Algorithm 2.

SR1 updating.

Initialization: $\gamma >1,\delta ={10}^{-8},\Xi >0$ a fixed real scalar, $x_k$, $x_{k-1}$, $\nabla f\left(x_k\right)$, and $\nabla f\left(x_{k-1}\right)$.

1: if $k=1$ then

2:  $B_k\leftarrow \Xi I$.

3: else

4:  Set $s_k\leftarrow x_k-x_{k-1}$ and $m_k\leftarrow \nabla f\left(x_k\right)-\nabla f\left(x_{k-1}\right)$.

5:  Compute ${\tau }_k\leftarrow \gamma \frac{{‖m_k‖}_2^2}{〈s_k,m_k〉}$. $\%〈a,b〉=b^{H}a$

6:  if $\tau <0$ then

7:   $B_k\leftarrow \Xi I$.

8:  else

9:   $H_0\leftarrow {\tau }_{k}I$.

10:   $u_k\leftarrow m_k-H_0{s}_k$.

11:   if $\left|〈u_k,s_k〉\right|\le \delta {‖s_k‖}_2{‖u_k‖}_2$ then

12:    $u_k\leftarrow 0$.

13:   end if

14:   $B_k\leftarrow H_0+\frac{u_k{u}_k^H}{〈m_k-H_0{s}_k,s_k〉}$.

15:  end if

16: end if

17: Return: $B_k$