Solver functions and classes. More...

Classes
class	Reconstruction
	To perform various reconstruction operations. More...

class	EigenProblemSolver
	Class to find eigenvalues and corresponding eigenvectors of a given matrix in a generalized eigenproblem, i.e. Find scalars l and non-null vectors v such that [K]{v} = l[M]{v} where [K] and [M] are symmetric matrices. The eigenproblem can be originated from a PDE. For this, we will refer to the matrices K and M as Stiffness and Mass matrices respectively. More...

class	FuncApprox
	To set function approximation methods. More...

class	GeoModel
	To set geometry modelling. More...

class	Integration
	Class for numerical integration methods. More...

class	Iter< T_ >
	Class to drive an iterative process. More...

class	LeastSquare
	To compute a least square approximation. More...

class	LinearSolver
	Class to solve systems of linear equations by direct or iterative methods. More...

class	LPSolver
	To solve a linear programming problem. More...

class	MyNLAS
	Abstract class to define by user specified function. More...

class	MyOpt
	Abstract class to define by user specified optimization function. More...

class	NLASSolver
	To solve a system of nonlinear algebraic equations of the form f(u) = 0. More...

class	ODESolver
	To solve a system of ordinary differential equations. More...

class	OptSolver
	To solve an optimization problem with bound constraints. More...

class	Prec< T_ >
	To set a preconditioner. More...

class	TimeStepping
	To solve time stepping problems, i.e. systems of linear ordinary differential equations of the form [A2]{y"} + [A1]{y'} + [A0]{y} = {b}. More...

Macros
#define	MAX_NB_INPUT_FIELDS 3
	Maximum number of fields for an equation.

#define	MAX_NB_MESHES 10
	Maximum number of meshes.

#define	TIME_LOOP(ts, t, ft, n)
	A macro to loop on time steps to integrate on time ts : Time step t : Initial time value updated at each time step ft : Final time value n : Time step index.

#define	TimeLoop
	A macro to loop on time steps to integrate on time.

#define	IterationLoop while (++theIteration<MaxNbIterations && Converged==false)
	A macro to loop on iterations for an iterative procedure.

Functions
ostream &	operator<< (ostream &s, const Muscl3DT &m)
	Output mesh data as calculated in class Muscl3DT.

template<class T_ >
int	BiCG (const SpMatrix< T_ > &A, const Prec< T_ > &P, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t &toler)
	Biconjugate gradient solver function.

template<class T_ >
int	BiCG (const SpMatrix< T_ > &A, int prec, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	Biconjugate gradient solver function.

template<class T_ >
int	CG (const SpMatrix< T_ > &A, const Prec< T_ > &P, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	Conjugate gradient solver function.

template<class T_ >
int	CG (const SpMatrix< T_ > &A, int prec, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	Conjugate gradient solver function.

template<class T_ >
int	CGS (const SpMatrix< T_ > &A, const Prec< T_ > &P, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	Conjugate Gradient Squared solver function.

template<class T_ >
int	CGS (const SpMatrix< T_ > &A, int prec, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	Conjugate Gradient Squared solver function.

ostream &	operator<< (ostream &s, const EigenProblemSolver &es)
	Output eigenproblem information.

template<class T_ >
int	GMRes (const SpMatrix< T_ > &A, const Prec< T_ > &P, const Vect< T_ > &b, Vect< T_ > &x, size_t m, int max_it, real_t toler)
	GMRes solver function.

template<class T_ >
int	GMRes (const SpMatrix< T_ > &A, int prec, const Vect< T_ > &b, Vect< T_ > &x, size_t m, int max_it, real_t toler)
	GMRes solver function.

template<class T_ >
int	GS (const SpMatrix< T_ > &A, const Vect< T_ > &b, Vect< T_ > &x, real_t omega, int max_it, real_t toler)
	Gauss-Seidel solver function.

template<class T_ >
int	Jacobi (const SpMatrix< T_ > &A, const Vect< T_ > &b, Vect< T_ > &x, real_t omega, int max_it, real_t toler)
	Jacobi solver function.

ostream &	operator<< (ostream &s, const ODESolver &de)
	Output differential system information.

template<class T_ , class M_ >
int	Richardson (const M_ &A, const Vect< T_ > &b, Vect< T_ > &x, real_t omega, int max_it, real_t toler, int verbose)
	Richardson solver function.

template<class T_ >
void	Schur (SkMatrix< T_ > &A, SpMatrix< T_ > &U, SpMatrix< T_ > &L, SpMatrix< T_ > &D, Vect< T_ > &b, Vect< T_ > &c)
	Solve a linear system of equations with a 2x2-block matrix.

template<class T_ , class M_ >
int	SSOR (const M_ &A, const Vect< T_ > &b, Vect< T_ > &x, int max_it, real_t toler)
	SSOR solver function.

ostream &	operator<< (ostream &s, TimeStepping &ts)
	Output differential system information.

