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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 combines the essential assertions from the three cases of the proof
// (a<b, a==b, and a>b), by combining the assertions in the files
// essential_asserts_collins.h (which covers a<b),
// essential_asserts__b_eq_0__b_eq_a.h and essential_asserts__b_gt_0__b_eq_a.h
// (which both cover a==b), and essential_asserts__a_ge_0__b_gt_a__improved.h
// (which covers a>b).
//
// As a result, this file realizes the final goal of providing all essential
// asserts for the Extended Euclidean algorithm (given preconditions a>=0 and
// b>=0).
#ifndef ESSENTIAL_ASSERTS_COMBINED
#define ESSENTIAL_ASSERTS_COMBINED 1
#ifndef NDEBUG
# include "helpers/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_combined(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 >= 0 && a >= 0);
T x0 = 1, y0 = 0, a0 = a;
T x1 = 0, y1 = 1, a1 = b;
while (a1 != 0) {
T q = a0/a1;
assert(0 <= 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(y1 == 1 || abs(y1) <= abs(y2));
x0=x1; y0=y1; a0=a1;
x1=x2; y1=y2; a1=a2;
}
if (a == 0 && b == 0) {
assert(x1 == 0);
assert(y1 == 1);
assert(x0 == 1);
assert(y0 == 0);
} else {
assert(gcd(a,b) >= 1);
assert(abs(x1) == b/gcd(a,b));
assert(abs(y1) == a/gcd(a,b));
assert(x0 == 1 || 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