14.3.5 BiCGStab(l)

Select an initial guess $\underline{\phi}\sp{0}$ $\hat{\underline{r}}\sp{0} = \tilde{\underline{r}} = \underline{r}\sp{0} = \underline{b} - \dunder{A} \underline{\phi}\sp{0}$ k=0 $\rho = 1, \omega = 1, \alpha = 0$ $\hat{\underline{\phi}}\sp{0} = \underline{\phi}\sp{0}$ do $\rho = -\omega \rho\sb{0}$ for j from 0 to l-1 $\rho\sb{1} = (\tilde{\underline{r}})\sp{T} \hat{\underline{r}}\sp{j}$ $\beta = \alpha \rho\sb{1} / \rho$ $\rho = \rho\sb{1}$ for i from 0 to j $\hat{\underline{u}}\sp{i} = \hat{\underline{r}}\sp{i} - \beta \hat{\underline{u}}\sp{i}$ for end $\hat{\underline{u}}\sp{j+1} = \dunder{A} \hat{\underline{u}}\sp{j}$ $\gamma = (\hat{\underline{u}}\sp{j+1})\sp{T} \tilde{\underline{r}}$ $\alpha = \rho / \gamma$ for i from 0 to j $\hat{\underline{r}}\sp{i} = \hat{\underline{r}}\sp{i} - \alpha \hat{\underline{u}}\sp{i}$ for end $\hat{\underline{r}}\sp{j+1} = \dunder{A} \hat{\underline{r}}\sp{j}$ $\hat{\underline{\phi}}\sp{0} = \hat{\underline{\phi}}\sp{0} + \alpha \hat{\underline{u}}\sp{0}$ for end

for j from 1 to l $\underline{q}\sp{j} = \hat{\underline{r}}\sp{j}$ for i from 1 to j-1 $\tau\sb{ij} = \frac{1}{\sigma\sb{i}}(\underline{q}\sp{j})\sp{T} \underline{q}\sp{i}$ $\underline{q}\sp{j} = \underline{q}\sp{j} - \tau\sb{ij} \underline{q}\sp{i}$ for end $\sigma\sb{j} = (\underline{q}\sp{j})\sp{T} \underline{q}\sp{j}$ $\gamma\sb{j}\sp{1} = \frac{1}{\sigma\sb{j}}(\hat{\underline{r}}\sp{0})\sp{T} \underline{q}\sp{j}$ for end

$\gamma\sb{l} = \gamma\sb{l}\sp{1}$ $\omega = \gamma\sb{l}$ for j down from l-1 to 1 $\gamma\sb{j} = \gamma\sb{j}\sp{1} - \sum\sb{i=j+1}\sp{l} \tau\sb{ji}\gamma\sb{i}$ for end for j from 1 to l-1 $\gamma\sb{j}\sp{2} = \gamma\sb{j+1} + \sum\sb{i=j+1}\sp{l-1} \tau\sb{ji}\gamma\sb{i+1}$ for end $\hat{\underline{\phi}}\sp{0} = \hat{\underline{\phi}}\sp{0} + \gamma\sb{1}\hat{\underline{r}}\sp{0}$ $\hat{\underline{r}}\sp{0} = \hat{\underline{r}}\sp{0} - \gamma\sb{l}\sp{1}\hat{\underline{r}}\sp{l}$ $\hat{\underline{u}}\sp{0} = \hat{\underline{u}}\sp{0} - \gamma\sb{l}\hat{\underline{u}}\sp{l}$ for j from l to l-1 $\hat{\underline{\phi}}\sp{0} = \hat{\underline{\phi}}\sp{0} + \gamma\sb{j}\sp{2}\hat{\underline{r}}\sp{j}$ $\hat{\underline{r}}\sp{0} = \hat{\underline{r}}\sp{0} - \gamma\sb{j}\sp{1}\hat{\underline{r}}\sp{j}$ $\hat{\underline{u}}\sp{0} = \hat{\underline{u}}\sp{0} - \gamma\sb{j}\hat{\underline{u}}\sp{j}$ for end k=k+l $\underline{u}\sp{k-1} = \hat{\underline{u}}\sp{0}$ $\underline{r}\sp{k} = \hat{\underline{r}}\sp{0}$ $\underline{\phi}\sp{k} = \hat{\underline{\phi}}\sp{0}$ while not converged