Utilities

Classes
class	Funct
	A simple class to parse real valued functions. More...
class	Gauss
	Calculate data for Gauss integration. More...
class	Point< T_ >
	Defines a point with arbitrary type coordinates. More...
class	Point2D< T_ >
	Defines a 2-D point with arbitrary type coordinates. More...
class	Tabulation
	To read and manipulate tabulated functions. More...
class	Timer
	To set time counting. More...
class	UserData< T_ >
	Abstract class to define by user various problem data. More...
Files
file	OFELI.h
	Header file that includes all kernel classes of the library.
file	OFELI_Config.h
	File that contains some macros.
file	OFELI_Const.h
	File that contains some widely used constants.
Defines
#define	OFELI_TOLERANCE   OFELI_EPSMCH*10000
#define	OFELI_IMAG   std::complex<double>(0.,1.);
#define	PARSE(exp, var)   theParser.Parse(exp,var)
#define	EVAL(d)   theParser.Eval(d)
Typedefs
typedef unsigned long	lsize_t
typedef double	real_t
typedef std::complex< double >	complex_t
Functions
void	banner (const string &prog=" ")
	Outputs a banner as header of any developed program.
void	BSpline (size_t n, size_t t, Vect< Point< double > > &control, Vect< Point< double > > &output, size_t num_output)
	Function to perform a B-spline interpolation.
void	getBamg (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Bamg format.
void	getEasymesh (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Easymesh format.
void	getEMC2 (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in EMC2 format.
void	getGambit (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Gambit neutral format.
void	getGmsh (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Gmsh format.
void	getMatlab (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a Matlab mesh data.
void	getNetgen (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Netgen format.
void	getTetgen (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Tetgen format.
void	getTriangle (const string &file, Mesh &mesh, size_t nb_dof=1)
	Construct an instance of class Mesh from a mesh file stored in Triangle format.
ostream &	operator<< (ostream &s, const std::complex< double > &x)
	Output a complex number.
ostream &	operator<< (ostream &s, const std::string &c)
	Output a string.
template<class T_ >
ostream &	operator<< (ostream &s, const vector< T_ > &v)
	Output a vector instance.
template<class T_ >
ostream &	operator<< (ostream &s, const valarray< T_ > &v)
	Output a valarray instance.
template<class T_ >
ostream &	operator<< (ostream &s, const std::pair< T_, T_ > &a)
	Output a pair instance.
template<class T_ >
bool	operator== (const Point< T_ > &a, const Point< T_ > &b)
	Operator ==.
template<class T_ >
Point< T_ >	operator+ (const Point< T_ > &a, const Point< T_ > &b)
	Operator +.
template<class T_ >
Point< T_ >	operator+ (const Point< T_ > &a, const T_ &x)
	Operator +.
template<class T_ >
Point< T_ >	operator- (const Point< T_ > &a)
	Unary Operator -.
template<class T_ >
Point< T_ >	operator- (const Point< T_ > &a, const Point< T_ > &b)
	Operator -.
template<class T_ >
Point< T_ >	operator- (const Point< T_ > &a, const T_ &x)
	Operator -.
template<class T_ >
Point< T_ >	operator* (const T_ &a, const Point< T_ > &b)
	Operator *.
template<class T_ >
Point< T_ >	operator* (const int &a, const Point< T_ > &b)
	Operator *.
template<class T_ >
Point< T_ >	operator* (const Point< T_ > &b, const T_ &a)
	Operator /.
template<class T_ >
Point< T_ >	operator* (const Point< T_ > &b, const int &a)
	Operator *.
template<class T_ >
T_	operator* (const Point< T_ > &b, const Point< T_ > &a)
	Operator *.
template<class T_ >
Point< T_ >	operator/ (const Point< T_ > &b, const T_ &a)
	Operator /.
bool	areClose (const Point< double > &a, const Point< double > &b, double toler=DBL_EPSILON *10000)
	Return true if both instances of class Point<double> are distant with less then toler.
double	SqrDistance (const Point< double > &a, const Point< double > &b)
	Return squared euclidean distance between points a and b.
double	Distance (const Point< double > &a, const Point< double > &b)
	Return euclidean distance between points a and b.
template<class T_ >
std::ostream &	operator<< (std::ostream &s, const Point< T_ > &a)
	Output point coordinates.
template<class T_ >
bool	operator== (const Point2D< T_ > &a, const Point2D< T_ > &b)
	Operator ==.
template<class T_ >
Point2D< T_ >	operator+ (const Point2D< T_ > &a, const Point2D< T_ > &b)
	Operator +.
template<class T_ >
Point2D< T_ >	operator+ (const Point2D< T_ > &a, const T_ &x)
