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* Definition of Lorene class Scalar
*
*/
/*
* Copyright (c) 2003-2004 Eric Gourgoulhon & Jerome Novak
*
* Copyright (c) 1999-2000 Jean-Alain Marck (for previous class Cmp)
* Copyright (c) 1999-2002 Eric Gourgoulhon (for previous class Cmp)
* Copyright (c) 1999-2001 Philippe Grandclement (for previous class Cmp)
* Copyright (c) 2000-2002 Jerome Novak (for previous class Cmp)
* Copyright (c) 2000-2001 Keisuke Taniguchi (for previous class Cmp)
*
* This file is part of LORENE.
*
* LORENE is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* LORENE is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with LORENE; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
#ifndef __SCALAR_H_
#define __SCALAR_H_
/*
* $Id: scalar.h,v 1.94 2015/12/18 15:52:51 j_novak Exp $
* $Log: scalar.h,v $
* Revision 1.94 2015/12/18 15:52:51 j_novak
* New method is_nan() for class Scalar.
*
* Revision 1.93 2015/09/10 13:28:05 j_novak
* New methods for the class Hot_Eos
*
* Revision 1.92 2014/10/13 08:52:36 j_novak
* Lorene classes and functions now belong to the namespace Lorene.
*
* Revision 1.91 2013/06/05 15:06:10 j_novak
* Legendre bases are treated as standard bases, when the multi-grid
* (Mg3d) is built with BASE_LEG.
*
* Revision 1.90 2013/01/11 15:44:54 j_novak
* Addition of Legendre bases (part 2).
*
* Revision 1.89 2012/12/19 13:59:56 j_penner
* added a few lines to the documentation for scalar_match_tau function
*
* Revision 1.88 2012/01/17 15:05:46 j_penner
* *** empty log message ***
*
* Revision 1.87 2012/01/17 10:16:27 j_penner
* functions added: sarra_filter_r, sarra_filter_r_all_domains, Heaviside
*
* Revision 1.86 2011/04/08 13:13:09 e_gourgoulhon
* Changed the comment of function val_point to indicate specifically the
* division by r^dzpuis in the compactified external domain.
*
* Revision 1.85 2008/09/29 13:23:51 j_novak
* Implementation of the angular mapping associated with an affine
* mapping. Things must be improved to take into account the domain index.
*
* Revision 1.84 2008/09/22 19:08:01 j_novak
* New methods to deal with boundary conditions
*
* Revision 1.83 2008/05/24 15:05:22 j_novak
* New method Scalar::match_tau to match the output of an explicit time-marching scheme with the tau method.
*
* Revision 1.82 2007/12/21 16:06:16 j_novak
* Methods to filter Tensor, Vector and Sym_tensor objects.
*
* Revision 1.81 2007/10/31 10:33:11 j_novak
* Added exponential filters to smooth Gibbs-type phenomena.
*
* Revision 1.80 2007/06/21 19:56:36 k_taniguchi
* Introduction of another filtre_r.
*
* Revision 1.79 2007/05/06 10:48:08 p_grandclement
* Modification of a few operators for the vorton project
*
* Revision 1.78 2007/01/16 15:05:59 n_vasset
* New constructor (taking a Scalar in mono-domain angular grid for
* boundary) for function sol_elliptic_boundary
*
* Revision 1.77 2006/05/26 09:00:09 j_novak
* New members for multiplication or division by cos(theta).
*
* Revision 1.76 2005/11/30 13:48:06 e_gourgoulhon
* Replaced M_PI/2 by 1.57... in argument list of sol_elliptic_sin_zec
* (in order not to require the definition of M_PI).
*
* Revision 1.75 2005/11/30 11:09:03 p_grandclement
* Changes for the Bin_ns_bh project
*
* Revision 1.74 2005/11/17 15:29:46 e_gourgoulhon
* Added arithmetics with Mtbl.
*
* Revision 1.73 2005/10/25 08:56:34 p_grandclement
* addition of std_spectral_base in the case of odd functions near the origin
*
* Revision 1.72 2005/09/07 13:10:47 j_novak
* Added a filter setting to zero all mulitpoles in a given range.
*
* Revision 1.71 2005/08/26 14:02:38 p_grandclement
* Modification of the elliptic solver that matches with an oscillatory exterior solution
* small correction in Poisson tau also...
*
* Revision 1.70 2005/08/25 12:14:07 p_grandclement
* Addition of a new method to solve the scalar Poisson equation, based on a multi-domain Tau-method
*
* Revision 1.69 2005/06/09 07:56:25 f_limousin
* Implement a new function sol_elliptic_boundary() and
* Vector::poisson_boundary(...) which solve the vectorial poisson
* equation (method 6) with an inner boundary condition.
*
* Revision 1.68 2005/06/08 12:35:18 j_novak
* New method for solving divergence-like ODEs.
*
* Revision 1.67 2005/05/20 14:42:30 j_novak
* Added the method Scalar::get_spectral_base().
*
* Revision 1.66 2005/04/04 21:28:57 e_gourgoulhon
* Added argument lambda to method poisson_angu
* to deal with the generalized angular Poisson equation:
* Lap_ang u + lambda u = source.
*
* Revision 1.65 2004/12/14 09:09:39 f_limousin
* Modif. comments.
*
* Revision 1.64 2004/11/23 12:41:53 f_limousin
* Intoduce function poisson_dir_neu(...) to solve a scalar poisson
* equation with a mixed boundary condition (Dirichlet + Neumann).
*
* Revision 1.63 2004/10/11 15:09:00 j_novak
* The radial manipulation functions take Scalar as arguments, instead of Cmp.
