#include <la_linsys_new.hh>

Inheritance diagram for LinSys:

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Collaboration diagram for LinSys:

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Classes
struct	SolveInfo

Public Types
enum	SetValuesMode { INSERT =INSERT_VALUES, ADD =ADD_VALUES, ALLOCATE, DONE, NONE }

Public Member Functions
	LinSys (const Distribution *rows_ds)

	LinSys (LinSys &other)

unsigned int	size ()

unsigned int	vec_lsize ()

virtual const Mat *	get_matrix ()

virtual const Vec *	get_rhs ()

void	set_matrix_changed ()

void	set_rhs_changed ()

virtual void	set_tolerances (double r_tol, double a_tol, unsigned int max_it)=0

bool	is_matrix_changed ()

bool	is_rhs_changed ()

virtual PetscErrorCode	set_matrix (Mat &, MatStructure)

virtual PetscErrorCode	set_rhs (Vec &)

virtual PetscErrorCode	mat_zero_entries ()

virtual PetscErrorCode	rhs_zero_entries ()

const Vec &	get_solution ()

void	set_solution (Vec sol_vec)

void	set_solution (double *sol_array)

void	set_solution ()

double *	get_solution_array ()

virtual void	start_allocation ()

virtual void	start_add_assembly ()

virtual void	start_insert_assembly ()

virtual void	finish_assembly ()=0

virtual void	mat_set_values (int nrow, int rows, int ncol, int cols, double *vals)=0

void	mat_set_value (int row, int col, double val)

virtual void	rhs_set_values (int nrow, int rows, double vals)=0

void	rhs_set_value (int row, double val)

void	set_values (int nrow, int rows, int ncol, int cols, PetscScalar mat_vals, PetscScalar rhs_vals)
	Set values in the system matrix and values in the right-hand side vector on corresponding rows. More...

void	set_local_system (LocalSystem &local)

void	set_local_system (LocalSystem &local, const std::vector< LongIdx > &local_to_global_map)

void	set_values (std::vector< int > &row_dofs, std::vector< int > &col_dofs, const arma::mat &matrix, const arma::vec &rhs, const arma::vec &row_solution, const arma::vec &col_solution)

void	add_constraint (int row, double value)

virtual void	apply_constrains (double scalar)=0

virtual SolveInfo	solve ()=0

double	get_residual_norm ()

virtual double	compute_residual ()=0

double	get_relative_accuracy ()

virtual double	get_absolute_accuracy ()

void	set_symmetric (bool flag=true)

bool	is_symmetric ()

void	set_positive_definite (bool flag=true)

void	set_negative_definite (bool flag=true)

bool	is_positive_definite ()

bool	is_negative_definite ()

bool	is_new ()

bool	is_preallocated ()

void	set_spd_via_symmetric_general (bool flag=true)

bool	is_spd_via_symmetric_general ()

virtual void	view (string)

virtual void	set_from_input (const Input::Record in_rec)

virtual double	get_solution_precision ()=0

virtual	~LinSys ()

Static Public Member Functions
static Input::Type::Abstract &	get_input_type ()

Protected Types
typedef std::pair< unsigned, double >	Constraint_

typedef std::vector< Constraint_ >	ConstraintVec_

Protected Attributes
double	r_tol_
	relative tolerance of linear solver More...

double	a_tol_
	absolute tolerance of linear solver More...

unsigned int	max_it_
	maximum number of iterations of linear solver More...

MPI_Comm	comm_

SetValuesMode	status_
	Set value status of the linear system. More...

const unsigned	lsize_
	local number of matrix rows (non-overlapping division of rows) More...

unsigned	size_
	global number of matrix rows, i.e. problem size More...

const Distribution *	rows_ds_
	final distribution of rows of MH matrix More...

bool	symmetric_

bool	positive_definite_

bool	negative_definite_

bool	spd_via_symmetric_general_

bool	matrix_changed_
	true if the matrix was changed since the last solve More...

bool	rhs_changed_
	true if the right hand side was changed since the last solve More...

Vec	solution_
	PETSc vector constructed with vb array. More...

double *	v_solution_
	local solution array pointing into Vec solution_ More...

bool	own_vec_
	Indicates if the solution vector has been allocated by this class. More...

bool	own_solution_
	Indicates if the solution array has been allocated by this class. More...

double	residual_norm_
	local solution array pointing into Vec solution_ More...

