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'fast Euclidean algorithm'
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| Title of object: |
fast Euclidean algorithm |
| Canonical Name: |
FastEuclideanAlgorithm |
| Type: |
Algorithm |
| Created on: |
2004-07-22 16:58:15 |
| Modified on: |
2005-03-28 20:27:13 |
| Classification: |
msc:11A05 |
| Keywords: |
Lehmer, Euclidean Algorithm, GCD, Donald Knuth |
| Synonyms: |
fast Euclidean algorithm=Half-GCD Algorithm
fast Euclidean algorithm=Lehmer's Algorithm |
Revision comment (for changes between this and next version):
| Changes for correction #6181 ('quote marks should look like ``this'', because ``this" doesn't work with latex2html'). |
Preamble:
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\usepackage{amssymb}
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Content:
Given two polynomials of degree $n$ with coefficients from a field $K$, the straightforward Eucliean Algorithm uses $O(n^2)$ field operations to compute their greatest common divisor. The Fast Euclidean Algorithm computes the same GCD in $O(\mathsf{M}(n) \log(n))$ field operations, where $\mathsf{M}(n)$ is the time to multiply two $n$-degree polynomials; with FFT multiplication the GCD can thus be computed in time $O(n \log^2(n) \log(\log(n)))$. The algorithm can also be used to compute any particular pair of coefficients from the Extended Euclidean Algorithm, although computing every pair of coefficients would involve $O(n^2)$ outputs and so the efficiency is not as helpful when all are needed.
The algorithm can be made to work over $\mathbb{Z}$ but it is very tricky. A newer version that is easier to understand was published by Damien Stehl\'e and Paul Zimmerman, ``A Binary Recursive Gcd Algorithm.''
Here we sketch the algorithm over $K[x]$. The basic idea is that the quotients $q_i$ computed by the Euclidean Algorithm can usually be computed by looking at only the first few coefficients of the polynomial; for example, if
$$A(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0, \quad B(x) = b_{n-1}x^{n-1} + \ldots +b_0$$
then
$$quo(A(x), B(x)) = \frac{a_n}{b_{n-1}}x+\frac{b_{n-1}a_{n-1}-a_nb_{n-2}}{b_{n-1}^2}$$
With more detailed analysis, we can show that in fact a divide-and-conquer approach can be used to calculate the GCD. First, we remove the terms whose degree is $n/2$ or less from both polynomials $A$ and $B$. Then, we recursively compute their GCD and Euclidean coefficients. We then apply the Euclidean coefficients to $A$ and $B$, and recursively complete the Euclidean Algorithm.
The full algorithm, and a comprehensive runtime analysis is given in ``Modern Computer Algebra'' by von zur Gathen and Gerhard. (More details to come, I highly suggest the book if you are interested and impatient) |
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