7.129 Bug in solve in Maple V.5 (3.3.99)

7.129.1 Norbert Van den Bergh

It looks as if we have a new Release 5.1 BUG (I only tried it under the Windows 95 version). Try this:

>restart: 
>solve(z*cos(a)^2+x*sin(a)*cosa+x^2,x);
 

The answer is what you would expect ... Now go for this one:

>restart; 
>solve(z*cos(a)^2+x*sin(a)*cos(a)+x^2,x);
 

It is corrected with Maple 6. (U. Klein)

7.129.2 Joe Riel (4.3.99)

I’m not sure I’d classify this as a bug (the null answer isn’t wrong) so much as a deficiency. R4 could get this one, but neither R5 nor R5.1 can. However, frontend will permit a solution, and a slightly nicer one then R4 using only solve:

> eq := z*cos(a)^2+x*sin(a)*cos(a)+x^2: 
> frontend(solve,[eq,{x}],[{},{x,z}]); 
                                      2 
  {x = (- 1/2 sin(a) + 1/2 sqrt(sin(a)  - 4 z)) cos(a)}, 
                                            2 
        {x = (- 1/2 sin(a) - 1/2 sqrt(sin(a)  - 4 z)) cos(a)}
 

7.129.3 Helmut Kahovec (5.3.99)

Yes, it is a bug: Maple converts \(\cos (a)\) into an expression of \(\sin (a)\) and thus introduces a RootOf expression. During the computation, Maple tests the left-hand side of the equation for a polynomial with coefficients of type rational or algnum, and this test returns false:

> restart; 
> eq:=z*cos(a)^2+x*sin(a)*cos(a)+x^2=0; 
 
                            2                      2 
              eq := z cos(a)  + x sin(a) cos(a) + x  = 0 
 
> solve({eq},{x});
 

eqns2 is an intermediate representation of eq:

> eqns2:={z-z*_S01^2+x*_S01*RootOf(_Z^2+_S01^2-1)+x^2}; 
 
                         2                   2       2         2 
     eqns2 := {z - z _S01  + x _S01 RootOf(_Z  + _S01  - 1) + x }
 

Now the type check mentioned earlier returns false:

> type(subs(I='RootOf(_Z^2+1)',eqns2[1]),polynom({rational,algnum})); 
 
                                false
 

However, if you replace \(\cos (a)\) with cosa then there will be no RootOf expression and the type check returns true:

> eq:=z*cos(a)^2+x*sin(a)*cosa+x^2=0; 
 
                             2                    2 
               eq := z cos(a)  + x sin(a) cosa + x  = 0 
 
> solve({eq},{x}); 
 
                       ... result skipped ... 
 
> eqns2:={z-z*_S01^2+x*_S01*cosa+x^2}; 
 
                                  2                  2 
              eqns2 := {z - z _S01  + x _S01 cosa + x } 
 
> type(subs(I='RootOf(_Z^2+1)',eqns2[1]),polynom({rational,algnum})); 
 
                                 true
 

As a workaround I suggest using frontend(): in this case there is no RootOf expression at all and you get a neat result without any simplification:

> eq:=z*cos(a)^2+x*sin(a)*cos(a)+x^2=0; 
 
                            2                      2 
              eq := z cos(a)  + x sin(a) cos(a) + x  = 0 
 
> frontend(solve,[{eq},{x}],[{`+`,`*`},{x}]); 
 
                                      2 
  {x = (- 1/2 sin(a) + 1/2 sqrt(sin(a)  - 4 z)) cos(a)}, 
 
                                            2 
        {x = (- 1/2 sin(a) - 1/2 sqrt(sin(a)  - 4 z)) cos(a)}
 

7.129.4 Bob Wright (8.3.99)

Apparently the same happens in Release V.0: but see below:

solve(z*cos(a)^2+x*sin(a)*cos(a)+x^2,x);
 

no solution found. But look at this:

> solve(z*cos(a)^2+x*sin(b)*cos(a)+x^2,x); 
 
                                 2 
  (- 1/2 sin(b) + 1/2 sqrt(sin(b)  - 4 z)) cos(a), 
 
                                       2 
        (- 1/2 sin(b) - 1/2 sqrt(sin(b)  - 4 z)) cos(a)