Conjugate gradients

 

The method of conjugate-gradients (Numerical Recipes, Sec. 10.6) can make dramatic improvements in the convergence.

While the theory behind conjugate-gradients is subtle, the implementation is straight-forward. The theory states that at step n, rather than choosing the search direction ( ) to be along the down-hill gradient

one should search along a direction which is a linear combination of the previous search direction ( ) and the down-hill gradient,

The ``magic'' linear-combination coefficient is

Because conjugate-gradients depends upon being relatively near a minimum, it pays to first do a few (say ten) standard steps of line-minimization first. Thus, you should replace the statement

    dir=(-1.)*grad;

in your code with the following code fragment

    if (i<=10) 
      dir=(-1.)*grad;
    else {
      beta= ...
      dir=beta*dir+(-1.)*grad;
    }

where beta is defined as above.

Hint: Don't forget to compute dot products by taking the real part of the corresponding ``dagger product''.

• Debugging
• Gathering data