.. _responses-mixed_hessians-id_quasi_hessians-sr1:

"""
sr1
"""


Use the Symmetric Rank 1 update method to compute quasi-Hessians


.. toctree::
   :hidden:
   :maxdepth: 1



**Specification**

- *Alias:* None

- *Arguments:* None


**Description**


The Symmetric Rank 1 (SR1) update (specified with the keyword ``sr1``)
will be used to compute quasi-Hessians.


.. math:: 

   B_{k+1} = B_k + \frac{(y_k - B_k s_k)(y_k - B_k s_k)^T}{(y_k - B_k s_k)^T s_k}

where :math:`B_k`  is the :math:`k^{th}`  approximation to the Hessian,
:math:`s_k = x_{k+1} - x_k`  is the step and
:math:`y_k = \nabla f_{k+1} - \nabla f_k`  is the corresponding yield
in the gradients.

*Notes*


- Initial scaling of :math:`\frac{y_k^T y_k}{y_k^T s_k} I`  is used for :math:`B_0`  prior to the first update.
- Numerical safeguarding is used to protect against numerically small denominators within the updates.
- This safeguarding skips the update if :math:`|(y_k - B_k s_k)^T s_k| < 10^{-6} ||s_k||_2 ||y_k - B_k s_k||_2` 


