LAPACK/ScaLAPACK Development

I used the routine zgsevd for finding the X of AX=0. But while S(16) is small (in order of 10^-8), AX is of order of 10^-1. Here, I've taken the 16 column of VT to be X. Indeed this X can't be the solution of AX=0. Am I doing something wrong? would you please help?

Also, is there any other method that I can use to find the nontrivial solutions of AX=0?

Also, if I use zgeev. when eigenvalue is zero, does the eigenvector show the nontrivial solution of AX=0? I don't know if zgeev is capable of showing nontrivial solution.
Thanks!

Usually, for computing the kernel of the matrix A,
AX=0,
we use SVD.