仅使用三个乘法的复数乘积

问题描述

我们进行复数乘法，如下所示:

We do complex number multiplication as follows:

(a + i * b) * (c + i * d) = (a * c - b * d) + i * (a * d + b * c)

结果的实部和虚部是

real part = (a * c - b * d)
imag part = (a * d + b * c)

这涉及四个实数乘法.只有三个实数乘法怎么办?

This involves four real multiplications. How can we do it with only three real multiplications?

推荐答案

您对两个数字感兴趣:A=ac−bd和B=ad+bc.计算三个实数乘法S1=ac，S2=bd和S3=(a+b)(c+d).现在您可以将结果计算为A=S1−S2和B=S3−S1−S2.

You are interested in two numbers : A=ac−bd and B=ad+bc. Compute three real multiplications S1=ac, S2=bd, and S3=(a+b)(c+d). Now you can compute the results asA=S1−S2 and B=S3−S1−S2.

此过程称为Karatsuba乘法，在算法分析中大量使用.

This process is called Karatsuba multiplication and used heavily in Algorithm analysis.

它用于查找最接近的一对点.

It is used to find the closest pair of points.

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