Schematic workflow of this library

In the following description, the loop for $N_R$ is omitted for simplicity and instead of $G_{i j}(z_k)$, the $N_L$-dimensional vector ${\bf x}_{k}$ is obtained by using the library.

The names of the routines is defined as follows.

• komega_bicg_init, komega_cocg_init, komega_cg_c_init, komega_cg_r_init

  Set the initial conditions such as the allocation of variables used in the library.

• komega_bicg_update, komega_cocg_update, komega_cg_c_update, komega_cg_r_update

  These routines are called in the iteration to update the solution vectors.

• komega_bicg_finalize, komega_cocg_finalize, komega_cg_c_finalize, komega_cg_r_finalize

  Release the allocated vectors in the library.

• komega_bicg_getcoef, komega_cocg_getcoef, komega_cg_c_getcoef, komega_cg_r_getcoef

  Get the $\alpha$, $\beta$, $z_{\rm seed}$, ${\bf r}^{\rm L}$ conserved at each iteration.

• komega_bicg_getvec, komega_cocg_getvec, komega_cg_c_getvec, komega_cg_r_getvec

  Get the vectors ${\boldsymbol r}$, ${\boldsymbol r}^{\rm old}$, ${\tilde {\boldsymbol r}}$, ${\tilde {\boldsymbol r}}^{\rm old}$.

• komega_bicg_restart, komega_cocg_restart, komega_cg_c_restart, CG_R_restart

Note

• Give the vector size $N_H$ corresponding to the size of the Hilbert space and the number of the frequency $z$.

• Allocate the two vectors (in the case of BiCG method, four vectors) with the size of $N_H$.

• Give the function for the Hamiltonian-vector production.

• Allocate the solution vectors. It is noted that the length of each solution vector is not always equal to $N_H$. In fact, the its length in the previous section is $N_L$. In this case, the length of the (bi-)conjugate gradient vector ${\bf p}_k (k=1,\cdots N_z)$ also becomes $N_L$. We have to prepare a code for projecting $N_H$-dimensional vector onto $N_L$dimensional space.

  $$\begin{aligned} {\bf r}^{\rm L} = {\hat P}^\dagger {\boldsymbol r}, \qquad {\hat P} \equiv ({\boldsymbol \varphi}_1, \cdots, {\boldsymbol \varphi}_{N_L}) \end{aligned}$$

• If the result converges (or a breakdown occurs), komega_*_update return the first element of status as a negative integer. Therefore, please exit loop when status(1) < 0 .

• The 2-norm is used for the convergence check in the routine komega_*_update. Therefore, if 2-norms of residual vectors at all shift points becomes smaller than threshold, this routine assumes the result is converged.

• We can obtain the history of $\alpha, \beta, {\bf r}^{\rm L}$ for restarting calculation. In this case, itermax must not be 0.

The schematic workflow of shifted BiCG library

Allocate ${\boldsymbol v}_{1 2}$, ${\boldsymbol v}_{1 3}$, ${\boldsymbol v}_2$, ${\boldsymbol v}_3$, $\{{\bf x}_k\}, {\bf r}^{\rm L}$ ${\boldsymbol v}_2 = {\boldsymbol \varphi_j}$

komega_bicg_init(N_H, N_L, N_z, x, z, itermax, threshold) start

Allocate ${\boldsymbol v}_3$, ${\boldsymbol v}_5$, $\{\pi_k\}$ , $\{\pi_k^{\rm old}\}$, $\{{\bf p}_k\}$

Copy $\{z_k\}$

If itermax $\neq$ 0 , allocate arrays to store $\alpha$, $\beta$, and:math:{bf r}^{rm L} at each iteration.

${\boldsymbol v}_4 = {\boldsymbol v}_2^*$ (an arbitrary vector), ${\boldsymbol v}_3 = {\boldsymbol v}_5 = {\bf 0}$,

${\bf p}_{k} = {\bf x}_k = {\bf 0}(k=1 \cdots N_z),\; \pi_k=\pi_k^{\rm old} = 1(k=1 \cdots N_z)$

