Schematic workflow of this library ================================== In the following description, the loop for :math:`N_R` is omitted for simplicity and instead of :math:`G_{i j}(z_k)`, the :math:`N_L`\ -dimensional vector :math:`{\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 :math:`\alpha`, :math:`\beta`, :math:`z_{\rm seed}`, :math:`{\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 :math:`{\boldsymbol r}`, :math:`{\boldsymbol r}^{\rm old}`, :math:`{\tilde {\boldsymbol r}}`, :math:`{\tilde {\boldsymbol r}}^{\rm old}`. - ``komega_bicg_restart``, ``komega_cocg_restart``, ``komega_cg_c_restart``, ``CG_R_restart`` .. note:: - Give the vector size :math:`N_H` corresponding to the size of the Hilbert space and the number of the frequency :math:`z`. - Allocate the two vectors (in the case of BiCG method, four vectors) with the size of :math:`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 :math:`N_H`. In fact, the its length in the previous section is :math:`N_L`. In this case, the length of the (bi-)conjugate gradient vector :math:`{\bf p}_k (k=1,\cdots N_z)` also becomes :math:`N_L`. We have to prepare a code for projecting :math:`N_H`\ -dimensional vector onto :math:`N_L`\ dimensional space. .. math:: \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 :math:`\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 :math:`{\boldsymbol v}_{1 2}`, :math:`{\boldsymbol v}_{1 3}`, :math:`{\boldsymbol v}_2`, :math:`{\boldsymbol v}_3`, :math:`\{{\bf x}_k\}, {\bf r}^{\rm L}` :math:`{\boldsymbol v}_2 = {\boldsymbol \varphi_j}` ``komega_bicg_init(N_H, N_L, N_z, x, z, itermax, threshold)`` start Allocate :math:`{\boldsymbol v}_3`, :math:`{\boldsymbol v}_5`, :math:`\{\pi_k\}` , :math:`\{\pi_k^{\rm old}\}`, :math:`\{{\bf p}_k\}` Copy :math:`\{z_k\}` If ``itermax`` :math:`\neq` ``0`` , allocate arrays to store :math:`\alpha`, :math:`\beta`, and:math:`{\bf r}^{\rm L}` at each iteration. :math:`{\boldsymbol v}_4 = {\boldsymbol v}_2^*` (an arbitrary vector), :math:`{\boldsymbol v}_3 = {\boldsymbol v}_5 = {\bf 0}`, :math:`{\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)` :math:`\rho = \infty,\; \alpha = 1,\; z_{\rm seed}=0` ( :math:`{\boldsymbol v}_2 \equiv {\boldsymbol r}`, :math:`{\boldsymbol v}_3 \equiv {\boldsymbol r}^{\rm old}`, :math:`{\boldsymbol v}_4 \equiv {\tilde {\boldsymbol r}}`, :math:`{\boldsymbol v}_5 \equiv {\tilde {\boldsymbol r}}^{\rm old}`. ) ``komega_bicg_init`` finish do iteration :math:`{\bf r}^{\rm L} = {\hat P}^\dagger {\boldsymbol v}_2` :math:`{\boldsymbol v}_{1 2} = {\hat H} {\boldsymbol v}_2`, :math:`{\boldsymbol v}_{1 4} = {\hat H} {\boldsymbol v}_4` [Or :math:`({\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 :math:`\circ` Seed equation :math:`\rho^{\rm old} = \rho,\; \rho = {\boldsymbol v}_4^* \cdot {\boldsymbol v}_2` :math:`\beta = \rho / \rho^{\rm old}` :math:`{\boldsymbol v}_{1 2} = z_{\rm seed} {\boldsymbol v}_2 - {\boldsymbol v}_{1 2}`, :math:`{\boldsymbol v}_{1 4} = z_{\rm seed}^* {\boldsymbol v}_4 - {\boldsymbol v}_{1 4}` :math:`\alpha^{\rm old} = \alpha,\; \alpha = \frac{\rho}{{\boldsymbol v}_3^* \cdot {\boldsymbol v}_{1 2} - \beta \rho / \alpha }` :math:`\circ` Shifted equation do :math:`k = 1 \cdots N_z` :math:`\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)` :math:`{\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}` :math:`{\bf x}_{k} = {\bf x}_{k} + \frac{\pi_k}{\pi_k^{\rm new}} \alpha {\bf p}_{k}` :math:`\pi_k^{\rm old} = \pi_k`, :math:`\pi_k = \pi_k^{\rm new}` end do :math:`k` :math:`{\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`, :math:`{\boldsymbol v}_3 = {\boldsymbol v}_2,\; {\boldsymbol v}_2 = {\boldsymbol v}_{1 