	Operator +.
template<class T_ >
Point2D< T_ >	operator- (const Point2D< T_ > &a)
	Unary Operator -.
template<class T_ >
Point2D< T_ >	operator- (const Point2D< T_ > &a, const Point2D< T_ > &b)
	Operator -.
template<class T_ >
Point2D< T_ >	operator- (const Point2D< T_ > &a, const T_ &x)
	Operator -.
template<class T_ >
Point2D< T_ >	operator* (const T_ &a, const Point2D< T_ > &b)
	Operator *.
template<class T_ >
Point2D< T_ >	operator* (const int &a, const Point2D< T_ > &b)
template<class T_ >
Point2D< T_ >	operator* (const Point2D< T_ > &b, const T_ &a)
	Operator /.
template<class T_ >
Point2D< T_ >	operator* (const Point2D< T_ > &b, const int &a)
	Operator *.
template<class T_ >
T_	operator* (const Point2D< T_ > &b, const Point2D< T_ > &a)
	Operator *.
template<class T_ >
Point2D< T_ >	operator/ (const Point2D< T_ > &b, const T_ &a)
	Operator /.
bool	areClose (const Point2D< double > &a, const Point2D< double > &b, double toler=DBL_EPSILON *10000)
	Return true if both instances of class Point2D<double> are distant with less then toler.
double	SqrDistance (const Point2D< double > &a, const Point2D< double > &b)
	Return squared euclidean distance between points a and b.
double	Distance (const Point2D< double > &a, const Point2D< double > &b)
	Return euclidean distance between points a and b.
template<class T_ >
std::ostream &	operator<< (std::ostream &s, const Point2D< T_ > &a)
	Output point coordinates.
template<class T_ >
void	qksort (std::vector< T_ > &a, int begin, int end)
	Function to sort a vector according to a key function.
template<class T_ , class C_ >
void	qksort (std::vector< T_ > &a, int begin, int end, C_ compare)
	Function to sort a vector according to a key function.
template<class T_ >
void	QuickSort (std::vector< T_ > &a, int begin, int end)
	Function to sort a vector.
template<class T_ >
void	qksort (std::valarray< T_ > &a, int begin, int end)
	Function to sort a vector.
template<class T_ , class C_ >
void	qksort (std::valarray< T_ > &a, int begin, int end, C_ compare)
	Function to sort a vector according to a key function.
void	saveField (Vect< double > &v, string output_file, int opt)
void	saveField (Vect< double > &v, const Mesh &mesh, string output_file, int opt)
void	saveGnuplot (string input_file, string output_file, string mesh_file)
	Convert an OFELI field file to an input Gnuplot file.
void	saveTecplot (string input_file, string output_file, string mesh_file)
	Convert an OFELI field file to an input Tecplot file.
void	saveVTK (string input_file, string output_file, string mesh_file)
	Convert an OFELI field file to an input VTK file.
void	saveGmsh (string input_file, string output_file, string mesh_file)
	Convert an OFELI field file to an input Gmsh file.
void	saveGmsh (const string &gp_file, const Mesh &mesh)
	This function outputs a Mesh instance in a file in Gmsh format.
void	saveGnuplot (const string &file, const Mesh &mesh)
	This function outputs a Mesh instance in a file in Gmsh format.
void	saveMatlab (const string &file, const Mesh &mesh)
	This function outputs a Mesh instance in a file in Matlab format.
void	saveTecplot (const string &file, const Mesh &mesh)
	This function outputs a Mesh instance in a file in Tecplot format.
void	saveVTK (const string &file, const Mesh &mesh, int format=0)
	This function outputs a Mesh instance in a file in VTK format.
ostream &	operator<< (ostream &s, const Tabulation &t)
	Output Tabulated function data.
int	Sgn (double a)
	Return sign of a.
double	Abs2 (complex_t a)
	Return square of modulus of complex number a.
double	Abs2 (double a)
	Return square of real number a.
double	Abs (double a)
	Return absolute value of a.
double	Abs (complex_t a)
	Return modulus of complex number a.
double	Abs (const Point< double > &p)
	Return Norm of vector a.
double	Conjg (double a)
	Return complex conjugate of real number a.
complex_t	Conjg (complex_t a)
	Return complex conjugate of complex number a.
double	Max (double a, double b, double c)
	Return maximum value of real numbers a, b and c.
int	Max (int a, int b, int c)
	Return maximum value of integer numbers a, b and c.
double	Min (double a, double b, double c)
	Return minimum value of real numbers a, b and c.
int	Min (int a, int b, int c)
	Return minimum value of integer numbers a, b and c.
double	Max (double a, double b, double c, double d)
	Return maximum value of integer numbers a, b, c and d.
int	Max (int a, int b, int c, int d)
	Return maximum value of integer numbers a, b, c and d.
