Example of Extended Euclidean AlgorithmRecall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to write 3 as a linear combination of 84 and 33: 3 = 18 − 15 [Now 3 is a linear combination of 18 and 15] = 18 − (33...
扩展Euclidean 算法(Extended Euclidean Algorithm,算法竞赛中又称为exgcd)是用来求解 “给定正整数x,y,求满足ax+by=gcd(x,y)的某个特解(a′,b′)” 的算法。 这里顺便一提 ax+by=gcd(x,y) 这个不定方程的通解,实际上,如果有特解(a′,b′),这个通解是 (a,b)=(a′+kygcd(x,y),b′−kxgcd(...
Lemma 12. The input pair and the output pair of a step of the Euclidean algorithm have the same GCD. Proof. Let S1 be the set of common divisors of the input (a, b), and let S2 be the set of common divisors of the output (b, r). Recall that a = bq + r, so r = a ...
Difficulty: 3. Solved by 283 people. The Extended Euclidean Algorithm gives a quick way to calculate the greatest common divisor between two numbers: given two whole numbers x x and y y with x > y x>y,GCD( x x, y y) = GCD( y y, x x% y y). Did Euclid believe in God? Eucl...
Buchberger's algorithm for computing strong standard bases (sBBA). A first criterion takes care of useless GCD-polynomials and goes back to Pan: Lemma 13 Pan (1989) Let f,g∈P such that lc(f)|lc(g) or lc(g)|lc(f). Then gpoly(f,g) reduces to zero w.r.t. {f,g}. Proof...
For a concrete example, take (a,b)=(14,8) over R=Z, for which n=2, q1=q2=1, q3=3, and r2=gcd(14,8)=2. Applying (c) above and thereafter the inductive proof of Lemma 2, we get the following factorization of A into n+2=4 idempotents:(14800)=(1100)(0021)(43−4−3...
Finally, the proposed algorithm is verified experimentally. It is demonstrated that cubic PHH-Bézier curves can accurately approximate H-Bézier curves but that the selection of the middle two control points of the PHH-Bézier curve has a profound impact on the quality of the approximation. ...