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sursumCorda

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More credible functions that handle real...

sursumCorda 1254
October 30 2024
0 0

Consider the following two systems over the reals:

sys__1:=(r*(387*r+52)+2<r*(226*q+121*s)+9*q*(q*(2*q-5)-3*s+2)+6*s,4*q**3+r*(27*r+4)+s**2=q*(q+18*r),q>=0,r>=0):
sys__2:=((392-1739*q)*r+4*(2-9*q)**2+2151*r**2<75*r*s,4*q**3+r*(27*r+4)+s**2=q*(q+18*r),q>=0,r>=0):

The following results indicate that both and are satisfiable 

QuantifierElimination:-QuantifierEliminate(exists([s,q,r],And(sys__1)));
                              true

QuantifierElimination:-QuantifierEliminate(exists([s,q,r],And(sys__1)));
                              true

but RealDomain:-solve simply returns an empty list (that is, no solution exists) in both cases

RealDomain:-solve(And(sys__1),[q,s,r]); # thus sys1 cannot be satisfied
                               []

RealDomain:-solve(And(sys__2),[q,s,r]); # thus sys2 cannot be satisfied
                               []

Evidently, the above two results conflict with each other, which suggests that there must be a bug in one of these two functions. 
As discussed in the previous problem, the use of RealDomain:-solve is unsafe, Nevertheless, it appears that using QuantifierElimination:-QuantifierEliminate is still not always safe (since the unsatisfiability is also checked by another software). 

Updated. The “inconsistency” problem still exists in Maple 2025: 
 

> 

restart;
> 

kernelopts('version');

`Maple 2025.1, X86 64 WINDOWS, Jun 12 2025, Build ID 1932578`

 (1)
> 

`#msub(mi("sys"),mi("1"))` := r*(387*r+52)+2 < r*(226*q+121*s)+9*q*(q*(2*q-5)-3*s+2)+6*s, 4*q^3+r*(27*r+4)+s^2 = q*(q+18*r), q >= 0, r >= 0
> 

`#msub(mi("sys"),mi("2"))` := (392-1739*q)*r+4*(2-9*q)^2+2151*r^2 < 75*r*s, 4*q^3+r*(27*r+4)+s^2 = q*(q+18*r), q >= 0, r >= 0
> 

RegularChains:-SemiAlgebraicSetTools:-QuantifierElimination(`&E`([q, r]), `&and`(sys__1))
> 

RegularChains:-SemiAlgebraicSetTools:-QuantifierElimination(`&E`([q, r]), `&and`(sys__2))

false

  

false

 (2)
> 

QuantifierElimination:-QuantifierEliminate(exists([q, r], And(sys__1)))
> 

QuantifierElimination:-QuantifierEliminate(exists([q, r], And(sys__2)))

s-RootOf(27*_Z^3-216*_Z^2+9*_Z+1, 17498784809929/2199023255552 .. 139990278479459/17592186044416) = 0

  

s-RootOf(13500*_Z^2-210681*_Z-9826, 144368837035598577819/9223372036854775808 .. 288737674071197155665/18446744073709551616) = 0

 (3)
> 

NULL


 

Download 239277-Which-Function-Is-More-Credible.mw

Maple should still try harder ...

sursumCorda 1254 Maple 2022
optimization extrema
January 06 2023
0 4

In an old post, vv reported a bug in simpl/max, which has been "fixed" in Maple 2018. However, it seem that such repairs are not complete enough.
For example, suppose it is required to find the (squared) distance between the origin and a point on x3 - x + y2 = ⅓ which is closest to the origin. In other words, one needs to minimize x²＋y² among the points on this curve, i.e., 

extrema(x^2 + y^2, {x^3 + y^2 - x = 1/3}, {x, y}, 's'); # in exact form

Unfortunately, an identical error message appears again: 

> 

restart;
> 

extrema(x^2+y^2, {x^3+y^2-x = -2/(3*sqrt(3))}, {x, y})

{4/3}

 (1)
> 

extrema(x^2+y^2, {x^3+y^2-x = 1/3}, {x, y})

Error, (in simpl/max) complex argument to max/min: 1/36*((36+12*I*3^(1/2))^(2/3)+12)^2/(36+12*I*3^(1/2))^(2/3)

  
> 

`~`[`^`](extrema(sqrt(x^2+y^2), {x^3+y^2-x = 1/3}, {x, y}), 2)

{4/3, 4/27}

 (2)
> 

extrema(x^2+1/3-x^3+x, {x^3+y^2-x = 1/3}, {x, y})

{4/3, 4/27}

 (3)
> 

MTM[limit](extrema(x^2+y^2, {x^3+y^2-x = a}, {x, y}), 1/3)

{4/3, 4/27}

 (4)

Download tryHarder.mws

How about changing the values of parameter ?

for a from -3 by 3/27 to 3 do
    try
        extrema(x^2 + y^2, {x^3 + y^2 - x = a}, {x, y}); 
    catch:
        print(a); 
    end;
od;
                               -1
                               --
                               3 

                               -2
                               --
                               9 

                               -1
                               --
                               9 

                               1
                               -
                               9

                               2
                               -
                               9

                               1
                               -
                               3

By the way, like extrema, Student[MultivariateCalculus]:-LagrangeMultipliers also executes the Lagrange Multiplier method, but strangely, 

Student[MultivariateCalculus][LagrangeMultipliers](y^2 + x^2, [x^3 + y^2 - x - 1/3], [x, y], output = plot):

does not cause any errors.

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