2}` :math:`{\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`, :math:`{\boldsymbol v}_5 = {\boldsymbol v}_4,\; {\boldsymbol v}_4 = {\boldsymbol v}_{1 4}` :math:`\circ` Seed switch Search :math:`k` which gives the smallest :math:`|\pi_k|` . :math:`\rightarrow z_{\rm seed},\; \pi_{\rm seed},\; \pi_{\rm seed}^{\rm old}` :math:`{\boldsymbol v}_2 = {\boldsymbol v}_2 / \pi_{\rm seed}`, :math:`{\boldsymbol v}_3 = {\boldsymbol v}_3 / \pi_{\rm seed}^{\rm old}`, :math:`{\boldsymbol v}_4 = {\boldsymbol v}_4 / \pi_{\rm seed}^{*}`, :math:`{\boldsymbol v}_5 = {\boldsymbol v}_5 / \pi_{\rm seed}^{\rm old *}` :math:`\alpha = (\pi_{\rm seed}^{\rm old} / \pi_{\rm seed}) \alpha`, :math:`\rho = \rho / (\pi_{\rm seed}^{\rm old} \pi_{\rm seed}^{\rm old})` :math:`\{\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 :math:`|{\boldsymbol v}_2| <` Threshold)) exit end do iteration ``komega_bicg_finalize`` start Deallocate :math:`{\boldsymbol v}_4`, :math:`{\boldsymbol v}_5`, :math:`\{\pi_k\}`, :math:`\{\pi_k^{\rm old}\}`, :math:`\{{\bf p}_k\}` ``komega_bicg_finalize`` finish The schematic workflow of shifted COCG library ---------------------------------------------- Allocate :math:`{\boldsymbol v}_1`, :math:`{\boldsymbol v}_2`, :math:`\{{\bf x}_k\}, {\bf r}^{\rm L}` :math:`{\boldsymbol v}_2 = {\boldsymbol \varphi_j}` ``komega_cocg_init(N_H, N_L, N_z, x, z, itermax, threshold)`` start Allocate :math:`{\boldsymbol v}_3`, :math:`\{\pi_k\}`, :math:`\{\pi_k^{\rm old}\}`, :math:`\{{\bf p}_k\}` Copy :math:`\{z_k\}` If ``itermax`` :math:`\neq` ``0`` , allocate arrays to store :math:`\alpha`, :math:`\beta`, and :math:`{\bf r}^{\rm L}` . :math:`{\boldsymbol v}_3 = {\bf 0}`, :math:`{\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)` :math:`\rho = \infty,\; \alpha = 1,\; \beta=0,\; z_{\rm seed}=0` ( :math:`{\boldsymbol v}_2 \equiv {\boldsymbol r}`, :math:`{\boldsymbol v}_3 \equiv {\boldsymbol r}^{\rm old}`. ) ``komega_cocg_init`` finish do iteration :math:`{\bf r}^{\rm L} = {\hat P}^\dagger {\boldsymbol v}_2` :math:`{\boldsymbol v}_1 = {\hat H} {\boldsymbol v}_2` ``komega_cocg_update(v_1, v_2, x, r_small, status)`` start :math:`\circ` Seed equationw :math:`\rho^{\rm old} = \rho,\; \rho = {\boldsymbol v}_2 \cdot {\boldsymbol v}_2` :math:`\beta = \rho / \rho^{\rm old}` :math:`{\boldsymbol v}_1 = z_{\rm seed} {\boldsymbol v}_2 - {\boldsymbol v}_1` :math:`\alpha^{\rm old} = \alpha,\; \alpha = \frac{\rho}{{\boldsymbol v}_2 \cdot {\boldsymbol v}_1 - \beta \rho / \alpha }` :math:`\circ` Shifted equations do :math:`k = 1 \cdots N_z` :math:`\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)` :math:`{\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}` :math:`{\bf x}_{k} = {\bf x}_{k} + \frac{\pi_k}{\pi_k^{\rm new}} \alpha {\bf p}_{k}` :math:`\pi_k^{\rm old} = \pi_k,\; \pi_k = \pi_k^{\rm new}` end do :math:`k` :math:`{\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` :math:`{\boldsymbol v}_3 = {\boldsymbol v}_2`, :math:`{\boldsymbol v}_2 = {\boldsymbol v}_1` :math:`\circ` Seed switch Search :math:`k` which gives the smallest `|\pi_k|` . :math:`\rightarrow z_{\rm seed},\; \pi_{\rm seed},\; \pi_{\rm seed}^{\rm old}` :math:`{\boldsymbol v}_2 = {\boldsymbol v}_2 / \pi_{\rm seed}`, :math:`{\boldsymbol v}_3 = {\boldsymbol v}_3 / \pi_{\rm seed}^{\rm old}` :math:`\alpha = (\pi_{\rm seed}^{\rm old} / \pi_{\rm seed}) \alpha`, :math:`\rho = \rho / (\pi_{\rm seed}^{\rm old} \pi_{\rm seed}^{\rm old})` :math:`\{\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 :math:`|{\boldsymbol v}_2| <` Threshold.)) exit end do iteration ``komega_cocg_finalize`` start Deallocate :math:`{\boldsymbol v}_3`, :math:`\{\pi_k\}`, :math:`\{\pi_k^{\rm old}\}`, :math:`\{{\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.