ConstraintVec_	constraints_

std::vector< double >	globalSolution_
	global solution in numbering for linear system More...

Input::Record	in_rec_

Static Protected Attributes
static constexpr double	default_r_tol_ = 1e-7

static constexpr double	default_a_tol_ = 1e-11

static constexpr unsigned int	default_max_it_ = 1000

Private Member Functions
LinSys *	block (i, j) MatrixArray *matrix() virtual solve(Vector solution

Private Attributes
LinSys Vector	RHS

Friends
class	SchurComplement

Detailed Description

LinSys - matrix and a particular way how to compute solution for given RHS, i.e. this class can perform action of the matrix inverse

This is abstract class for members of possible hierarchical tree of the whole system.

Definition at line 169 of file la_linsys_new.hh.

Member Typedef Documentation

◆ Constraint_

typedef std::pair<unsigned,double> LinSys::Constraint_

protected

Definition at line 113 of file linsys.hh.

◆ ConstraintVec_

typedef std::vector< Constraint_ > LinSys::ConstraintVec_

protected

Definition at line 114 of file linsys.hh.

Member Enumeration Documentation

◆ SetValuesMode

enum LinSys::SetValuesMode

Enumerator
INSERT
ADD
ALLOCATE
DONE
NONE

Definition at line 96 of file linsys.hh.

Constructor & Destructor Documentation

◆ LinSys() [1/2]

LinSys::LinSys ( const Distribution * rows_ds )

inline

Constructor. Constructor of abstract class should not be called directly, but is used for initialization of member common to all particular solvers.

Parameters

comm	- MPI communicator

TODO: Vector solution_ is now initialized to NULL, but it should be rather allocated in the constructor instead of the method set_solution().

Definition at line 127 of file linsys.hh.

◆ LinSys() [2/2]

LinSys::LinSys ( LinSys & other )

inline

Copy constructor.

Definition at line 146 of file linsys.hh.

◆ ~LinSys()

virtual LinSys::~LinSys ( )

inlinevirtual

Definition at line 651 of file linsys.hh.

Member Function Documentation

◆ add_constraint()

void LinSys::add_constraint	(	int	row,
		double	value
	)

inline

Adds Dirichlet constrain.

Parameters

row	- global number of row that should be eliminated.
value	- solution value at the given row

Definition at line 513 of file linsys.hh.

◆ apply_constrains()

virtual void LinSys::apply_constrains ( double scalar )

pure virtual

Apply constrains to assembled matrix. Constrains are given by pairs: global row index, value. i.e. typedef pair<unsigned int, double> Constrain;

What is th meaning of ( const double factor ) form Cambridge code?

Implemented in LinSys_PETSC, and LinSys_BDDC.

◆ block()

LinSys* LinSys::block	(	i	,
		j
	)

private

◆ compute_residual()

virtual double LinSys::compute_residual ( )

pure virtual

Explicitly compute residual and its norm for current solution.

Implemented in LinSys_PETSC, SchurComplement, LinSys_BDDC, and LinSys_PERMON.

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◆ finish_assembly()

virtual void LinSys::finish_assembly ( )

pure virtual

Finish assembly of the whole system. For PETSC this should call MatEndAssembly with MAT_FINAL_ASSEMBLY

Implemented in LinSys_PETSC, and LinSys_BDDC.

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◆ get_absolute_accuracy()

virtual double LinSys::get_absolute_accuracy ( )

inlinevirtual

Returns information on absolute solver accuracy

Reimplemented in LinSys_PETSC, and LinSys_PERMON.

Definition at line 553 of file linsys.hh.

◆ get_input_type()

it::Abstract & LinSys::get_input_type ( )

static

Definition at line 29 of file linsys.cc.

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◆ get_matrix()

virtual const Mat* LinSys::get_matrix ( )

inlinevirtual

Returns PETSC matrix (only for PETSC solvers)

If matrix is changed, method set_matrix_changed() must be called. Example: @CODE MatDiagonalSet(schur->get_matrix(), new_diagonal, ADD_VALUES); schur->set_matrix_changed(); @ENDCODE

Reimplemented in LinSys_PETSC.

Definition at line 186 of file linsys.hh.

