance-critical code
  • Mathematical computations
  • Efficient power calculation
  • Divide-and-conquer learning

Example Scenarios:

  • Cryptography (modular exponentiation)
  • Scientific calculations
  • Algorithm optimization
  • Mathematical libraries
  • Performance-critical applications

9. Important Considerations

Divide-and-Conquer:

  • Breaks problem in half
  • Reduces recursive calls
  • Logarithmic time complexity
  • Efficient algorithm

Exponent Types:

  • Handles positive exponents
  • Handles negative exponents
  • Handles zero exponent
  • Comprehensive coverage

Precision:

  • Uses double for decimal results
  • Handles fractional bases
  • Maintains precision
  • Suitable for various inputs

10. return 0;

This ends the program successfully.

Summary

  • Power calculation: optimized recursion using divide-and-conquer approach.
  • Even exponent: base^exp = (base^(exp/2))^2, odd: base^exp = base × (base^((exp-1)/2))^2.
  • Time complexity: O(log n) instead of O(n), much more efficient.
  • Handles negative exponents: base^(-exp) = 1 / base^exp.
  • Understanding optimized power calculation demonstrates efficient algorithm design.
  • Essential for performance-critical applications and mathematical computations.

This program is fundamental for learning optimized recursion, understanding divide-and-conquer, and preparing for efficient algorithm design in C++ programs.

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