$\rho = \infty,\; \alpha = 1,\; z_{\rm seed}=0$

( ${\boldsymbol v}_2 \equiv {\boldsymbol r}$, ${\boldsymbol v}_3 \equiv {\boldsymbol r}^{\rm old}$, ${\boldsymbol v}_4 \equiv {\tilde {\boldsymbol r}}$, ${\boldsymbol v}_5 \equiv {\tilde {\boldsymbol r}}^{\rm old}$. )

komega_bicg_init finish

do iteration

${\bf r}^{\rm L} = {\hat P}^\dagger {\boldsymbol v}_2$

${\boldsymbol v}_{1 2} = {\hat H} {\boldsymbol v}_2$, ${\boldsymbol v}_{1 4} = {\hat H} {\boldsymbol v}_4$ [Or $({\boldsymbol v}_{1 2}, {\boldsymbol v}_{1 4}) = {\hat H} ({\boldsymbol v}_2, {\boldsymbol v}_4)$ ]

komega_bicg_update(v_12, v_2, v_14, v_4, x, r_small, status) start

$\circ$ Seed equation

$\rho^{\rm old} = \rho,\; \rho = {\boldsymbol v}_4^* \cdot {\boldsymbol v}_2$

$\beta = \rho / \rho^{\rm old}$

${\boldsymbol v}_{1 2} = z_{\rm seed} {\boldsymbol v}_2 - {\boldsymbol v}_{1 2}$, ${\boldsymbol v}_{1 4} = z_{\rm seed}^* {\boldsymbol v}_4 - {\boldsymbol v}_{1 4}$

$\alpha^{\rm old} = \alpha,\; \alpha = \frac{\rho}{{\boldsymbol v}_3^* \cdot {\boldsymbol v}_{1 2} - \beta \rho / \alpha }$

$\circ$ Shifted equation

do $k = 1 \cdots N_z$

$\pi_k^{\rm new} = [1+\alpha(z_k-z_{\rm seed})]\pi_k - \frac{\alpha \beta}{\alpha^{\rm old}}(\pi_k^{\rm old} - \pi_k)$

${\bf p}_{k} = \frac{1}{\pi_k} {\bf r}^{\rm L} + \frac{\pi^{\rm old}_k \pi^{\rm old}_k}{\pi_k \pi_k} \beta {\bf p}_{k}$

${\bf x}_{k} = {\bf x}_{k} + \frac{\pi_k}{\pi_k^{\rm new}} \alpha {\bf p}_{k}$

$\pi_k^{\rm old} = \pi_k$, $\pi_k = \pi_k^{\rm new}$

end do $k$

${\boldsymbol v}_{1 2} = \left( 1 + \frac{\alpha \beta}{\alpha^{\rm old}} \right) {\boldsymbol v}_2 - \alpha {\boldsymbol v}_{1 2} - \frac{\alpha \beta}{\alpha^{\rm old}} {\boldsymbol v}_3$, ${\boldsymbol v}_3 = {\boldsymbol v}_2,\; {\boldsymbol v}_2 = {\boldsymbol v}_{1 2}$

${\boldsymbol v}_{1 4} = \left( 1 + \frac{\alpha^* \beta^*}{\alpha^{{\rm old}*}} \right) {\boldsymbol v}_4 - \alpha^* {\boldsymbol v}_{1 4} - \frac{\alpha^* \beta^*}{\alpha^{{\rm old} *}} {\boldsymbol v}_5$, ${\boldsymbol v}_5 = {\boldsymbol v}_4,\; {\boldsymbol v}_4 = {\boldsymbol v}_{1 4}$

$\circ$ Seed switch

Search $k$ which gives the smallest $|\pi_k|$ . $\rightarrow z_{\rm seed},\; \pi_{\rm seed},\; \pi_{\rm seed}^{\rm old}$

${\boldsymbol v}_2 = {\boldsymbol v}_2 / \pi_{\rm seed}$, ${\boldsymbol v}_3 = {\boldsymbol v}_3 / \pi_{\rm seed}^{\rm old}$, ${\boldsymbol v}_4 = {\boldsymbol v}_4 / \pi_{\rm seed}^{*}$, ${\boldsymbol v}_5 = {\boldsymbol v}_5 / \pi_{\rm seed}^{\rm old *}$

$\alpha = (\pi_{\rm seed}^{\rm old} / \pi_{\rm seed}) \alpha$, $\rho = \rho / (\pi_{\rm seed}^{\rm old} \pi_{\rm seed}^{\rm old})$

$\{\pi_k = \pi_k / \pi_{\rm seed},\; \pi_k^{\rm old} = \pi_k^{\rm old} / \pi_{\rm seed}^{\rm old}\}$

komega_bicg_update finish

if(status(1) < 0 (This indicates $|{\boldsymbol v}_2| <$ Threshold)) exit

end do iteration

komega_bicg_finalize start

Deallocate ${\boldsymbol v}_4$, ${\boldsymbol v}_5$, $\{\pi_k\}$, $\{\pi_k^{\rm old}\}$, $\{{\bf p}_k\}$

komega_bicg_finalize finish

The schematic workflow of shifted COCG library

Allocate ${\boldsymbol v}_1$, ${\boldsymbol v}_2$, $\{{\bf x}_k\}, {\bf r}^{\rm L}$ ${\boldsymbol v}_2 = {\boldsymbol \varphi_j}$

komega_cocg_init(N_H, N_L, N_z, x, z, itermax, threshold) start

Allocate ${\boldsymbol v}_3$, $\{\pi_k\}$, $\{\pi_k^{\rm old}\}$, $\{{\bf p}_k\}$

Copy $\{z_k\}$

If itermax $\neq$ 0 , allocate arrays to store $\alpha$, $\beta$, and ${\bf r}^{\rm L}$ .