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◆ get_relative_accuracy()

double LinSys::get_relative_accuracy ( )

inline

Returns information on relative solver accuracy

Definition at line 546 of file linsys.hh.

◆ get_residual_norm()

double LinSys::get_residual_norm ( )

inline

Returns norm of the initial right hand side

Definition at line 534 of file linsys.hh.

◆ get_rhs()

virtual const Vec* LinSys::get_rhs ( )

inlinevirtual

Returns RHS vector (only for PETSC solvers)

If vector is changed, method set_rhs_changed() must be called. Example: @CODE VecScale(schur->get_rhs(), -1.0); schur->set_rhs_changed(); @ENDCODE

Reimplemented in LinSys_PETSC.

Definition at line 202 of file linsys.hh.

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◆ get_solution()

const Vec& LinSys::get_solution ( )

inline

Returns PETSC vector with solution. Underlying array can be provided on construction.

Definition at line 281 of file linsys.hh.

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◆ get_solution_array()

double* LinSys::get_solution_array ( )

inline

Returns PETSC subarray with solution. Underlying array can be provided on construction.

Definition at line 324 of file linsys.hh.

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◆ get_solution_precision()

virtual double LinSys::get_solution_precision ( )

pure virtual

Get precision of solving

Implemented in LinSys_PETSC, SchurComplement, LinSys_BDDC, and LinSys_PERMON.

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◆ is_matrix_changed()

bool LinSys::is_matrix_changed ( )

inline

Returns true if the system matrix has changed since the last solve.

Definition at line 232 of file linsys.hh.

◆ is_negative_definite()

bool LinSys::is_negative_definite ( )

inline

Definition at line 599 of file linsys.hh.

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◆ is_new()

bool LinSys::is_new ( )

inline

TODO: In fact we want to know if the matrix is already preallocated However to do this we need explicit finalisation of preallocating cycle.

Definition at line 604 of file linsys.hh.

◆ is_positive_definite()

bool LinSys::is_positive_definite ( )

inline

Definition at line 596 of file linsys.hh.

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◆ is_preallocated()

bool LinSys::is_preallocated ( )

inline

Definition at line 608 of file linsys.hh.

◆ is_rhs_changed()

bool LinSys::is_rhs_changed ( )

inline

Returns true if the system RHS has changed since the last solve.

Definition at line 238 of file linsys.hh.

◆ is_spd_via_symmetric_general()

bool LinSys::is_spd_via_symmetric_general ( )

inline

Definition at line 624 of file linsys.hh.

◆ is_symmetric()

bool LinSys::is_symmetric ( )

inline

Definition at line 569 of file linsys.hh.

◆ mat_set_value()

void LinSys::mat_set_value	(	int	row,
		int	col,
		double	val
	)

inline

Shortcut for assembling just one element into the matrix. Similarly we can provide method accepting armadillo matrices.

Definition at line 368 of file linsys.hh.

◆ mat_set_values()

virtual void LinSys::mat_set_values	(	int	nrow,
		int *	rows,
		int	ncol,
		int *	cols,
		double *	vals
	)

pure virtual

Assembly full rectangular submatrix into the system matrix. Should be virtual, implemented differently in particular solvers.

Implemented in LinSys_PETSC, and LinSys_BDDC.

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◆ mat_zero_entries()

virtual PetscErrorCode LinSys::mat_zero_entries ( )

inlinevirtual

Clears entries of the matrix

Reimplemented in LinSys_PETSC, and LinSys_BDDC.

Definition at line 263 of file linsys.hh.

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◆ rhs_set_value()

void LinSys::rhs_set_value	(	int	row,
		double	val
	)

inline

Shorcut for assembling just one element into RHS vector.

Definition at line 380 of file linsys.hh.

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◆ rhs_set_values()

virtual void LinSys::rhs_set_values	(	int	nrow,
		int *	rows,
		double *	vals
	)

pure virtual

Set values of the system right-hand side. Should be virtual, implemented differently in particular solvers.

Implemented in LinSys_PETSC, and LinSys_BDDC.

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◆ rhs_zero_entries()

virtual PetscErrorCode LinSys::rhs_zero_entries ( )

inlinevirtual

Clears entries of the right-hand side

Reimplemented in LinSys_PETSC, and LinSys_BDDC.

Definition at line 272 of file linsys.hh.