double	Min (double a, double b, double c, double d)
	Return minimum value of real numbers a, b, c and d.
int	Min (int a, int b, int c, int d)
	Return minimum value of integer numbers a, b, c and d.
double	Arg (complex_t x)
	Return argument of complex number x.
complex_t	Log (complex_t x)
	Return principal determination of logarithm of complex number x.
template<class T_ >
T_	Sqr (T_ &x)
	Return square of value x.
template<class T_ >
void	Scale (T_ a, const vector< T_ > &x, vector< T_ > &y)
	Mutiply vector x by a and save result in vector y. x and y are instances of class vector<T_>
template<class T_ >
void	Scale (T_ a, const valarray< T_ > &x, valarray< T_ > &y)
	Mutiply vector x by a and save result in vector y. x and y are instances of class valarray<T_>
template<class T_ >
void	Scale (T_ a, vector< T_ > &x)
	Mutiply vector x by a. x is an instance of class vector<T_>
template<class T_ >
void	Scale (T_ a, valarray< T_ > &x)
	Mutiply vector x by a. x is an instance of class valarray<T_>
template<class T_ >
void	Xpy (size_t n, T_ *x, T_ *y)
	Add array x to y.
template<class T_ >
void	Xpy (const vector< T_ > &x, vector< T_ > &y)
	Add vector x to y. x and y are instances of class vector<T_>
template<class T_ >
void	Xpy (const valarray< T_ > &x, valarray< T_ > &y)
	Add vector x to y. x and y are instances of class valarray<T_>
template<class T_ >
void	Axpy (size_t n, T_ a, T_ *x, T_ *y)
	Multiply array x by a and add result to y. n is the arrays length.
template<class T_ >
void	Axpy (T_ a, const vector< T_ > &x, vector< T_ > &y)
	Multiply vector x by a and add result to y. x and y are instances of class vector<T_>
template<class T_ >
void	Axpy (T_ a, const valarray< T_ > &x, valarray< T_ > &y)
	Multiply vector x by a and add result to y. x and y are instances of class valarray<T_>
template<class T_ >
void	Copy (size_t n, T_ *x, T_ *y)
	Copy array x to y. n is the arrays length.
template<class T_ >
void	Copy (const valarray< T_ > &x, valarray< T_ > &y)
	Copy vector x to y.
double	Error2 (const valarray< double > &x, const valarray< double > &y)
	Return absolute L2 error between vectors x and y.
double	RError2 (const valarray< double > &x, const valarray< double > &y)
	Return absolute L2 error between vectors x and y.
double	ErrorMax (const valarray< double > &x, const valarray< double > &y)
	Return absolute Max. error between vectors x and y.
double	RErrorMax (const valarray< double > &x, const valarray< double > &y)
	Return relative Max. error between vectors x and y.
template<class T_ >
T_	Dot (size_t n, T_ *x, T_ *y)
	Return dot product of arrays x and y. n is the arrays length.
template<class T_ >
T_	Dot (const vector< T_ > &x, const vector< T_ > &y)
	Return dot product of vectors x and y. x and y are instances of class vector<T_>
real_t	Dot (const valarray< real_t > &x, const valarray< real_t > &y)
	Return dot product of vectors x and y. x and y are instances of class valarray<T_>
complex_t	Dot (const valarray< complex_t > &x, const valarray< complex_t > &y)
	Return dot product of vectors x and y. x and y are instances of class valarray<T_>
template<class T_ >
T_	Dot (const Point< T_ > &x, const Point< T_ > &y)
	Return dot product of x and y.
complex_t	CDot (size_t n, const complex_t *x, const complex_t *y)
	Return dot product of complex arrays x and y. n is the arrays length.
complex_t	CDot (const valarray< complex_t > &x, valarray< complex_t > &y)
	Return dot product of complex vectors x and y.
double	Nrm2 (size_t n, double *x)
	Return 2-norm of array x.
double	Nrm2 (const valarray< double > &x)
	Return 2-norm of vector x.
template<class T_ >
double	Nrm2 (const Point< T_ > &a)
	Return 2-norm of a.
bool	Equal (double x, double y, double toler=DBL_EPSILON)
	Function to return true if numbers x and y are close up to a given tolerance toler.
char	itoc (int i)
	Function to convert an integer to a character.
std::string	itos (int i)
	Function to convert an integer to a string.
std::string	dtos (double d)
	Function to convert a double real to a string.
std::string	ftos (float f)
	Function to convert a float real to a string.
template<class T_ >
T_	stringTo (const std::string &s)
	Function to convert a string to a template type parameter.
void	RTrim (char *s)
	Function to remove blanks at the end of a string.
void	LTrim (char *s)
	Function to remove blanks at the beginning of a string.
void	Trim (char *s)
	Function to remove blanks at the beginning and end of a string.
template<class T_ >
void	Swap (T_ &a, T_ &b)
	Swap elements a and b.

