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I'm trying to get maple to solve some inequalities for me, and in particular to tell me when they have no solution. Unfortunately, it does not seem to be doing what I expect.

The following inequalities have no solution.

\begin{align*} 0 &\lt f[1], &0 &< f[2], \\0 &< f[3], &0 &< f[4], \\ 8 &< f[1]+f[3], &6 &< f[2]+f[4],\\ 6 &< f[2]+f[3], &7 &< f[1]+f[4],\\ 5&=f[1]+f[2], & 11/2&=f[3]+f[4] \end{align*}

This is because the left most bottom two expressions imply $1+f[1]<f[3]$ whereas the right-most bottom two expressions imply that $f[3]<f[1]-3/2$.

Yet if I put the following into maple 15

solve({f[1]+f[2] = 5, f[3]+f[4] = 11/2,  0 <f[1], 0 < f[2], 0 < f[3], 0 < f[4], 
         6 < f[2]+f[3], 6 < f[2]+f[4], 7 < f[1]+f[4], 8 < f[1]+f[3]},
                                                       {f[1],f[2],f[3],f[4]});

then it gives the following "solution":

{f[1] = 5-f[2], f[2] = f[2], f[3] = 11/2-f[4], f[4] = f[4]}

Who is being stupid here? Me or maple?

Simon Willerton
  • 178

2 Answers2

6

Same happens in Maple 16. When the system with inequalities is inconsistent in a non-obvious way, Maple sometimes disregards all inequalities and returns a solution based on equations only. Probably should be a bug report; the Maple help file claims that systems of linear equations and inequalities are fully supported.

Here is a simpler example of this kind:

solve({ a + b = 1, a > 0, b > 1, c > 0 }, {a,b,c});

output: {a = -b+1, b = b, c = c}

The inconsistency of inequalities is non-obvious since it arises when the equation is taken into account.

Oddly enough, dropping the irrelevant inequality $c>0$ causes Maple to give correct answer: no solutions.

solve({a+b = 1, a>0, b>1}, {a,b,c});

output: nothing

Definitely a bug.

1

In 16.01 on 64bit Linux on an Intel i5, after about 5 minutes,

SolveTools:-SemiAlgebraic({f1+f2 = 5, f3+f4 = 11/2,
                           0 <f1, 0 < f2, 0 < f3, 0 < f4,
                           6 < f2+f3, 6 < f2+f4,
                           7 < f1+f4, 8 < f1+f3},
                           [f1,f2,f3,f4]);

                         []

which, with [] as output, means no solutions.

More quickly,

SolveTools:-SemiAlgebraic({ a + b = 1, a > 0, b > 1, c > 0 }, {a,b,c});

                                   []

However, the computation time grows quickly with problem size, for this approach.

The solve command is using an older, faster, and buggier algorithm here.

acer
  • 5,343