E-parameters

Explicit

Preconditioned Methods

${\pi }_i$

${\tau }_i$

${\delta }_i$

$1+\beta -\alpha M_{r}A$

$-\beta$

$\alpha M_{r}b$

Explicit Richardson (Lipitakis and Evans, 1987)

$1+{\beta }_i-{\alpha }_i{M}_{r}A$

$-{\beta }_i$

${\alpha }_i{M}_{r}b$

Explicit Chebychev (Lipitakis and Evans, 1987)

${\left({\rho }_{i}I-G_1\right)}^{-1}{\left({\rho }_{i}I-G_2\right)}^{-1}$

0

$2{\rho }_i{\left({\rho }_{i}I+G_i\right)}^{-1}$

Generalized AGE method (Lipitakis and Evans, 1981)

$1+{\omega }_{i+1}{\left({\rho }_{i}I+G_1\right)}^{-1}{\left({\rho }_{i}I+G_2\right)}^{-1}A$

0

$-{\omega }_{i+1}{\left({\rho }_{i}I+G_1\right)}^{-1}{\left({\rho }_{i}I+G_2\right)}^{-1}b$

Richardson + AGE method (Evans, 1985)

${\omega }_i{H}_i$

$1-{\omega }_i$

$2{\omega }_i{\rho }_i{\left({\rho }_{i}I+G_1\right)}^{-1}{\left({\rho }_{i}I+G_2\right)}^{-1}b$

Chebychev + AGE method

(Evans, 1985)

${\left(I+\theta G_1\right)}^{-1}{\left(I+\theta G_2\right)}^{-1}\left\{\left(I+\theta G_1\right)\left(I+\theta G_2\right)-\eta A\right\}$

0

$\eta b$

General fractional step iteration (Marchuk, 1975)

${\displaystyle {\prod }_{i=0}^{\nu -1}\left(I-\alpha M_{\mu }A\right)}$

0

${\alpha }_i{M}_{\mu }b$

Multiple explicit Jacobi (Lipitakis, 1984)

${\rho }_{i+1}\left(1+{\gamma }_{i+1}{M}_{\mu }A\right)$

$1-{\rho }_{i+1}$

${\rho }_{i+1}{\gamma }_{i+1}{M}_{\mu }b$

Explicit Preconditioned Conjugate Gradients of second order (Lipitakis et al., 2017)