---

Detailed Description

A module that groups utility functions and classes

---

Define Documentation

#define OFELI_TOLERANCE   OFELI_EPSMCH*10000

Default tolerance for an iterative process = OFELI_EPSMCH * 10000

#define OFELI_IMAG   std::complex<double>(0.,1.);

= Unit imaginary number (i)

#define PARSE	(		exp,
			var
	)		theParser.Parse(exp,var)

A macro that parses a regular expression exp using the variables in the string var. For instance, to parse the function sin(x+y) one must declare PARSE("sin(x+y)","x,y")

#define EVAL

(

d

)

theParser.Eval(d)

A macro that evaluates a parsed regular expression For instance, with a declaration PARSE("sin(x+y)","x,y") the data x=1 and y=2 using this function must be evaluated as follows: EVAL(d) with d[0]=1, d[1]=2

---

Typedef Documentation

lsize_t

This type stands for type unsigned long

real_t

This type stands for type double

complex_t

This type stands for type std::complex<double>

---

Function Documentation

void banner

(

const string &

prog = " "

)

Outputs a banner as header of any developed program.

Parameters:

[in]

prog

Calling program name. Enables writing a copyright notice accompanying the program.

BSpline	(	size_t	n,
		size_t	t,
		Vect< Point< double > > &	control,
		Vect< Point< double > > &	output,
		size_t	num_output
	)

Function to perform a B-spline interpolation.

This program is adapted from a free program ditributed by Keith Vertanen (vertankd@cda.mrs.umn.edu) in 1994.

Parameters:

[in]	n	Number of control points minus 1.
[in]	t	Degree of the polynomial plus 1.
[in]	control	Control point array made up of Point stucture.
[out]	output	Vector in which the calculated spline points are to be put.
[in]	num_output	How many points on the spline are to be calculated.

Note:

Condition: n+2>t (No curve results if n+2<=t) Control vector contains the number of points specified by n Output array is the proper size to hold num_output point structures

void getBamg	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Bamg format.

Parameters:

[in]

file

Name of a file written in the Bamg format.

Note:

Bamg is a 2-D mesh generator. It allows to construct adapted meshes from a given metric. It was developed at INRIA, France. Available information can be found in the site:
www-rocq1.inria.fr/gamma/cdrom/www/bamg/eng.htm

Parameters:

[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getEasymesh	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Easymesh format.

Parameters:

[in]

file

Name of a file (without extension) written in Easymesh format. Actually, the function Easymesh2MDF attempts to read mesh data from files file.e, file.n and file.s produced by Easymesh.

Note:

Easymesh is a free program that generates 2-D, unstructured, Delaunay and constrained Delaunay triangulations in general domains. It can be downloaded from the site:
http://www-dinma.univ.trieste.it/nirftc/research/easymesh/Default.htm

Parameters:

[in]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getEMC2	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in EMC2 format.

