[Paper Review] An algorithm for dividing two complex numbers

Aleksandr Cariow|arXiv (Cornell University)|Aug 26, 2016

Numerical Methods and Algorithms4 references3 citations

TL;DR

This paper presents a novel algorithm for dividing two complex numbers that reduces the number of real multiplications from four (in the standard method) to three, while increasing additions from three to six. The method achieves computational efficiency by rationalizing the division process through algebraic manipulation, offering a performance advantage in hardware implementations and numerical computation contexts where multiplication is costly.

ABSTRACT

In this work a rationalized algorithm for calculating the quotient of two complex numbers is presented which reduces the number of underlying real multiplications. The performing of a complex number division using the naive method takes 4 multiplications, 3 additions, 2 squarings and 2 divisions of real numbers while the proposed algorithm can compute the same result in only 3 multiplications ( or multipliers in hardware implementation case), 6 additions, 2 squarings and 2 divisions of real numbers.

Motivation & Objective

• To reduce the computational cost of complex number division, particularly the number of real multiplications required.
• To improve efficiency in hardware implementations where multiplication operations are expensive.
• To provide a rationalized alternative to the standard method of complex division that maintains accuracy while reducing arithmetic operations.
• To optimize the balance between multiplications, additions, squarings, and divisions in complex arithmetic.
• To present a practical algorithm suitable for use in embedded systems and high-performance computing environments.

Proposed method

• The algorithm rewrites the standard complex division formula using algebraic rationalization to minimize the number of real multiplications.
• It expresses the quotient of two complex numbers as a rationalized fraction, avoiding direct computation of four multiplications.
• The method computes the real and imaginary parts of the quotient using only three real multiplications, achieved through factoring and substitution techniques.
• The algorithm retains two squarings (for the denominator magnitude), two divisions, and increases additions to six for improved operation count balance.
• The approach is derived from the identity (a + bi)/(c + di) = [(a c + b d) + i(b c - a d)] / (c² + d²), optimized via strategic grouping.
• The method is designed to be directly implementable in hardware, where minimizing multipliers is critical for area and power efficiency.

Experimental results

Research questions

• RQ1 Can the number of real multiplications required for complex division be reduced below the standard four?
• RQ2 What trade-offs in additions and other operations arise when minimizing multiplications in complex division?
• RQ3 How can algebraic rationalization be applied to derive a more efficient complex division algorithm?
• RQ4 To what extent does the proposed algorithm improve performance in hardware or embedded systems?
• RQ5 Is the reduction in multiplications sufficient to justify the increase in additions for practical applications?

Key findings

• The proposed algorithm reduces the number of real multiplications from four to three, which is a significant improvement in computational efficiency.
• The algorithm requires six additions, compared to three in the naive method, representing a trade-off between addition and multiplication operations.
• The number of squarings remains at two, and divisions are unchanged at two, both of which are inherent to the complex division process.
• The method maintains numerical accuracy while optimizing for multiplication count, making it suitable for resource-constrained environments.
• The algorithm is particularly advantageous in hardware implementations where multipliers are expensive in terms of area and power.
• The overall operation count is optimized for scenarios where multiplication is the dominant computational cost.