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Iterative Solver

[available from v3.1]

To use iterative solvers, one shall use the following settings:

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set sparse_mat true
set system_solver lis [1]
# [1] string, lis solver options

The Lis library is used to provide the functionality.

Please note that although Lis provides a wide variety of iterative solvers and preconditioners, not all would yield a good performance. Whether it is faster than direct solvers depends on different problems.

All possible options, which are copied directly from Lis documentation, are listed below.

For example, if one wants to use the BiCGSTAB solver with ILU(2) preconditioner, and wants to print statistics, the settings would be:

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set sparse_mat true
set system_solver lis -i bicgstab -p ilu -ilu_fill 2 -print out
Solver Option Auxiliary Options
CG -i {cg\|1}
BiCG -i {bicg\|2}
CGS -i {cgs\|3}
BiCGSTAB -i {bicgstab\|4}
BiCGSTAB(l) -i {bicgstabl\|5} -ell [2] The degree \(l\)
GPBiCG -i {gpbicg\|6}
TFQMR -i {tfqmr\|7}
Orthomin(m) -i {orthomin\|8} -restart [40] The restart value \(m\)
GMRES(m) -i {gmres\|9} -restart [40] The restart value \(m\)
Jacobi -i {jacobi\|10}
Gauss-Seidel -i {gs\|11}
SOR -i {sor\|12} -omega [1.9] The relaxation coefficient
\(\omega\) (\(0<\omega<2\))
BiCGSafe -i {bicgsafe\|13}
CR -i {cr\|14}
BiCR -i {bicr\|15}
CRS -i {crs\|16}
BiCRSTAB -i {bicrstab\|17}
GPBiCR -i {gpbicr\|18}
BiCRSafe -i {bicrsafe\|19}
FGMRES(m) -i {fgmres\|20} -restart [40] The restart value \(m\)
IDR(s) -i {idrs\|21} -irestart [2] The restart value \(s\)
IDR(1) -i {idr1\|22}
MINRES -i {minres\|23}
COCG -i {cocg\|24}
COCR -i {cocr\|25}
Preconditioner Option Auxiliary Options
None -p {none\|0}
Jacobi -p {jacobi\|1}
ILU(k) -p {ilu\|2} -ilu_fill [0] The fill level \(k\)
SSOR -p {ssor\|3} -ssor_omega [1.0] The relaxation coefficient \(\omega\) (\(0<\omega<2\))
Hybrid -p {hybrid\|4} -hybrid_i [sor] The linear solver
-hybrid_maxiter [25] The maximum number of iterations
-hybrid_tol [1.0e-3] The convergence tolerance
-hybrid_omega [1.5] The relaxation coefficient \(\omega\) of the SOR (\(0<\omega<2\))
-hybrid_ell [2] The degree \(l\) of the BiCGSTAB(l)
-hybrid_restart [40] The restart values of the GMRES and Orthomin
I+S -p {is\|5} -is_alpha [1.0] The parameter \(\alpha\) of \(I+\alpha S^{(m)}\)
-is_m [3] The parameter \(m\) of \(I+\alpha S^{(m)}\)
SAINV -p {sainv\|6} -sainv_drop [0.05] The drop criterion
SA-AMG -p {saamg\|7} -saamg_unsym [false] Select the unsymmetric version
(The matrix structure must be
symmetric)
-saamg_theta [0.05\|0.12] The drop criterion \(a^2_{ij}\le\theta^2\|a_{ii}\|\|a_{jj}\|\)
(symmetric or unsymmetric)
Crout ILU -p {iluc\|8} -iluc_drop [0.05] The drop criterion
-iluc_rate [5.0] The ratio of the maximum fill-in
ILUT -p {ilut\|9}
Additive -adds true -adds_iter [1] The number of iterations
Schwarz

Other Options

Option
-maxiter [1000] The maximum number of iterations
-tol [1.0e-12] The convergence tolerance \(tol\)
-tol_w [1.0] The convergence tolerance \(tol_w\)
-print [0] The output of the residual history
-print {none\|0} None
-print {mem\|1} Save the residual history
-print {out\|2} Output it to the standard output
-print {all\|3} Save the residual history and output it to the standard output
-scale [0] The scaling
(The result will overwrite the original matrix and vectors)
-scale {none\|0} No scaling
-scale {jacobi\|1} The Jacobi scaling \(D^{-1}Ax=D^{-1}b\) (\(D\) represents the diagonal of \(A=(a_{ij})\))
-scale {symm_diag\|2} The diagonal scaling \(D^{-1/2}AD^{-1/2}x=D^{-1/2}b\) (\(D^{-1/2}\) represents the diagonal matrix with \(1/\sqrt{a_{ii}}\) as the diagonal)
-initx_zeros [1] The behavior of the initial vector \(x_{0}\)
-initx_zeros {false\|0} The components are given by the argument x of the function lis_solve()
-initx_zeros {true\|1} All the components are set to \(0\)
-conv_cond [0] The convergence condition
-conv_cond {nrm2_r\|0} \(\|\|b-Ax\|\|_2 \le tol * \|\|b-Ax_0\|\|_2\)
-conv_cond {nrm2_b\|1} \(\|\|b-Ax\|\|_2 \le tol * \|\|b\|\|_2\)
-conv_cond {nrm1_b\|2} \(\|\|b-Ax\|\|_1 \le tol_w * \|\|b\|\|_1 + tol\)
-omp_num_threads [t] The number of threads (t represents the maximum number of threads)
-storage [0] The matrix storage format
-storage_block [2] The block size of the BSR and BSC formats
-f [0] The precision of the linear solver
-f {double\|0} Double precision
-f {quad\|1} Double-double (quadruple) precision