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// --- This file is distributed under the MIT Open Source License, as detailed
// in the file "LICENSE.TXT" in the root of this repository ---
// This file is a trimmed copy of extended_euclidean__a_ge_0__b_gt_a.h.
// It contains only the essential assertion results, without any of the proofs.
#ifndef ESSENTIAL_ASSERTS__A_GE_0__B_GT_A
#define ESSENTIAL_ASSERTS__A_GE_0__B_GT_A 1
#ifndef NDEBUG
# include "assert_helper_gcd.h"
#endif
#include <assert.h>
#include <limits>
#if defined(assert_precondition)
# error "assert_precondition was already defined"
#endif
// assert aliases will help self-document the code
#define assert_precondition assert
template <typename T>
void essential_asserts__a_ge_0__b_gt_a(T a, T b, T* pGcd, T* pX, T* pY)
{
static_assert(std::numeric_limits<T>::is_integer, "");
static_assert(std::numeric_limits<T>::is_signed, "");
assert_precondition(b > a);
assert_precondition(a >= 0);
assert(gcd(a,b) >= 1);
T x0 = 1, y0 = 0, a0 = a;
T x1 = 0, y1 = 1, a1 = b;
// The peeled-out first iteration:
{
T q = a0/a1;
assert(q == 0);
T a2 = a0 - q*a1;
assert(q*a1 == 0);
assert(0 <= a2 && a2 < a1);
assert(q <= a0/2);
T x2 = x0 - q*x1;
assert(abs(q*x1) <= abs(x2));
assert(abs(x1) <= abs(x2));
T y2 = y0 - q*y1;
assert(abs(q*y1) <= abs(y2));
assert(y1 == 1);
x0=x1; y0=y1; a0=a1;
x1=x2; y1=y2; a1=a2;
}
while (a1 != 0) {
T q = a0/a1;
assert(1 <= q && q <= a0);
T a2 = a0 - q*a1;
assert(0 < (q*a1) && (q*a1) <= a0);
assert(0 <= a2 && a2 < a1);
if (a2 != 0) assert(q <= a0/2);
T x2 = x0 - q*x1;
assert(abs(q*x1) <= abs(x2));
assert(abs(x1) <= abs(x2));
T y2 = y0 - q*y1;
assert(abs(q*y1) <= abs(y2));
assert(abs(y1) <= abs(y2));
x0=x1; y0=y1; a0=a1;
x1=x2; y1=y2; a1=a2;
}
assert(abs(x1) == b/gcd(a,b));
assert(abs(y1) == a/gcd(a,b));
assert(abs(x0) <= (b/gcd(a,b))/2);
assert(y0 == 1 || abs(y0) <= (a/gcd(a,b))/2);
assert(a0 == gcd(a,b));
assert(a*x0 + b*y0 == gcd(a,b));
*pX = x0;
*pY = y0;
*pGcd = a0;
}
#undef assert_precondition
#endif