Parameters:

file	[in] Name of a file written in the EMC2 format.

Note:

EMC2 is a portable, interactive and graphic software Edition of two dimensional geometry and mesh. It can be downloaded from the site:
http://www-rocq1.inria.fr/gamma/cdrom/www/emc2/eng.htm

Parameters:

[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getGambit	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Gambit neutral format.

Note:

Gambit is a commercial mesh generator associated to the CFD code Fluent. Informations about Gambit can be found in the site:
http://www.fluent.com/software/gambit/

Parameters:

[in]	file	Name of a file written in the Gambit neutral format.
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getGmsh	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Gmsh format.

Note:

Gmsh is a free mesh generator that can be downloaded from the site:
http://www.geuz.org/gmsh/

Parameters:

[in]	file	Name of a file written in the Gmsh format.
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getMatlab	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a Matlab mesh data.

Note:

Matlab is a language of scientific computing including visualization. It is developed by MathWorks. Available information can be found in the site:
http://www.mathworks.com/products/matlab/

Parameters:

[in]	file	Name of a file created by Matlab by executing the script file Matlab2OFELI.m
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getNetgen	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Netgen format.

Note:

Netgen is a tetrahedral mesh generator that can be downloaded from the site:
http://www.hpfem.jku.at/netgen/

Parameters:

[in]	file	Name of a file written in the Netgen format.
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. [ default = 1 ]

void getTetgen	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Tetgen format.

Note:

Tetgen is a free three-dimensional mesh generator that can be downloaded in the site:
http://tetgen.berlios.de/

Parameters:

[in]	file	Name of a file written in the Tetgen format.
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

void getTriangle	(	const string &	file,
		Mesh &	mesh,
		size_t	nb_dof = 1
	)

Construct an instance of class Mesh from a mesh file stored in Triangle format.

Note:

TRIANGLE is a C program that can generate meshes, Delaunay triangulations and Voronoi diagrams for 2D pointsets that can be downloaded in the site:
http://people.scs.fsu.edu/~burkardt/c_src/triangle/triangle.html/

Parameters:

[in]	file	Name of a file written in the Tetgen format.
[out]	mesh	Mesh instance created by the function.
[in]	nb_dof	Number of degrees of freedom for each node. This information is not provided, in general, by mesh generators. Its default value here is 1.

bool operator==	(	const Point< T_ > &	a,
		const Point< T_ > &	b
	)

Operator ==.

Return true if a=b, false if not.

Point< T_ > operator+	(	const Point< T_ > &	a,
		const Point< T_ > &	b
	)

Operator +.

Return sum of two points a and b.

Point< T_ > operator+	(	const Point< T_ > &	a,
		const T_ &	x
	)

Operator +.

Translate by x.

Point< T_ > operator-

(

const Point< T_ > &

a

)

Unary Operator -.

Return minus a

Point< T_ > operator-	(	const Point< T_ > &	a,
		const Point< T_ > &	b
	)

Operator -.

Return point a minus point b.

Point< T_ > operator-	(	const Point< T_ > &	a,
		const T_ &	x
	)

Operator -.

Translate a by -x.

Point< T_ > operator*	(	const T_ &	a,
		const Point< T_ > &	b
	)

Operator *.

Return point b premultiplied by constant a.

Point< T_ > operator*	(	const int &	a,
		const Point< T_ > &	b
	)

Operator *.

Return point b divided by integer constant a

Point< T_ > operator*	(	const Point< T_ > &	b,
		const T_ &	a
	)

Operator /.

Return point b postmultiplied by constant a.

Point< T_ > operator*	(	const Point< T_ > &	b,
		const int &	a
	)

Operator *.

Return point b postmultiplied by constant a.

T_ operator*	(	const Point< T_ > &	b,
		const Point< T_ > &	a
	)

Operator *.

Return point b postmultiplied by integer constant a.

Point< T_ > operator/	(	const Point< T_ > &	b,
		const T_ &	a
	)

Operator /.

Return point b divided by constant a

bool operator==	(	const Point2D< T_ > &	a,
		const Point2D< T_ > &	b
	)

Operator ==.

Return true if a=b, false if not.

Point2D< T_ > operator+	(	const Point2D< T_ > &	a,
		const Point2D< T_ > &	b
	)

Operator +.