Detailed Description

Solver functions and classes.

Macro Definition Documentation

◆ IterationLoop

#define IterationLoop while (++theIteration<MaxNbIterations && Converged==false)

A macro to loop on iterations for an iterative procedure.

It uses the following global variables defined in OFELI: theIteration, MaxNbIterations, Converged

Warning: The variable theIteration must be zeroed before using this macro

◆ MAX_NB_INPUT_FIELDS

#define MAX_NB_INPUT_FIELDS 3

Maximum number of fields for an equation.

Useful for coupled problems

◆ MAX_NB_MESHES

#define MAX_NB_MESHES 10

Maximum number of meshes.

Useful for coupled problems

◆ TimeLoop

#define TimeLoop

Value:

          OFELI::NbTimeSteps = int(OFELI::theFinalTime/OFELI::theTimeStep);   \
          for (OFELI::theTime=OFELI::theTimeStep, theStep=1;                  \
               theTime<OFELI::theFinalTime+0.001*OFELI::theTimeStep;          \
               OFELI::theTime+=OFELI::theTimeStep, ++OFELI::theStep)

A macro to loop on time steps to integrate on time.

It uses the following global variables defined in OFELI: theStep, theTime, theTimeStep, theFinalTime

Function Documentation

◆ BiCG() [1/2]

template<class T_ >

int BiCG	(	const SpMatrix< T_ > &	A,
		const Prec< T_ > &	P,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t &	toler
	)

Biconjugate gradient solver function.

This function uses the preconditioned Biconjugate Conjugate Gradient algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function BiCG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	P	Preconditioner (Instance of class Prec).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
	toler	[in] Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ BiCG() [2/2]

template<class T_ >

int BiCG	(	const SpMatrix< T_ > &	A,
		int	prec,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

Biconjugate gradient solver function.

This function uses the preconditioned Biconjugate Conjugate Gradient algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	prec	Enum variable selecting a preconditioner, among the values `IDENT_PREC`, `DIAG_PREC`, `ILU_PREC` or `SSOR_PREC`
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
	toler	[in] Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ CG() [1/2]

template<class T_ >

int CG	(	const SpMatrix< T_ > &	A,
		const Prec< T_ > &	P,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

Conjugate gradient solver function.

This function uses the preconditioned Conjugate Gradient algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	P	Preconditioner (Instance of class Prec).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ CG() [2/2]

template<class T_ >

int CG	(	const SpMatrix< T_ > &	A,
		int	prec,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

Conjugate gradient solver function.

This function uses the preconditioned Conjugate Gradient algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of abstract class SpMatrix).
[in]	prec	Enum variable selecting a preconditioner, among the values `IDENT_PREC`, `DIAG_PREC`, `ILU_PREC` or `SSOR_PREC`
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ CGS() [1/2]

template<class T_ >

int CGS	(	const SpMatrix< T_ > &	A,
		const Prec< T_ > &	P,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

Conjugate Gradient Squared solver function.

This function uses the preconditioned Conjugate Gradient Squared algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CGS
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	P	Preconditioner (Instance of class Prec).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations

Template Parameters

<T_>	Data type (real_t, float, complex<real_t>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ CGS() [2/2]

template<class T_ >

int CGS	(	const SpMatrix< T_ > &	A,
		int	prec,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

Conjugate Gradient Squared solver function.

This function uses the preconditioned Conjugate Gradient Squared algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	prec	Enum variable selecting a preconditioner, among the values `IDENT_PREC`, `DIAG_PREC`, `ILU_PREC` or `SSOR_PREC`
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations

Template Parameters

<T_>	Data type (real_t, float, complex<real_t>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ GMRes() [1/2]

template<class T_ >

int GMRes	(	const SpMatrix< T_ > &	A,
		const Prec< T_ > &	P,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		size_t	m,
		int	max_it,
		real_t	toler
	)

GMRes solver function.

This function uses the preconditioned GMRES algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CMRes
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	P	Preconditioner (Instance of class Prec).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	m	Number of subspaces to generate for iterations.
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ GMRes() [2/2]

template<class T_ >

int GMRes	(	const SpMatrix< T_ > &	A,
		int	prec,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		size_t	m,
		int	max_it,
		real_t	toler
	)

GMRes solver function.

This function uses the preconditioned GMRES algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function CG
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	prec	Enum variable selecting a preconditioner, among the values `IDENT_PREC`, `DIAG_PREC`, `ILU_PREC` or `SSOR_PREC`
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	m	Number of subspaces to generate for iterations.
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (double, float, complex<double>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ GS()

template<class T_ >

int GS	(	const SpMatrix< T_ > &	A,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		real_t	omega,
		int	max_it,
		real_t	toler
	)

Gauss-Seidel solver function.

