Why does the SOLVE function return the wrong answers for some equations?

2009 6 27
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Why does the SOLVE function return the wrong answers for some equations?
SOLVE returns incorrect symbolic solutions to some symbolic equations. For example, attempting to solve the equation z^6 = i results in incorrect answers:
solve('z^6-i')
ans =
[ (1/4*5^(1/2)-1/4+1/4*i*2^(1/2)*(5+5^(1/2))^(1/2))*(-i)^(1/2)]
[ (-1/4*5^(1/2)-1/4+1/4*i*2^(1/2)*(5-5^(1/2))^(1/2))*(-i)^(1/2)]
[ (-1/4*5^(1/2)-1/4-1/4*i*2^(1/2)*(5-5^(1/2))^(1/2))*(-i)^(1/2)]
[ (1/4*5^(1/2)-1/4-1/4*i*2^(1/2)*(5+5^(1/2))^(1/2))*(-i)^(1/2)]
[ 1/2*2^(1/2)-1/2*i*2^(1/2)]
[ -1/2*2^(1/2)+1/2*i*2^(1/2)]
Checking the accuracy of these values by performing
double(ans.^6)
shows that the first 4 answers are invalid:
ans =
-0.9511 + 0.3090i
-0.5878 - 0.8090i
0.5878 - 0.8090i
0.9511 + 0.3090i
0 + 1.0000i
0 + 1.0000i

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This issue has been fixed in Symbolic Math Toolbox 3.0.1 (R13+). If you are using an earlier versions:
This bug may be occurring in certain types of polynomials with complex coefficients, so you should watch for further such errors using these polynomials. As a workaround, you can use SOLVE with a similar problem involving a real equation whose roots include those of the polynomial with complex coefficients you are attempting to solve.
If you were working with the polynomial z^6-i, to find the appropriate polynomial you would multiply by the conjugate, z^6+i, to get z^12+1. The solutions to this equation will include those of z^6-i.
The following code finds the six solutions for this problem:
x = solve(z^12+1); % x holds 12 solution strings
y = double(x.^6); % contents of y are now y = i or y = -i
ix = (imag(y) > 0); % find indices where y = i
soln = x(ix); % soln holds the six solution strings where y = i
If you are looking for approximate answers to a polynomial, you can also use SYM2POLY and ROOTS:
syms z
f = z^6-i; % your polynomial here
p = sym2poly(f);
roots(p)
ans =
-0.9659 - 0.2588i
-0.7071 + 0.7071i
-0.2588 - 0.9659i
0.2588 + 0.9659i
0.7071 - 0.7071i
0.9659 + 0.2588i

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