Algorithm 1.

Conjugate Gradient Method

Require: Matrix M, vector v and initialization $d_{\text{HR}}^{(0)}$.
1:	Let $r_0=v-M{d}_{\text{HR}}^{(0)}$, p0 = r0 and k = 0
2:	loop
3:	${\alpha }_k=\frac{r_k^{\top }{r}_k}{p_k^{\top }M{p}_k}$
4:	$d_{\text{HR}}^{(k+1)}=d_{\text{HR}}^{(k)}+{\alpha }_k{p}_k$
5:	rk+1 = rk – αkMpk
6:	if $\frac{\Vert r_{k+1}\Vert }{\Vert d_{\text{HR}}^{(0)}\Vert }<\text{tol}$ then
7:	end loop
8:	end if
9:	${\beta }_k=\frac{r_{k+1}^{\top }{r}_{k+1}}{r_k^{\top }{r}_k}$
10:	pk+1 = rk+1 + βkpk
11:	k = k +1
12:	end loop
13:	return $d_{\text{HR}}^{(k+1)}$