gcd

Arguments

P

array of decimal integers, encoded integers, or of polynomials.

gpcd

single element of P type: the greatest common divisor of all P components.

U

Square matrix of the P type, with integer components or of minimal degrees. Its last column B = U(:,$) holds Bezout coefficients, such that B(1)*P(1) + B(2)*P(2) + .... + B($)*P($) == gpcd.

Description

[gpcd, U] = gcd(P) computes the greatest common divisor gpcd of components of P, and an unimodular matrix U.

If P components are decimal or encoded integers, they are priorly converted into int64 signed integers.

If P has an unsigned inttype uint8, uint16, uint32 or uint64, U gets the corresponding signed inttype.

When P are integers, the returned GCD is always positive.

When a second output is expected, an unimodular matrix U of the P type is returned, such that

• size(U) == [length(P) length(P)]
• matrix(P,1,-1)*U = [0...0 gpcd] with length(P)-1 leading zeros
• det(U) is 1 or -1.

Its last column provides Bezout coefficients.

gcd([0 0]) returns 0.

For big P values (smaller but of the order of 2^63, depending also on the number of input values), results may be corrupted by integer overflow and wrapping (int8(127)+1 == -128).

Examples

// With polynomials s = %s; p = [s s*(s+1)^2 2*s^2+s^3]; [GCD, U] = gcd(p) p*U // With encoded integers V = uint16([2^2*3^5 2^3*3^2 2^2*3^4*5]) [GCD, U] = gcd(V) typeof(GCD) typeof(U) V*U // With decimal integers V = [2^2*3^5 2^3*3^2 2^2*3^4*5]; [GCD, U] = gcd(V) type(GCD) type(U) V*U gcd([0 60]) gcd([0 0])

See also

• bezout — GCD of two polynomials or two integers, by the Bezout method
• lcm — least common (positive) multiple of integers or of polynomials
• factor — factor function
• prod — product of array elements
• hermit — Hermite form

History