pccg()

function [out, Elist]=pccg(W,Nit,cgform)

Input:

• W: $\prod S \times N_s$ matrix containing initial guess for eigensolutions
• Nit: Number of iterations desired
• cgform: Version of cg: 1=Fletcher-Reeves, 2=Polak-Ribiere, 3=Hestenes-Stiefel

Output:

• out: Result of Nit iterations of preconditioned conjugate gradients
• 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), ``conjugate gradient test'' (angle cosine, measured through the preconditioner, of the current gradient and the previous gradient)

Implement the preconditioned conjugate-gradients algorithm discussed in lecture,

1. Initialize $W_0$
2. Evaluate gradient: $g_n=\nabla_W E(W_n)$
3. If not first iteration, do ``linmin test'': check $(g_n \cdot d_{n-1})/\sqrt{(g_n \cdot g_n) (d_{n-1} \cdot d_{n-1})}$ for $d$ being the previous search direction
4. If not first iteration, do ``cg test'': check $(g_n \cdot K g_{n-1})/\sqrt{(g_n \cdot K g_n) (g_{n-1} \cdot K g_{n-1})}$ for $d$ being the previous search direction
5. Set search direction: $d_n=-K g_n+\beta d_{n-1}$, where
   • Fletcher-Reeves: $\beta=(g_n \cdot K g_n)/(g_{n-1} \cdot K g_{n-1})$
   • Polak-Ribiere: $\beta=([g_n-g_{n-1}] \cdot K g_n)/(g_{n-1} \cdot K g_{n-1})$
   • Hestenes-Stiefel: $\beta=([g_n-g_{n-1}] \cdot K g_n)/([g_n-g_{n-1}] \cdot d_{n-1})$
6. Evaluate gradient at trial step: $g_t=\nabla_W E(W_n+\alpha_t d_n)$
7. Compute estimate of best step: $\alpha=\alpha_t (g \cdot d)/([g-g_tt] \cdot d_n)$
8. $W_{n+1}=W_n+\alpha d_n$
9. Repeat (2-8)

with a function of the above prototype in the file ``pccg.m''.
Hint: You may wish to use the ``if ... elseif ... elseif ... end'' octave construct.