euclidian-ring.rd

[index]

# EuclidianRing

(G.C.D. module)

This is the module for getting G.C.D. (the greatest common divisor) from divmod. This is included to Integer or Algebra::Polynomial.

## File Name:

- euclidian-ring.rb

## Methods:

`gcd(other)`

  Returns the greatest common divisor of self and other.

`gcd_all(other0 [, other1[, ...]])`

  Returns the greatest common divisor of self and other0, other1,...

`gcd_coeff(other)`

  Returns the array of the greatest common divisor of self and other and the coefficients for getting it.

  Example:

      require "polynomial"
      require "rational"
      P = Algebra.Polynomial(Rational, "x")
      x = P.var
      f = (x + 2) * (x**2 - 1)**2
      g = (x + 2)**2 * (x - 1)**3
      gcd, a, b = f.gcd_coeff(g)
      p gcd #=> 4x^3 - 12x + 8
      p a   #=> -x + 2
      p b   #=> x - 1
      p gcd == a*f + b*g #=> true

`gcd_ext(other)`

  Same as gcd_coeff

`gcd_coeff_all(other0 [, other1[, ...]])`

  Returns the array of the greatest common divisor self and other0, other1,.. and the coefficients for getting it.

  Example:

      require "polynomial"
      require "rational"
      P = Algebra.Polynomial(Rational, "x")
      x = P.var
      f = (x + 2) * (x**2 - 1)**2
      g = (x + 2)**2 * (x - 1)**3
      h = (x + 2) * (x + 1) * (x - 1)
      gcd, a, b, c = f.gcd_coeff_all(g, h)
      p gcd #=> -8x^2 - 8x + 16
      p a   #=> -x + 2
      p b   #=> x - 1
      p c   #=> -4
      p gcd == a*f + b*g + c*h #=> true

`gcd_ext_all(other0 [, other1[, ...]])`

  Same as gcd_coeff_all

`lcm(b)`

  Return of the least common multiple of self and other.

`lcm_all(other0 [, other1[, ...]])`

  Return of the least common multiple of self and other0, other1,...