lcm

Arguments

p

matrix of polynomials (type 2), or of decimal or encoded integers (types 1 or 8).

pp

a polynomial or a decimal integer: Positive Least Common Multiple of p components.

fact

matrix of polynomials, or of decimal integers (type 1), with the size of p, such that fact(i)= pp./p(i).

Description

pp=lcm(p) computes the LCM pp of p components.

If p are polynomials, pp is a polynomial and fact is also a matrix of polynomials.

If p is a set of integers,

• if they are encoded integers, they are then converted into decimal integers before processing.
• Any int64 or uint64 input |integers| > 2^53 will be truncated and lcm() will return a wrong result.
• If some of them are negative, the returned value pp of their LCM is always positive.

The least common multiple of an array p of real numbers can be obtained by converting it to a polynomial before calling lcm, through p = inv_coeff(p, 0).

Examples

With polynomials:

s = %s;
p = [s , s*(s+1) , s^2-1]
[pp, fact] = lcm(p)
p .* fact == pp

--> p = [s , s*(s+1) , s^2-1]
 p  =
           2       2
   s   s +s   -1 +s

--> [pp, fact] = lcm(p)
 fact  =
       2
  -1 +s   -1 +s   s

 pp  =
       3
  -s +s

--> p .* fact == pp
 ans  =
  T T T

With encoded integers:

// Prime numbers: 2  3  5  7  11  13  17  19  23  29  31  37  41  43  47
V = int16([2*3 3*7 ; 7*5  3*5])
[pp, fact] = lcm(V)

--> V = int16([2*3 3*7 ; 7*5  3*5])
 V  =
   6  21
  35  15

--> [pp, fact] = lcm(V)
 pp  =
   210.

 fact  =
   35.   10.
   6.    14.

With decimal integers:

V = [2*3 3*7 ; 7*5  3*5]
[pp, fact] = lcm(V)

With big integers:

V = [3*2^51 , 3*5]
[pp, fact] = lcm(V)    // OK

--> V = [3*2^51 , 3*5]
 V  =
   6.755D+15   15.

--> [pp, fact] = lcm(V)
 fact  =
   5.   2.252D+15

 pp  =
   3.378D+16

When the numerical encoding is overflown, truncature occurs and results turn wrong:

V = [3*2^52 , 3*5]
[pp, fact] = lcm(V)

--> V = [3*2^52 , 3*5]
 V  =
   1.351D+16   15.

--> [pp, fact] = lcm(V)
 fact  =
   15.   1.351D+16

 pp  =
   2.027D+17

See also

• gcd — Greatest (positive) Common Divisor
• bezout — Bezout法により、2つの多項式または2つの整数の最大公約数を計算します
• factor — 素因数分解関数

History