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gcd

Greatest (positive) Common Divisor

Syntax

gpcd = gcd(P)
[gpcd, U] = gcd(P)

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])

• 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

 Version Description 6.0.1 int64 and uint64 input integers are now supported. The input P may be any array instead of a row vector. For input integers possibly negative, the returned GCD is now always positive.