Rank One Corrections

A way to construct H_k.

• F symmetric, so H_k symmetric.
• Preserving symmetry:

  H_k+1 = H_k + a_k z_k z_k^T

• Choose a_k and z_k so that

  H_k+1 q_k = p_k

• Explicit formula

  $$H_{k+1} = H_k + \frac{(p_k-H_kq_k)(p_k-H_kq_k)^T} {q_k^T(p_k - h_k q_k)}$$

• Can show that H_k+1 q_i = p_i, i<k.

The method then is

• d_k = - H_k g_k
• , line search.
• Update H_k as above

A problem is that H_k is not guaranteed to be positive definite.