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More solve curiosities

Doug Meade 3073
September 19 2007
0
It's a new day, but not a new problem. Yesterday I asked about solving inequalities with abs. Today I am looking at solving an equation with abs. Try the following: 
restart;
q :=  sin(x)/x - 1:
test4 := y -> [ y,
                solve( q=y, x ),
                [solve( abs(q)=y, x )],
                fsolve( q=y, x ),
                fsolve( abs(q)=y, x ) ]:
test4(  0  );
test4( 0.1 );
test4( 0.5 );
test4( 1/2 );
test4(  1  );

[                         /sin(x)           \                -13]
[ 0,   0, [0],      fsolve|------ - 1 = 0, x|, 1.869896012 10   ]
[                         \  x              /                   ]

[                         /sin(x)             \               ]
[0.1, 0., [0., 0.], fsolve|------ - 1 = 0.1, x|, -0.7866830720]
[                         \  x                /               ]

[                         /sin(x)             \              ]
[0.5, 0., [0., 0.], fsolve|------ - 1 = 0.5, x|, -1.895494267]
[                         \  x                /              ]

[ 1                                         /sin(x)       1   \              ]
[ - , 0, [0, RootOf(_Z - 2 sin(_Z))], fsolve|------ - 1 = -, x|, -1.895494267]
[ 2                                         \  x          2   /              ]

[                      /sin(x)           \              ]
[ 1,  0, [0, 0], fsolve|------ - 1 = 1, x|, -3.141592654]
[                      \  x              /              ]
At first, it is impressive that Maple seems to be able to solve sin(x)/x=1 and get x=0. But, looking at the results with other values of the RHS, it become obvious that Maple returns 0 for almost any RHS. Putting an abs in the problem causes Maple to report the "solution" at x=0 twice. Based on yesterday's experience, I presume the multiple solutions are arising from Maple's attempt to separate the abs into two pieces. If so, then x=0 is only in one of the two subintervals but is never in the domain of the function, so should never be returned as an answer from solve. Moving to fsolve, it's shocking (to me) that fsolve is unable to find a solution only in the cases where abs is used. Each of these results is disappointing. Comments? Explanations? Solutions? Hoping for some enlightenment, Doug

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