A fast algorithm for multiplying polynomials. The naïve algorithm multiplies term by term, yielding time complexity of O(m*n) (where m,n are the number of terms in each polynomial, and assuming unit time for arithmetic operations).

But we can do better here as well! For simplicity, I show the case where n=m=2k+1. We wish to multiply P(x) by Q(x), so we write

P(x) = P1(x) + x2k ⋅ P2(x)
Q(x) = Q1(x) + x2k ⋅ Q2(x)
and we wish to calculate
P(x)*Q(x) = P1(x)*Q1(x) + x2k ⋅ (P1(x)*Q2(x) + P2(x)*Q1(x)) + x2k+1 ⋅ P2(x)*Q2(x).

So define 3 multiples:

• A(x) = P1(x)*Q1(x)
• B(x) = P2(x)*Q2(x)
• C(x) = (P1(x)+P2(x))*(Q1(x)+Q2(x))
Then
P(x)*Q(x) = A(x) + x2k ⋅ (C(x)-A(x)-B(x)) + x2k+1 ⋅ B(x).
Multiply each of the 3 products using a similar algorithm, until some small value of k is reached (say, k=1). The net effect is to translate the 4 multiplications required by the naïve algorithm into just 3 multiplications.

A similar calculation to that of the Strassen algorithm for matrix multiplication shows that the algorithm runs in time O(3k) = O(nlg2 3) = O(n1.585).

Similar algorithms exist for matrix multiplication and complex number multiplication.

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