getgrad()

function grad=getgrad(W)

Input:

• W: Expansion coefficients for $N_s$ unconstrained wave functions, stored as an $\prod S_k\,\times\,N_s$ matrix

Output:

• grad: $\prod S_k \, \times N_s$ matrix containing the derivatives $\partial E/\partial W^*_{i,n}$

Global variable:

• gbl_f: State fillings stored as an $N_s\,\times\,1$ column vector.

Modify getgrad() as above to employ the appropriate formula for the case of variable fillings,

$$\nabla_{W^\dagger} E = \left(H(W)-O(W) U^{-1} \left(W^\dagge... ...O(W) \left(U^{-\frac{1}{2}} Q_U(\tilde{H}F-F\tilde{H})\right),$$

where $U\equiv W^\dagger O(W)$, $\tilde{H} \equiv U^{-\frac{1}{2}} \left( W^\dagger H(W) \right) U^{-\frac{1}{2}}$, $F\equiv \mbox{\bf Diag\,} \vec f$, and $Q_U({\bf A})$ is the operator which the function in Appendix A computes.¹ Note that the parentheses have been arranged here again to minimize the size and cost of the calculation of intermediate expressions and that $U^{-\frac{1}{2}}$ may be computed in octave as sqrtm(inv(U)). (Don't forget the ``m'' in ``sqrtm''! It specifies the matrix-square root as opposed to simply taking the square root of each element of the matrix.)