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| 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. |
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. |
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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.''
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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."
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| 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 |
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 |
| $$A(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0, \quad B(x) = b_{n-1}x^{n-1} + \ldots +b_0$$ |
$$A(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0, \quad B(x) = b_{n-1}x^{n-1} + \ldots +b_0$$ |
| then |
then |
| $$quo(A(x), B(x)) = \frac{a_n}{b_{n-1}}x+\frac{b_{n-1}a_{n-1}-a_nb_{n-2}}{b_{n-1}^2}$$ |
$$quo(A(x), B(x)) = \frac{a_n}{b_{n-1}}x+\frac{b_{n-1}a_{n-1}-a_nb_{n-2}}{b_{n-1}^2}$$ |
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| 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. |
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. |
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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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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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