${\boldsymbol v}_3 = {\bf 0}$,

${\bf p}_{k} = {\bf x}_k = {\bf 0}(k=1 \cdots N_z),\; \pi_k=\pi_k^{\rm old} = 1(k=1 \cdots N_z)$

$\rho = \infty,\; \alpha = 1,\; \beta=0,\; z_{\rm seed}=0$

( ${\boldsymbol v}_2 \equiv {\boldsymbol r}$, ${\boldsymbol v}_3 \equiv {\boldsymbol r}^{\rm old}$. )

komega_cocg_init finish

do iteration

${\bf r}^{\rm L} = {\hat P}^\dagger {\boldsymbol v}_2$

${\boldsymbol v}_1 = {\hat H} {\boldsymbol v}_2$

komega_cocg_update(v_1, v_2, x, r_small, status) start

$\circ$ Seed equationw

$\rho^{\rm old} = \rho,\; \rho = {\boldsymbol v}_2 \cdot {\boldsymbol v}_2$

$\beta = \rho / \rho^{\rm old}$

${\boldsymbol v}_1 = z_{\rm seed} {\boldsymbol v}_2 - {\boldsymbol v}_1$

$\alpha^{\rm old} = \alpha,\; \alpha = \frac{\rho}{{\boldsymbol v}_2 \cdot {\boldsymbol v}_1 - \beta \rho / \alpha }$

$\circ$ Shifted equations

do $k = 1 \cdots N_z$

$\pi_k^{\rm new} = [1+\alpha(z_k-z_{\rm seed})]\pi_k - \frac{\alpha \beta}{\alpha^{\rm old}}(\pi_k^{\rm old} - \pi_k)$

${\bf p}_{k} = \frac{1}{\pi_k} {\bf r}^{\rm L} + \frac{\pi^{\rm old}_k \pi^{\rm old}_k}{\pi_k \pi_k} \beta {\bf p}_{k}$

${\bf x}_{k} = {\bf x}_{k} + \frac{\pi_k}{\pi_k^{\rm new}} \alpha {\bf p}_{k}$

$\pi_k^{\rm old} = \pi_k,\; \pi_k = \pi_k^{\rm new}$

end do $k$

${\boldsymbol v}_1 = \left( 1 + \frac{\alpha \beta}{\alpha^{\rm old}} \right) {\boldsymbol v}_2 - \alpha {\boldsymbol v}_1 - \frac{\alpha \beta}{\alpha^{\rm old}} {\boldsymbol v}_3$

${\boldsymbol v}_3 = {\boldsymbol v}_2$, ${\boldsymbol v}_2 = {\boldsymbol v}_1$

$\circ$ Seed switch

Search $k$ which gives the smallest |pi_k| . $\rightarrow z_{\rm seed},\; \pi_{\rm seed},\; \pi_{\rm seed}^{\rm old}$

${\boldsymbol v}_2 = {\boldsymbol v}_2 / \pi_{\rm seed}$, ${\boldsymbol v}_3 = {\boldsymbol v}_3 / \pi_{\rm seed}^{\rm old}$

$\alpha = (\pi_{\rm seed}^{\rm old} / \pi_{\rm seed}) \alpha$, $\rho = \rho / (\pi_{\rm seed}^{\rm old} \pi_{\rm seed}^{\rm old})$

$\{\pi_k = \pi_k / \pi_{\rm seed},\; \pi_k^{\rm old} = \pi_k^{\rm old} / \pi_{\rm seed}^{\rm old}\}$

komega_cocg_update finish

if(status(1) < 0 (This indicates $|{\boldsymbol v}_2| <$ Threshold.)) exit

end do iteration

komega_cocg_finalize start

Deallocate ${\boldsymbol v}_3$, $\{\pi_k\}$, $\{\pi_k^{\rm old}\}$, $\{{\bf p}_k\}$

komega_cocg_finalize finish

The schematic workflow of shifted CG library

The workflow is the same as that of the shifted COCG library.