Re: InequalitySolve with algebraic numbers and Simplify

*To*: mathgroup at smc.vnet.net*Subject*: [mg23004] Re: [mg22987] InequalitySolve with algebraic numbers and Simplify*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Mon, 10 Apr 2000 02:22:31 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

on 00.4.9 2:45 PM, Gianluca Gorni at gorni at dimi.uniud.it wrote: > > Hello! > > I had a system of *linear* inequalities in two variables x,y, > with simple algebraic numbers as coefficients, > to solve with InequalitySolve[], and I naively assumed > that the solution would be a set of likewise *linear* > inequalities. So I was surprised at results like this: > > Needs["Algebra`InequalitySolve`"] > > InequalitySolve[{y <= x*Sqrt[2], y <= x}, {x, y}] > > x <= 0 && y <= -(Sqrt[2]*Sqrt[x^2]) || x > 0 && y <= x > > Strictly speaking the result is correct, but it does not look > good, because of that Sqrt[x^2]. I tried with > Experimental`CylindricalAlgebraicDecomposition, > but it gives exactly the same answer. > > Trying to reduce the results to a more manageable form, I > met some somewhat disappointing behaviour of Simplify[] > with assumptions (of Mathematica version 4): > > In: Simplify[Sqrt[x^2], x == 1 + Sqrt[2]] > > Out: Sqrt[x^2] > > It seems that Mathematica doesn't notice that 1+Sqrt[2] is real and > positive! So let's teach Mathematica that it is real, at least: > > In: Simplify[Sqrt[x^2], {Element[x, Reals], x == 1 + Sqrt[2]}] > > Out: x > > Somehow I would have expected the answer to be 1+Sqrt[2], what > about you? Even if x has a smaller LeafCount than 1+Sqrt[2]. > > Best regards, > > Gianluca Gorni It seems to me that you have come across some unpleasant quirks in Mathematica if not exactly bugs. As for the first one, I don't now exactly the cause of it but in general it seems that Mathematica generally has problems with expressions like Sqrt[x^2] in logical expressions. For example, compare the following behaviour: In[1]:= And[x < 3, x < 2] // Simplify Out[1]= x < 2 In[2]:= And[x < 4, x > 5] // Simplify Out[2]= False In[3]:= And[x > 3, x^2 < 9] // Simplify Out[3]= False In[4]:= And[x > 3, Sqrt[x^2] < 3] // Simplify Out[4]= 2 x > 3 && Sqrt[x ] < 3 That last one should have been false, of course. On the other Simplify with assumptions behaves correctly in such cases: In[5]:= Simplify[x > 3, Sqrt[x^2] < 3] Out[5]= False This makes me feel that it should not be impossible to fix the first problem you pointed out. For example one might define a function: mysimplify[expr_] := expr /. And[LessEqual[x_, 0], y_] /; Not[FreeQ[y, x]] :> And[LessEqual[x, 0], Simplify[y, LessEqual[x, 0]]] Now, In[7]:= Experimental`CylindricalAlgebraicDecomposition[{y <= x*Sqrt[2], y <= x}, {x,y}] Out[7]= 2 x <= 0 && y <= -Sqrt[2] Sqrt[x ] || x > 0 && y <= x We can improve this by applying mysimplify In[8]:= mysimplify[%] Out[8]= x <= 0 && y <= Sqrt[2] x || x > 0 && y <= x gives the kind of answer one would expect. Of course one would need something a lot more sophisticated to deal with all the cases in which this sort of problem may arise. The second one has indeed something to do with the ComplexityFunction. Mathematica does indeed give; In[9]:= Simplify[Sqrt[x^2], { x == 1 + Sqrt[2]}] Out[9]= 2 Sqrt[x ] Suppose however, we define a complexity function f by: cf[expr_] := Count[expr, _?(Not[NumericQ[#]] &), {0, Infinity}] Now we notice something curious: In[11]:= Simplify[Sqrt[x^2], { x == 1 + Sqrt[2]}, ComplexityFunction -> cf] Out[11]= Sqrt[3 + 2 Sqrt[2]] That's a bit weird. If we try FullSimplify (Simplify does not work here!) we indeed get: In[12]:= FullSimplify[%] Out[12]= 1 + Sqrt[2] Why did we not get this answer the first time? Both 1+Sqrt[2] and Sqrt[3 + 2 Sqrt[2]] have the same value of cf, namely 0, but of course the firs tone has much smaller LeafCount. It seems that Simplify never considered the simpler answer. This happens even if we modify cf so as to make cf[1+Sqrt[2]] explicitly smaller than cf[Sqrt[3 + 2 Sqrt[2]]: In[16]:= Clear[cf] In[17]:= cf[expr_] := 10*Count[expr, _?(Not[NumericQ[#]] &), {0, Infinity}] + LeafCount[expr] In[18]:= cf[1 + Sqrt[2]] Out[18]= 7 In[19]:= cf[Sqrt[3 + 2*Sqrt[2]]] Out[19]= 13 However: In[20]:= Simplify[Sqrt[x^2], { x == 1 + Sqrt[2]}, ComplexityFunction -> cf] Out[20]= Sqrt[3 + 2 Sqrt[2]] Fortunately In[21]:= FullSimplify[Sqrt[x^2], { x == 1 + Sqrt[2]}, ComplexityFunction -> cf] Out[21]= 1 + Sqrt[2] Note also that with the previous complexity function cf, which is zero on both 1+Sqrt[2] and Sqrt[3 + 2 Sqrt[2]]) even FullSimplify chooses the latter answer!. All of it does seem weirs and in need of improvement. -- Andrzej Kozlowski Toyama International University Toyama, Japan

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