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◆ set_from_input()

virtual void LinSys::set_from_input ( const Input::Record in_rec )

inlinevirtual

Sets basic parameters of LinSys defined by user in input file and used to calculate

Reimplemented in LinSys_PETSC, LinSys_BDDC, and SchurComplement.

Definition at line 640 of file linsys.hh.

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◆ set_local_system() [1/2]

void LinSys::set_local_system ( LocalSystem & local )

inline

Definition at line 391 of file linsys.hh.

◆ set_local_system() [2/2]

void LinSys::set_local_system	(	LocalSystem &	local,
		const std::vector< LongIdx > &	local_to_global_map
	)

inline

Sets local system, considering that the dof indices are locallized on current processor.

Parameters

local_to_global_map - maps the local dof indices to global ones

Definition at line 411 of file linsys.hh.

◆ set_matrix()

virtual PetscErrorCode LinSys::set_matrix	(	Mat &	,
		MatStructure
	)

inlinevirtual

Sets PETSC matrix (only for PETSC solvers)

Reimplemented in LinSys_PETSC.

Definition at line 245 of file linsys.hh.

◆ set_matrix_changed()

void LinSys::set_matrix_changed ( )

inline

Sets matrix changed flag.

Definition at line 211 of file linsys.hh.

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◆ set_negative_definite()

void LinSys::set_negative_definite ( bool flag = true )

inline

Provides user knowledge about negative definiteness.

Definition at line 587 of file linsys.hh.

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◆ set_positive_definite()

void LinSys::set_positive_definite ( bool flag = true )

inline

Provides user knowledge about positive definiteness.

Definition at line 575 of file linsys.hh.

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◆ set_rhs()

virtual PetscErrorCode LinSys::set_rhs ( Vec & )

inlinevirtual

Sets RHS vector (only for PETSC solvers)

Reimplemented in LinSys_PETSC.

Definition at line 254 of file linsys.hh.

◆ set_rhs_changed()

void LinSys::set_rhs_changed ( )

inline

Sets rhs changed flag (only for PETSC solvers)

Definition at line 217 of file linsys.hh.

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◆ set_solution() [1/3]

void LinSys::set_solution ( )

inline

Create PETSc solution

Definition at line 313 of file linsys.hh.

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◆ set_solution() [2/3]

void LinSys::set_solution ( double * sol_array )

inline

Create PETSc solution

Definition at line 302 of file linsys.hh.

◆ set_solution() [3/3]

void LinSys::set_solution ( Vec sol_vec )

inline

Set PETSc solution

Definition at line 289 of file linsys.hh.

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◆ set_spd_via_symmetric_general()

void LinSys::set_spd_via_symmetric_general ( bool flag = true )

inline

Provides user knowledge about positive definiteness via symmetric general approach. This is useful for solving Darcy flow by mixed hybrid method, where blocks on subdomains are saddle point but interface among subdomains is only at the block of Lagrange multipliers and is symmetric positive definite. Problem condensed to interface can thus be solved by PCG method, although original problem is saddle point.

Definition at line 618 of file linsys.hh.

◆ set_symmetric()

void LinSys::set_symmetric ( bool flag = true )

inline

Provides user knowledge about symmetry.

Definition at line 560 of file linsys.hh.

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◆ set_tolerances()

virtual void LinSys::set_tolerances	(	double	r_tol,
		double	a_tol,
		unsigned int	max_it
	)

pure virtual

Set relative tolerance, absolute tolerance, and maximum number of iterations of the linear solver.

For each of these three parameters we first look for the value at user input if not set we use the value provided to this method and finally the default values are set by the call of this method in the constructor.

Implemented in SchurComplement, LinSys_PETSC, and LinSys_BDDC.

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◆ set_values() [1/2]

void LinSys::set_values	(	int	nrow,
		int *	rows,
		int	ncol,
		int *	cols,
		PetscScalar *	mat_vals,
		PetscScalar *	rhs_vals
	)

inline

Set values in the system matrix and values in the right-hand side vector on corresponding rows.

Definition at line 385 of file linsys.hh.