This function uses the relaxed Gauss-Seidel algorithm to solve a linear system with a sparse matrix.
The global variable Verbosity enables choosing output message level

Verbosity < 2 : No output message
Verbosity > 1 : Notify executing the function GS
Verbosity > 2 : Notify convergence with number of performed iterations or divergence
Verbosity > 3 : Output each iteration number and residual
Verbosity > 6 : Print final solution if convergence
Verbosity > 10 : Print obtained solution at each iteration

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	omega	Relaxation parameter.
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations

Template Parameters

<T_>	Data type (real_t, float, complex<real_t>, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ Jacobi()

template<class T_ >

int Jacobi	(	const SpMatrix< T_ > &	A,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		real_t	omega,
		int	max_it,
		real_t	toler
	)

Jacobi solver function.

Parameters

[in]	A	Problem matrix (Instance of class SpMatrix).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	omega	Relaxation parameter.
[in]	max_it	Maximum number of iterations.
[in,out]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Parameters

<T_>	Data type (real_t, float, complex<real_t>, ...)
<M_>	Matrix storage class

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ operator<<()

ostream & operator<<	(	ostream &	s,
		TimeStepping &	ts
	)

Output differential system information.

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ Richardson()

template<class T_ , class M_ >

int Richardson	(	const M_ &	A,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		real_t	omega,
		int	max_it,
		real_t	toler,
		int	verbose
	)

Richardson solver function.

Parameters

[in]	A	Problem matrix problem (Instance of abstract class M_).
[in]	b	Right-hand side vector (class Vect)
	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	omega	Relaxation parameter.
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).
[in]	verbose	Information output parameter (`0`: No output, `1`: Output iteration information, `2` and greater: Output iteration information and solution at each iteration.

Returns: nb_it Number of performed iterations,

Template Parameters

<T_>	Data type (real_t, float, complex<real_t>, ...)
<M_>	Matrix storage class

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ Schur()

template<class T_ >

void Schur	(	SkMatrix< T_ > &	A,
		SpMatrix< T_ > &	U,
		SpMatrix< T_ > &	L,
		SpMatrix< T_ > &	D,
		Vect< T_ > &	b,
		Vect< T_ > &	c
	)

Solve a linear system of equations with a 2x2-block matrix.

The linear system is of the form

     | A  U | |x|   |b|
     |      | | | = | |
     | L  D | |y|   |c|

Parameters

[in]	A	Instance of class SkMatrix class for the first diagonal block. The matrix must be invertible and factorizable (Do not use SpMatrix class) where `A`, `U`, `L`, `D` are instances of matrix classes,
[in]	U	Instance of class SpMatrix for the upper triangle block. The matrix can be rectangular
[in]	L	Instance of class SpMatrix for the lower triangle block. The matrix can be rectangular
[in]	D	Instance of class SpMatrix for the second diagonal block. The matrix must be factorizable (Do not use SpMatrix class)
[in,out]	b	Vector (Instance of class Vect) that contains the first block of right-hand side on input and the first block of the solution on output. `b` must have the same size as the dimension of `A`.
[in,out]	c	Vect instance that contains the second block of right-hand side on output and the first block of the solution on output. `c` must have the same size as the dimension of `D`.

Template Argument:

Template Parameters

<T_>	data type (real_t, float, ...)

Author: Rachid Touzani

Copyright: GNU Lesser Public License

◆ SSOR()

template<class T_ , class M_ >

int SSOR	(	const M_ &	A,
		const Vect< T_ > &	b,
		Vect< T_ > &	x,
		int	max_it,
		real_t	toler
	)

SSOR solver function.

Parameters

[in]	A	Problem matrix (Instance of abstract class M_).
[in]	b	Right-hand side vector (class Vect)
[in,out]	x	Vect instance containing initial solution guess in input and solution of the linear system in output (If iterations have succeeded).
[in]	max_it	Maximum number of iterations.
[in]	toler	Tolerance for convergence (measured in relative weighted 2-Norm).

Returns: Number of performed iterations,

Template Arguments:

T_ data type (double, float, ...)
M_ Matrix storage class

Author
Rachid Touzani

Copyright
GNU Lesser Public License

Classes

Macros

Functions

Detailed Description

Macro Definition Documentation

◆ IterationLoop

◆ MAX_NB_INPUT_FIELDS

◆ MAX_NB_MESHES

◆ TimeLoop

Function Documentation

◆ BiCG() [1/2]

◆ BiCG() [2/2]

◆ CG() [1/2]

◆ CG() [2/2]

◆ CGS() [1/2]

◆ CGS() [2/2]

◆ GMRes() [1/2]

◆ GMRes() [2/2]

◆ GS()

◆ Jacobi()

◆ operator<<()

◆ Richardson()

◆ Schur()

◆ SSOR()