lm()

function [out, Elist]=lm(W,Nit)

Input:

• W: $\prod S \times N_s$ matrix containing initial guess for eigensolutions
• Nit: Number of iterations desired

Output:

• out: Result of Nit iterations of line minimization
• Elist: Nit$\times$1 column vector containing result of getE() after each iteration
• DISPLAY: result of each getE(), ``linmin test'' (angle cosine between previous search direction and current gradient)

Implement the line minimization algorithm discussed in lecture,

1. Initialize $W_0$
2. Evaluate gradient: $g=\nabla_W E(W_n)$
3. If not first iteration, do ``linmin test'': check $(g \cdot d)/\sqrt{(g \cdot g) (d \cdot d)}$ for $d$ being the previous search direction
4. Set search direction: $d=-g$
5. Evaluate gradient at trial step: $g_t=\nabla_W E(W_n+\alpha_t d)$
6. Compute estimate of best step: $\alpha=\alpha_t (g \cdot d)/([g-gt] \cdot d)$
7. $W_{n+1}=W_n+\alpha d$
8. Repeat (2-7)

with a function of the above prototype in the file ``lm.m''. You may wish to experiment with the trial step size. You will find that the algorithm is not very sensitive to this parameter. A trial step size of $\alpha_t=3 \times 10^{-5}$ to work well.
Hint: As on the previous assignment, because of our trick of coding gradients as complex matrices, we must evaluate the dot products according to the formula

$$a \cdot b = 2 {\mbox{Re\,}}{\mbox{Tr\,}}\left(a^\dagger b\right).$$

As a matter of convenience, note that the ``extra'' factor of two, while very important for getting the values of dot products correct, actually cancels out of all of the expressions above.