Schur decomposition of a symmetric matrix

[senatorcheng]

As we all know, every real matrix A has a Schur decomposition A = OSO' where O is orthogonal, S is a lower (or upper) triangular matrix.
In the case that A is symmetric, the S matrix is diagonal.

But why isn't there a subroutine in LAPACK doing exactly Schur decomposition? The closet routine is DSYTD2, but it only reduces symmetric A to a tridiagonal.
I cannot understand why DSYTD2 only reduces A to tridiagonal when it CAN be made diagonal !

Can anyone give me a hint?

[senatorcheng]

I have figured it out myself.

The symmetric matrix can be made tridiagonal by successively applying Householder reflections. Although this tridiagonal matrix is theoretically orthogonally similar to a diagonal matrix, to do so we need QR iteration.

So the hard part is the latter part.