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Re: Common Divisor’s computing -> Non-Linear Programming task
Posted:
Jun 22, 2009 6:52 PM
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Author: Yuly Shipilevsky
Algorithms for finding the Common Multiples and Least Common Multiple .
(Reducing of the Common Multiples computing to the solution of a
Non-Linear Programming/Optimization problem).
First algorithm.
Let N be a set of positive integers, N = {1, 2, . . . }
Let N^M denotes a Cartesian product of M sets N,
Let m denotes an arbitrary M-tucle (m1 , . . . , mM) contained in N^M.
Let CM be a Common Multiple of X1, . . . , XM.
There exists an unique M-tucle m0=(m01 , . . . , m0M) that corresponds to the
M-tucle X=( X1, . . . , XM) so that:
(1) CM = m0iXi , 1<= i <= M.
Let lcm(X1, . . . , XM) denotes the Least Common Multiple of X1, . . . , XM.
Lets define the following function:
(2) S2(m,X) = SUM_i,j=1...M,i>j[miXi - mjXj]^2 ,
where m=(m1 , . . . , mM) is contained in N^M
Proposition 1 (Minimum Principle).
for any
M-tucle m contained in N^M the following is true:
min_m { S2(m,X) | N^M } = S2(m0,X) = 0,
(the minimum is being searched amongst all the m
contained in N^M),
S2(m,X) > = S2(m0,X) = 0,
If and only if m0 contained in N^M, corresponds to the
M-tucle X=( X1, . . . , XM) and satisfies (1).
Proof.
It follows from Xs Common Multiples existence and (1) , (2) .
On the base of Proposition 1 we can suggest a
Theorem 1 (Reducing of Common Multiples computing to the solution of a Integer Programming/Optimization problem).
To compute the Least Common Multiples of X1, . . . , XM do the following steps:
(a) Find M-tucles m contained in N^M and minimizing S2
(S2 -> min | N^M)
(b) For each and every M-tucle, determined at step (a) find the
corresponding Common Multiple of X using (1)
(the minimum is being searched amongst all the m
contained in N^M).
Proof.
It follows from the Proposition 1.
Lemma 1.
If S_R_ N^M = {m: mi >= 1, mi - real numbers,
1<= i <=M } ,
then
min_m { S2(m,X) | N^M } =
min_m { S2(m,X) | S_R_ N^M} = 0
Proof.
It follows from the Proposition 1 and (1) , (2).
On the base of Lemma 1 and Theorem 1 we can suggest a
Theorem 2 (Reducing of Common Multiples computing to the solution of a Non-Linear Programming
/Optimization problem).
To compute the Common Multiples of X1, . . . , XM do the following steps:
(a) Find M-tucles m contained in S_R_N^M and minimizing S2
(S2 -> min | S_R_N^M)
(b) For each and every M-tucle, determined at step (a) find the
corresponding Common Multiple of X using (1)
(the minimum is being searched amongst all the m
contained in S_R_N^M).
Second algorithm.
Lets define the following function:
(3) S3(m,X) = SUM_i,j=1...M,i>j [miXi - mjXj]^2 + mkXk ,
where m=(m1 , . . . , mM) is contained in N^M,
1 <= k <= M.
Proposition 2 (Minimum Principle).
The following is true:
lcm(X1, . . . , XM) = min_m { S3(m,X) | N^M } = S3(m0,X)
(the minimum is being searched amongst all the m
contained in N^M).
If and only if m0 corresponds to the
M-tucle X=( X1, . . . , XM) , satisfies
(1) and corresponds to the Least Common Multiple
lcm(X1, . . . , XM).
Proof.
It follows from Xs Common Multiples existence and (1) , (3) .
On the base of Proposition 2 we can suggest a
Theorem 3. (Reducing of lcm computing to the solution of a Integer Programming
/Optimization problem).
To compute the Least Common Multiple of X1, . . . , XM do:
(a) Find an M-tucle m0 contained in N^M and minimizing S3
(S3 -> min | N^M)
(b) Find the Least Common Multiple corresponding to the M-tucle m0 determined at step (a)
as follows:
lcm(X1, . . . , XM) = min_m { S3(m,X)
| N^M } = S3(m0,X).
(the minimum is being searched amongst all the m
contained in N^M).
Lemma 2.
If S_R_ N^M = {m: mi >= 1 , mi - real numbers,
1<= i <=M } ,
then
min_m { S3(m,X) | N^M } = min_m { S3(m,X) | S_R_ N^M}
Proof.
It follows from Proposition 2, the fact that
SUM_i,j=1...M,i>j [miXi - mjXj]^2 >= 0 and
continuosity and monotonicity of mkXk function.
On the base of Lemma 2 and Theorem 3 we can suggest a
Theorem 4. (Reducing of lcm computing to the solution of a Non-Linear Programming
/Optimization problem).
To compute the Least Common Multiple of X1, . . . , XM do:
(a) Find an M-tucle m0 contained in S_R_N^M and minimizing S3
(S3 -> min | S_R_N^M)
(b) Find the Least Common Multiple corresponding to the M-tucle m0 determined at step (a)
as follows:
lcm(X1, . . . , XM) = min_m { S3(m,X) |
S_R_N^M } = S3(m0,X).
(the minimum is being searched amongst all the m
contained in S_R_N^M).
Example.
Finding the Least Common Multiple of 9636, 1456 and 2098 is being reduced to the
following Non-Linear Programming task:
S3(m , X) = ( 9636 m1 - 1456 m2 ) ^2 +
( 1456 m2 - 2098m3 ) ^2 + 9636 m1 -> min
m =(m1, m2, m3) ,
X=(9636, 1456, 2098),
m1, m2, m3 - real numbers,
satisfying the following constraints:
m1 >= 1,
m2 > = 1 ,
m3 >= 1
lcm(9636, 1456, 2098) = min_m { S3(m , X) |
m1 >= 1 , m2 >= 1 , m3 >= 1 }
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