MaplePrimes - answers and comments on Question, The integer solutions of a linear inequality system

By slices with DirectSearch

Markiyan Hirnyk — Mon, 09 Jul 2012 18:36:18 Z

Here is my procedure to do it.
We find the nonnegative integer solutions of the equation sum(s*x[s], s = 1 .. i) - N = 0,
where k-2*i <= N and N <= k-i, with the DirectSearch package
http://www.mapleprimes.com/posts/101374-DirectSearch-Optimization-Package-Version-2 .

>a:=proc (i::posint, k::posint)
local s,N;
if 2*i <= k and 2 <= i then
seq(DirectSearch:-SolveEquations([sum(s*x[s], s = 1 .. i) - N = 0],
{seq(0 <= x[s], s = 1 .. i)}, assume = integer, AllSolutions), N = k - 2*i ..k - i)
else
print(Invalid Input)
end if
end proc
For instance,
>a(3,7);
x[2] = 1, x[3] = 0], (4, 4) = 246})

This takes a time. If the AllSolutions option is omitted, then only a part of the solutions is obtained.

Procedure

Kitonum — Mon, 09 Jul 2012 18:50:48 Z

The procedure P solves your problem:

P:=proc(k, i)

local L, s, M, K;

if not (k>=2*i and 2*i>=4) then

error `should be k>=2*i>=4` fi;

if not type(k, posint) or not type(i, posint) then

error `should be k is integer and i is integer` fi;

L:=[];

for s from k-2*i to k-i do

M:=combinat[composition](s+i, i);

K:=seq([seq(M[j,l]-1, l=1..i)], j=1..nops(M));

L:=[op(L), K];

od;

end proc;

Example:

P(8, 3);

Thanks

kwgl — Thu, 12 Jul 2012 07:21:37 Z

Dear Markiyan and Kitonum, thank you two! I am not sure how to install DirectSearch on Mac, and so did not use the package. Inspired by your algorithms, finally I chose the following codes to use:

with(combinat):

Timing

Markiyan Hirnyk — Mon, 09 Jul 2012 19:08:51 Z

282.190

Error, (in unknown) object too large
35.708

I am not strong in programming.

PS. A good code is a commented code.

@Markiyan Hirnyk Unfortunately, there

Kitonum — Mon, 09 Jul 2012 22:19:54 Z

@Markiyan Hirnyk

Unfortunately, there is a limit in Maple on the length of the list. Run the following code (all permutations of 10 elements):

combinat[permute](10);

There are ways around this restriction, for example, dividing the list into several parts.

Something to adjust

Markiyan Hirnyk — Mon, 09 Jul 2012 23:51:00 Z

The triple [x[1]=1.x[2]=2,x[3]=2] does not satisfy 11 = 1+2*2+3*2 = x[1]+2*x[2]+3*x[3] <= k-i = 8- 3 =5.

Let us suppose the inverse reverse order. Then the triple [x[1]= 2, x[2]=2 , x[3] =1] does not also satisfy it.