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◆ set_values() [2/2]

void LinSys::set_values	(	std::vector< int > &	row_dofs,
		std::vector< int > &	col_dofs,
		const arma::mat &	matrix,
		const arma::vec &	rhs,
		const arma::vec &	row_solution,
		const arma::vec &	col_solution
	)

inline

Add given dense matrix to preallocation. Shortcut to assembly into matrix and RHS in one call, possibly apply Dirichlet boundary conditions. row_dofs - are global indices of rows of dense matrix and rows of dense vector @rhs in global system col_dofs - are global indices of columns of the matrix, and possibly

Application of Dirichlet conditions: 1) Rows with negative dofs are set to zero. 2) Cols with negative dofs are eliminated. 3) If there are entries on global diagonal. We determine value K either from diagonal of local matrix, or (if it is zero) from diagonal average.

Caveats:

can not set dirichlet condition on zero dof
Armadillo stores matrix in column first form (Fortran like) which makes it not well suited for passing local matrices.

Definition at line 451 of file linsys.hh.

◆ size()

unsigned int LinSys::size ( )

inline

Returns global system size.

Definition at line 162 of file linsys.hh.

◆ solve()

virtual SolveInfo LinSys::solve ( )

pure virtual

Solve the system and return convergence reason.

Implemented in LinSys_PETSC, SchurComplement, LinSys_BDDC, and LinSys_PERMON.

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◆ start_add_assembly()

virtual void LinSys::start_add_assembly ( )

inlinevirtual

Switch linear system into adding assembly. (the only one supported by triplets ??)

Reimplemented in LinSys_PETSC.

Definition at line 340 of file linsys.hh.

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◆ start_allocation()

virtual void LinSys::start_allocation ( )

inlinevirtual

Switch linear system into allocating assembly. (only for PETSC_MPIAIJ_preallocate_by_assembly)

Reimplemented in LinSys_PETSC.

Definition at line 332 of file linsys.hh.

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◆ start_insert_assembly()

virtual void LinSys::start_insert_assembly ( )

inlinevirtual

Switch linear system into insert assembly. (not currently used)

Reimplemented in LinSys_PETSC.

Definition at line 348 of file linsys.hh.

◆ vec_lsize()

unsigned int LinSys::vec_lsize ( )

inline

Returns local system size. (for partitioning of solution vectors) for PETSC_MPIAIJ it is also partitioning of the matrix

Definition at line 171 of file linsys.hh.

◆ view()

virtual void LinSys::view ( string )

inlinevirtual

Output the system in the Matlab format possibly with given ordering. Rather we shoud provide output operator <<, since it is more flexible.

Reimplemented in LinSys_PETSC, and LinSys_PERMON.

Definition at line 632 of file linsys.hh.

Friends And Related Function Documentation

◆ SchurComplement

friend class SchurComplement

friend

Definition at line 91 of file linsys.hh.

Member Data Documentation

◆ a_tol_

double LinSys::a_tol_

protected

absolute tolerance of linear solver

Definition at line 664 of file linsys.hh.

◆ comm_

MPI_Comm LinSys::comm_

protected

Definition at line 667 of file linsys.hh.

◆ constraints_

ConstraintVec_ LinSys::constraints_

protected

Definition at line 690 of file linsys.hh.

◆ default_a_tol_

constexpr double LinSys::default_a_tol_ = 1e-11

staticconstexprprotected

Definition at line 660 of file linsys.hh.

◆ default_max_it_

constexpr unsigned int LinSys::default_max_it_ = 1000

staticconstexprprotected

Definition at line 661 of file linsys.hh.

◆ default_r_tol_

constexpr double LinSys::default_r_tol_ = 1e-7

staticconstexprprotected

Definition at line 659 of file linsys.hh.

◆ globalSolution_

std::vector<double> LinSys::globalSolution_

protected

global solution in numbering for linear system

Definition at line 692 of file linsys.hh.

◆ in_rec_

Input::Record LinSys::in_rec_

protected

Definition at line 694 of file linsys.hh.

◆ lsize_

const unsigned LinSys::lsize_

protected

local number of matrix rows (non-overlapping division of rows)

Definition at line 670 of file linsys.hh.

◆ matrix_changed_

bool LinSys::matrix_changed_

protected

true if the matrix was changed since the last solve

Definition at line 680 of file linsys.hh.

◆ max_it_

unsigned int LinSys::max_it_

protected

maximum number of iterations of linear solver

Definition at line 665 of file linsys.hh.

◆ negative_definite_

bool LinSys::negative_definite_

protected

Definition at line 677 of file linsys.hh.

