Linear solvers#

Warning

Some capabilities may require the installation of GEMSEO with all its features and some others may depend on plugins.

Warning

All the features of the wrapped libraries may not be exposed through GEMSEO.

Note

The algorithm settings can be passed to a function of the form

function(..., settings_model: AlgorithmSettings | None = None, **settings: Any)

either one by one:

function(..., setting_name_1=setting_name_1, setting_name_2=setting_name_2, ...)

or using the argument name "settings_model" and the Pydantic model associated with the algorithm:

settings_model = AlgorithmSettings(setting_name_1=setting_name_1, setting_name_2=setting_name_2, ...)
function(..., settings_model=settings_model)

BICG#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

BI-Conjugate Gradient

Pydantic model of the settings for BICG#
from gemseo.settings.linear_solvers import BICG_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.bicg.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

BICGSTAB#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Bi-Conjugate Gradient STABilized

Pydantic model of the settings for BICGSTAB#
from gemseo.settings.linear_solvers import BICGSTAB_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.bicgstab.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

CG#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Conjugate Gradient

Pydantic model of the settings for CG#
from gemseo.settings.linear_solvers import CG_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.cg.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

CGS#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Conjugate Gradient Squared

Pydantic model of the settings for CGS#
from gemseo.settings.linear_solvers import CGS_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.cgs.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

DEFAULT#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Default solver (LGMRES)

Pydantic model of the settings for DEFAULT#
from gemseo.settings.linear_solvers import LGMRES_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lgmres.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • inner_m : <class 'int'>, optional

    Number of inner GMRES iterations per each outer iteration.

    By default it is set to 30.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • outer_k : <class 'int'>, optional

    Number of vectors to carry between inner GMRES iterations.

    By default it is set to 3.

  • outer_v : list[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]], optional

    List of tuples (v, Av) used to augment the Krylov subspace.

    By default it is set to [].

  • prepend_outer_v : <class 'bool'>, optional

    Whether to put outer_v augmentation vectors before Krylov iterates.

    By default it is set to False.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_outer_Av : <class 'bool'>, optional

    Whether A @ v should be additionnaly stored in outer_v list.

    By default it is set to True.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

GCROT#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Generalized Conjugate Residual with Optimal Truncation

Pydantic model of the settings for GCROT#
from gemseo.settings.linear_solvers import GCROT_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.gcrotmk.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • CU : collections.abc.Iterable[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]] | None, optional

    List of tuples (c, u) required to form the matrices C and U.

    If None start from empty matrices.

    By default it is set to None.

  • discard_C : <class 'bool'>, optional

    Whether to discard the C-vectors at the end.

    By default it is set to True.

  • k : <class 'int'>, optional

    Number of vectors to carry between inner FGMRES iterations.

    If None use the same value as m.

    By default it is set to 30.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • m : <class 'int'>, optional

    Number of inner FGMRES iterations per each outer iteration.

    By default it is set to 30.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • truncate : typing.Literal['oldest', 'smallest'], optional

    The vectors from the previous cycle to drop.

    By default it is set to oldest.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

GMRES#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Generalized Minimum RESidual

Pydantic model of the settings for GMRES#
from gemseo.settings.linear_solvers import GMRES_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.gmres.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • restart : <class 'int'>, optional

    Number of iterations between restarts.

    By default it is set to 20.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

LGMRES#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Loose Generalized Minimum RESidual

Pydantic model of the settings for LGMRES#
from gemseo.settings.linear_solvers import LGMRES_Settings

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lgmres.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • inner_m : <class 'int'>, optional

    Number of inner GMRES iterations per each outer iteration.

    By default it is set to 30.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • outer_k : <class 'int'>, optional

    Number of vectors to carry between inner GMRES iterations.

    By default it is set to 3.

  • outer_v : list[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]], optional

    List of tuples (v, Av) used to augment the Krylov subspace.

    By default it is set to [].

  • prepend_outer_v : <class 'bool'>, optional

    Whether to put outer_v augmentation vectors before Krylov iterates.

    By default it is set to False.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_outer_Av : <class 'bool'>, optional

    Whether A @ v should be additionnaly stored in outer_v list.

    By default it is set to True.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.

TFQMR#

Module: gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg

Transpose-Free Quasi-Minimal Residual

Pydantic model of the settings for TFQMR#
from gemseo.settings.linear_solvers import BaseSciPyLinalgSettingsBase

More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.tfqmr.html.

Optional settings

  • atol : <class 'float'>, optional

    The absolute tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 0.0.

  • callback : typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional

    The user-supplied function to call after each iteration.

    It is called as callback(xk), where xk is the current solution vector. If None, no function is called.

    By default it is set to None.

  • M : scipy.sparse.linalg._interface.LinearOperator | numpy.ndarray | scipy.sparse._base.sparray | None, optional

    The preconditioner approximating the inverse of A.

    By default it is set to None.

  • maxiter : <class 'int'>, optional

    Maximum number of iterations.

    By default it is set to 1000.

  • rtol : <class 'float'>, optional

    The relative tolerance.

    Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).

    By default it is set to 1e-12.

  • save_when_fail : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • store_residuals : <class 'bool'>, optional

    Whether to store the residual norms at each iterations.

    By default it is set to False.

  • use_ilu_precond : <class 'bool'>, optional

    Whether to use an incomplete LU factorization as preconditioner.

    By default it is set to False.

  • x0 : numpy.ndarray | None, optional

    Starting guess for the solution.

    If None, start from a matrix of zeros.

    By default it is set to None.