Input: $A\mathrm{,}B\mathrm{,}J$ , $w^{\left(0\right)}={\left[z_1^{\left(0\right)}\mathrm{,}{z}_2^{\left(0\right)}\right]}^{\text{T}}\mathrm{,}{\alpha }^{\left(0\right)}\mathrm{,}{\beta }^{\left(0\right)}$ , $k_{max}$ ,

1) for $k=0,1,2,\cdots$ until convergence do

2) Solve $Mu=B_{1}w$ for $u$ :

Find the lu factorization of $M$ i.e., $\left[L\mathrm{,}U\right]=lu\left(M\right)$ .

Solve the lower triangular system $Lc=B_{1}w$ for $c$ .

Solve the Upper triangular system $Uu=c$ for $u$ .

3) Compute $n_{\alpha }=w^{\text{T}}{B}_{1}Ju$ and $n_{\beta }=w^{\text{T}}{B}_{1}u$

4) Form the matrix

$T=\left[\begin{array}{cc}{n}_{\beta }& -n_{\alpha }\\ n_{\alpha }& n_{\beta }\end{array}\right]\mathrm{.}$

5) Find the LU factorization of $T$ , that is $\left[L\mathrm{,}U\right]=lu\left(T\right)$

6) Solve $Lc=\left[\begin{array}{c}\frac{1}{2}\left(w^{\text{T}}{B}_{1}w+1\right)\\ \frac{n_w^{\text{T}}w}{\left(1+{\Vert u\Vert }^2\right)}\end{array}\right]$ for $c$ .

7) Solve $U\omega =c$ for $\omega =\left[\begin{array}{c}\Delta \alpha \\ \Delta \beta \end{array}\right]$ .

8) Form the vector $w^{+}=\Delta \alpha u-\Delta \beta Ju$ .

9) Update

$\left[\begin{array}{c}{\alpha }^{+}\\ {\beta }^{+}\end{array}\right]=\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]+\left[\begin{array}{c}\Delta \alpha \\ \Delta \beta \end{array}\right].$

10) end for

Output: $v^{\left(k_{max}\right)}=\left[w^{\left(k_{max}\right)}\mathrm{,}{\alpha }^{\left(k_{max}\right)}\mathrm{,}{\beta }^{\left(k_{max}\right)}\right]$ .