Calculation Techniques
Core Calculation Methods
1. Addition and Subtraction
class BigNumber {
public:
BigNumber add(const BigNumber& other) {
std::vector<int> result;
int carry = 0;
int maxLength = std::max(digits.size(), other.digits.size());
for (int i = 0; i < maxLength; ++i) {
int sum = carry;
if (i < digits.size()) sum += digits[i];
if (i < other.digits.size()) sum += other.digits[i];
result.push_back(sum % 10);
carry = sum / 10;
}
if (carry > 0) {
result.push_back(carry);
}
return BigNumber(result);
}
};
2. Multiplication Techniques
graph TD
A[Multiplication Methods]
A --> B[Naive Algorithm]
A --> C[Karatsuba Algorithm]
A --> D[FFT-based Multiplication]
Karatsuba Multiplication
BigNumber karatsuba_multiply(BigNumber x, BigNumber y) {
int n = std::max(x.size(), y.size());
// Base case
if (n < 10) {
return naive_multiply(x, y);
}
// Split numbers
int mid = n / 2;
BigNumber a, b, c, d;
split_number(x, a, b, mid);
split_number(y, c, d, mid);
// Recursive multiplication
BigNumber ac = karatsuba_multiply(a, c);
BigNumber bd = karatsuba_multiply(b, d);
BigNumber ad_plus_bc = karatsuba_multiply(a+b, c+d) - ac - bd;
return ac * pow(10, 2*mid) + ad_plus_bc * pow(10, mid) + bd;
}
Division Strategies
Method |
Complexity |
Precision |
Long Division |
O(nĀ²) |
High |
Newton-Raphson |
O(log n) |
Very High |
Recursive Division |
O(n log n) |
Moderate |
3. Advanced Division Algorithm
BigNumber divide(BigNumber dividend, BigNumber divisor) {
if (divisor == 0) {
throw std::runtime_error("Division by zero");
}
BigNumber quotient, remainder;
// Implement long division algorithm
while (dividend >= divisor) {
dividend -= divisor;
quotient++;
}
remainder = dividend;
return quotient;
}
Modular Arithmetic
Modular Exponentiation
BigNumber modular_pow(BigNumber base, BigNumber exponent, BigNumber modulus) {
BigNumber result = 1;
base %= modulus;
while (exponent > 0) {
if (exponent % 2 == 1) {
result = (result * base) % modulus;
}
exponent >>= 1;
base = (base * base) % modulus;
}
return result;
}
Optimization Considerations
- Minimize unnecessary computations
- Use efficient memory management
- Implement lazy evaluation techniques
- Leverage compiler optimizations
Practical Challenges
graph LR
A[Calculation Challenges]
A --> B[Precision Limits]
A --> C[Performance Overhead]
A --> D[Memory Constraints]
Conclusion
Mastering big number calculation techniques requires understanding various algorithms and their trade-offs. LabEx recommends continuous practice and exploring advanced mathematical libraries for complex computations.