I have done some debugging of LAPACK code to see how it calls other functions. There is blocked householder reflector one (dgebrd) and unblocked one(dgebd2). The dgebd2 algorithm is similar to golub,vanloan one except if you directly follow that (Algorithm 5.1.1) you will get singular vectors with different signs. golub uses parlet's theory to decide on the sign of reflector vector but LAPACK compares with diagonal element like BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA ). I haven't done the blocked one which will be really large matrix. you can find journal or online link to learn that process. The code base is like more than 5000 lines and with really bad indentation, go to statements takes ages to find out which branch my program is going.