Return sum of two points a and b.

Point2D< T_ > operator+	(	const Point2D< T_ > &	a,
		const T_ &	x
	)

Operator +.

Translate by x.

Point2D< T_ > operator-

(

const Point2D< T_ > &

a

)

Unary Operator -.

Return minus a

Point2D< T_ > operator-	(	const Point2D< T_ > &	a,
		const Point2D< T_ > &	b
	)

Operator -.

Return point a minus point b.

Point2D< T_ > operator-	(	const Point2D< T_ > &	a,
		const T_ &	x
	)

Operator -.

Translate a by -x.

Point2D< T_ > operator*	(	const T_ &	a,
		const Point2D< T_ > &	b
	)

Operator *.

Return point b premultiplied by constant a.

Point2D< T_ > operator*	(	const int &	a,
		const Point2D< T_ > &	b
	)

Operator *.

Return point b divided by integer constant a

Point2D< T_ > operator*	(	const Point2D< T_ > &	b,
		const T_ &	a
	)

Operator /.

Return point b postmultiplied by constant a.

Point2D< T_ > operator*	(	const Point2D< T_ > &	b,
		const int &	a
	)

Operator *.

Return point b postmultiplied by constant a.

T_ operator*	(	const Point2D< T_ > &	b,
		const Point2D< T_ > &	a
	)

Operator *.

Return point b postmultiplied by integer constant a.

Point2D< T_ > operator/	(	const Point2D< T_ > &	b,
		const T_ &	a
	)

Operator /.

Return point b divided by constant a

void qksort	(	std::vector< T_ > &	a,
		int	begin,
		int	end
	)

Function to sort a vector according to a key function.

qksort uses the famous quick sorting algorithm.

Parameters:

[in]	a	Vector to sort.
[in]	begin	index of starting iterator
[in]	end	index of ending iterator

The calling program must provide the template function bool _less(const T_ &x, const T_ &y); that returns true if object x is ordered lower than object y and false if not.

void qksort	(	std::vector< T_ > &	a,
		int	begin,
		int	end,
		C_	compare
	)

Function to sort a vector according to a key function.

qksort uses the famous quick sorting algorithm.

Parameters:

[in,out]	a	Vector to sort.
[in]	begin	index of starting iterator
[in]	end	index of ending iterator
[in]	compare	A function object that implements the ordering. The user must provide this function that returns a boolean function that is true if the first argument is less than the second and false if not.

void QuickSort	(	std::vector< T_ > &	a,
		int	begin,
		int	end
	)

Function to sort a vector.

qksort uses the famous quick sorting algorithm.

Parameters:

[in,out]	a	Vector to sort.
[in]	begin	index of starting iterator
[in]	end	index of ending iterator

The calling program must provide an overloading of the operator < for the type T_

void qksort	(	std::valarray< T_ > &	a,
		int	begin,
		int	end
	)

Function to sort a vector.

qksort uses the famous quick sorting algorithm.

Parameters:

[in,out]	a	Vector to sort.
[in]	begin	index of starting index (default value is 0)
[in]	end	index of ending index (default value is the vector size - 1)

void qksort	(	std::valarray< T_ > &	a,
		int	begin,
		int	end,
		C_	compare
	)

Function to sort a vector according to a key function.

qksort uses the famous quick sorting algorithm.

Parameters:

[in,out]	a	Vector to sort.
[in]	begin	index of starting index (0 for the beginning of the vector)
[in]	end	index of ending index
[in]	compare	A function object that implements the ordering. The user must provide this function that returns a boolean function that is true if the first argument is less than the second and false if not.

void saveField	(	Vect< double > &	v,
		string	output_file,
		int	opt
	)

Parameters:

[in]	v	Vect instance to save
[in]	output_file	Output file where to save the vector
[in]	opt	Option to choose file format to save. This is to be chosen among enumerated values: GMSH GNUPLOT MATLAB TECPLOT VTK

void saveField	(	Vect< double > &	v,
		const Mesh &	mesh,
		string	output_file,
		int	opt
	)

Parameters:

[in]	v	Vect instance to save
[in]	mesh	Mesh instance
[in]	output_file	Output file where to save the vector
[in]	opt	Option to choose file format to save. This is to be chosen among enumerated values: GMSH GNUPLOT MATLAB TECPLOT VTK

void saveGnuplot	(	string	input_file,
		string	output_file,
		string	mesh_file
	)

Convert an OFELI field file to an input Gnuplot file.