◆ own_solution_

bool LinSys::own_solution_

protected

Indicates if the solution array has been allocated by this class.

Definition at line 686 of file linsys.hh.

◆ own_vec_

bool LinSys::own_vec_

protected

Indicates if the solution vector has been allocated by this class.

Definition at line 685 of file linsys.hh.

◆ positive_definite_

bool LinSys::positive_definite_

protected

Definition at line 676 of file linsys.hh.

◆ r_tol_

double LinSys::r_tol_

protected

relative tolerance of linear solver

Definition at line 663 of file linsys.hh.

◆ residual_norm_

double LinSys::residual_norm_

protected

local solution array pointing into Vec solution_

Definition at line 688 of file linsys.hh.

◆ RHS

LinSys Vector LinSys::RHS

private

Definition at line 173 of file la_linsys_new.hh.

◆ rhs_changed_

bool LinSys::rhs_changed_

protected

true if the right hand side was changed since the last solve

Definition at line 681 of file linsys.hh.

◆ rows_ds_

const Distribution* LinSys::rows_ds_

protected

final distribution of rows of MH matrix

Definition at line 673 of file linsys.hh.

◆ size_

unsigned LinSys::size_

protected

global number of matrix rows, i.e. problem size

Definition at line 671 of file linsys.hh.

◆ solution_

Vec LinSys::solution_

protected

PETSc vector constructed with vb array.

Definition at line 683 of file linsys.hh.

◆ spd_via_symmetric_general_

bool LinSys::spd_via_symmetric_general_

protected

Definition at line 678 of file linsys.hh.

◆ status_

SetValuesMode LinSys::status_

protected

Set value status of the linear system.

Definition at line 668 of file linsys.hh.

◆ symmetric_

bool LinSys::symmetric_

protected

Definition at line 675 of file linsys.hh.

◆ v_solution_

double* LinSys::v_solution_

protected

local solution array pointing into Vec solution_

Definition at line 684 of file linsys.hh.

The documentation for this class was generated from the following files:

/opt/flow123d/flow123d/src/la/la_linsys_new.hh
/opt/flow123d/flow123d/src/la/linsys.hh
/opt/flow123d/flow123d/src/la/linsys.cc

Classes

Public Types

Public Member Functions

Static Public Member Functions

Protected Types

Protected Attributes

Static Protected Attributes

Private Member Functions

Private Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ Constraint_

◆ ConstraintVec_

Member Enumeration Documentation

◆ SetValuesMode

Constructor & Destructor Documentation

◆ LinSys() [1/2]

◆ LinSys() [2/2]

◆ ~LinSys()

Member Function Documentation

◆ add_constraint()

◆ apply_constrains()

◆ block()

◆ compute_residual()

◆ finish_assembly()

◆ get_absolute_accuracy()

◆ get_input_type()

◆ get_matrix()

◆ get_relative_accuracy()

◆ get_residual_norm()

◆ get_rhs()

◆ get_solution()

◆ get_solution_array()

◆ get_solution_precision()

◆ is_matrix_changed()

◆ is_negative_definite()

◆ is_new()

◆ is_positive_definite()

◆ is_preallocated()

◆ is_rhs_changed()

◆ is_spd_via_symmetric_general()

◆ is_symmetric()

◆ mat_set_value()

◆ mat_set_values()

◆ mat_zero_entries()

◆ rhs_set_value()

◆ rhs_set_values()

◆ rhs_zero_entries()

◆ set_from_input()

◆ set_local_system() [1/2]

◆ set_local_system() [2/2]

◆ set_matrix()

◆ set_matrix_changed()

◆ set_negative_definite()

◆ set_positive_definite()

◆ set_rhs()

◆ set_rhs_changed()

◆ set_solution() [1/3]

◆ set_solution() [2/3]

◆ set_solution() [3/3]

◆ set_spd_via_symmetric_general()

◆ set_symmetric()

◆ set_tolerances()

◆ set_values() [1/2]

◆ set_values() [2/2]

◆ size()

◆ solve()

◆ start_add_assembly()

◆ start_allocation()

◆ start_insert_assembly()

◆ vec_lsize()

◆ view()

Friends And Related Function Documentation

◆ SchurComplement

Member Data Documentation

◆ a_tol_

◆ comm_

◆ constraints_

◆ default_a_tol_