Gnuplot is a command-line driven program for producing 2D and 3D plots. It is under the GNU General Public License. Available information can be found in the site:
http://www.gnuplot.info/

Parameters:

[in]	input_file	Input file (OFELI XML file containing a field).
[in]	output_file	Output file (gnuplot format file)
[in]	mesh_file	File containing mesh data

void saveTecplot	(	string	input_file,
		string	output_file,
		string	mesh_file
	)

Convert an OFELI field file to an input Tecplot file.

Tecplot is high quality post graphical commercial processing program developed by Amtec. Available information can be found in the site: http://www.tecplot.com

Parameters:

[in]	input_file	Input file (OFELI XML file containing a field).
[in]	output_file	Output file (gnuplot format file)
[in]	mesh_file	File containing mesh data

saveVTK	(	string	input_file,
		string	output_file,
		string	mesh_file
	)

Convert an OFELI field file to an input VTK file.

The Visualization ToolKit (VTK) is an open source, freely available software system for 3D computer graphics. Available information can be found in the site:
http://public.kitware.com/VTK/

Parameters:

[in]	input_file	Input file (OFELI XML file containing a field).
[in]	output_file	Output file (gnuplot format file)
[in]	mesh_file	File containing mesh data

void saveGmsh	(	string	input_file,
		string	output_file,
		string	mesh_file
	)

Convert an OFELI field file to an input Gmsh file.

Gmsh is a free mesh generator and postprocessor that can be downloaded from the site:
http://www.geuz.org/gmsh/

Parameters:

[in]	input_file	Input file (OFELI XML file containing a field).
[in]	output_file	Output file (gnuplot format file)
[in]	mesh_file	File containing mesh data

void saveGmsh	(	const string &	file,
		const Mesh &	mesh
	)

This function outputs a Mesh instance in a file in Gmsh format.

Note:

Gmsh is a free mesh generator that can be downloaded from the site: http://www.geuz.org/gmsh/

Parameters:

[out]	file	Output file in Gmsh format.
[in]	mesh	Mesh instance to save.

void saveGnuplot	(	const string &	file,
		const Mesh &	mesh
	)

This function outputs a Mesh instance in a file in Gmsh format.

Note:

Gnuplot is a command-line driven program for producing 2D and 3D plots. It is under the GNU General Public License. Available information can be found in the site:
http://www.gnuplot.info/

Parameters:

[out]	file	Output file in Gnuplot format.
[in]	mesh	Mesh instance to save.

void saveMatlab	(	const string &	file,
		const Mesh &	mesh
	)

This function outputs a Mesh instance in a file in Matlab format.

Note:

Matlab is a language of scientific computing including visualization. It is developed by MathWorks. Available information can be found in the site:
http://www.mathworks.com/products/matlab/

Parameters:

[out]	file	Output file in Matlab format.
[in]	mesh	Mesh instance to save.

void saveTecplot	(	const string &	file,
		const Mesh &	mesh
	)

This function outputs a Mesh instance in a file in Tecplot format.

Note:

Tecplot is high quality post graphical commercial processing program developed by Amtec. Available information can be found in the site:
http://www.tecplot.com

Parameters:

[out]	file	Output file in Tecplot format.
[in]	mesh	Mesh instance to save.

void saveVTK	(	const string &	file,
		const Mesh &	mesh,
		int	format = 0
	)

This function outputs a Mesh instance in a file in VTK format.

Note:

The Visualization ToolKit (VTK) is an open source, freely available software system for 3D computer graphics. Available information can be found in the site:
http://public.kitware.com/VTK/

Parameters:

[out]	file	Output file in VTK format.
[in]	mesh	Mesh instance to save.
[in]	format	Option for output format.

double Nrm2	(	size_t	n,
		double *	x
	)

Return 2-norm of array x.

Parameters:

[in]	n	is Array length
[in]	x	Array to treat

bool Equal	(	double	x,
		double	y,
		double	toler = DBL_EPSILON
	)

Function to return true if numbers x and y are close up to a given tolerance toler.

Default value of tolerance is the constant OFELI_EPSMCH