* Added a conversion operator from Scalar to Cmp.
* The Cmp radial manipulation function make conversion to Scalar, call to the
* Map_radial version with a Scalar argument and back.
*
* Revision 1.62 2004/08/24 09:14:40 p_grandclement
* Addition of some new operators, like Poisson in 2d... It now requieres the
* GSL library to work.
*
* Also, the way a variable change is stored by a Param_elliptic is changed and
* no longer uses Change_var but rather 2 Scalars. The codes using that feature
* will requiere some modification. (It should concern only the ones about monopoles)
*
* Revision 1.61 2004/07/27 08:24:26 j_novak
* Modif. comments
*
* Revision 1.60 2004/07/26 16:02:21 j_novak
* Added a flag to specify whether the primitive should be zero either at r=0
* or at r going to infinity.
*
* Revision 1.59 2004/07/06 13:36:27 j_novak
* Added methods for desaliased product (operator |) only in r direction.
*
* Revision 1.58 2004/06/22 08:49:57 p_grandclement
* Addition of everything needed for using the logarithmic mapping
*
* Revision 1.57 2004/06/14 15:24:23 e_gourgoulhon
* Added method primr (radial primitive).
*
* Revision 1.56 2004/06/11 14:29:56 j_novak
* Scalar::multipole_spectrum() is now a const method.
*
* Revision 1.55 2004/05/24 14:07:31 e_gourgoulhon
* Method set_domain now includes a call to del_deriv() for safety.
*
* Revision 1.54 2004/05/07 11:26:10 f_limousin
* New method filtre_r(int*)
*
* Revision 1.53 2004/03/22 13:12:43 j_novak
* Modification of comments to use doxygen instead of doc++
*
* Revision 1.52 2004/03/17 15:58:47 p_grandclement
* Slight modification of sol_elliptic_no_zec
*
* Revision 1.51 2004/03/11 12:07:30 e_gourgoulhon
* Added method visu_section_anim.
*
* Revision 1.50 2004/03/08 15:45:38 j_novak
* Modif. comment
*
* Revision 1.49 2004/03/05 15:09:40 e_gourgoulhon
* Added method smooth_decay.
*
* Revision 1.48 2004/03/01 09:57:02 j_novak
* the wave equation is solved with Scalars. It now accepts a grid with a
* compactified external domain, which the solver ignores and where it copies
* the values of the field from one time-step to the next.
*
* Revision 1.47 2004/02/27 09:43:58 f_limousin
* New methods filtre_phi(int) and filtre_theta(int).
*
* Revision 1.46 2004/02/26 22:46:26 e_gourgoulhon
* Added methods derive_cov, derive_con and derive_lie.
*
* Revision 1.45 2004/02/21 17:03:49 e_gourgoulhon
* -- Method "point" renamed "val_grid_point".
* -- Method "set_point" renamed "set_grid_point".
*
* Revision 1.44 2004/02/19 22:07:35 e_gourgoulhon
* Added argument "comment" in method spectral_display.
*
* Revision 1.43 2004/02/11 09:47:44 p_grandclement
* Addition of a new elliptic solver, matching with the homogeneous solution
* at the outer shell and not solving in the external domain (more details
* coming soon ; check your local Lorene dealer...)
*
* Revision 1.42 2004/01/28 16:46:22 p_grandclement
* Addition of the sol_elliptic_fixe_der_zero stuff
*
* Revision 1.41 2004/01/28 13:25:40 j_novak
* The ced_mult_r arguments have been suppressed from the Scalar::*dsd* methods.
* In the div/mult _r_dzpuis, there is no more default value.
*
* Revision 1.40 2004/01/28 10:39:17 j_novak
* Comments modified.
*
* Revision 1.39 2004/01/27 15:10:01 j_novak
* New methods Scalar::div_r_dzpuis(int) and Scalar_mult_r_dzpuis(int)
* which replace div_r_inc*. Tried to clean the dzpuis handling.
* WARNING: no testing at this point!!
*
* Revision 1.38 2004/01/23 13:25:44 e_gourgoulhon
* Added methods set_inner_boundary and set_outer_boundary.
* Methods set_val_inf and set_val_hor, which are particular cases of
* the above, have been suppressed.
*
* Revision 1.37 2004/01/22 16:10:09 e_gourgoulhon
* Added (provisory) method div_r_inc().
*
* Revision 1.36 2003/12/16 06:32:20 e_gourgoulhon
* Added method visu_box.
*
* Revision 1.35 2003/12/14 21:46:35 e_gourgoulhon
* Added argument start_dx in visu_section.
*
* Revision 1.34 2003/12/11 16:19:38 e_gourgoulhon
* Added method visu_section for visualization with OpenDX.
*
* Revision 1.33 2003/12/11 14:48:47 p_grandclement
* Addition of ALL (and that is a lot !) the files needed for the general elliptic solver ... UNDER DEVELOPEMENT...
*
* Revision 1.32 2003/11/13 13:43:53 p_grandclement
* Addition of things needed for Bhole::update_metric (const Etoile_bin&, double, double)
*
* Revision 1.31 2003/11/06 14:43:37 e_gourgoulhon
* Gave a name to const arguments in certain method prototypes (e.g.
* constructors) to correct a bug of DOC++.
*
* Revision 1.30 2003/11/04 22:55:50 e_gourgoulhon
* Added new methods mult_cost(), mult_sint() and div_sint().
*
* Revision 1.29 2003/10/29 13:09:11 e_gourgoulhon
* -- Added integer argument to derivative functions dsdr, etc...
* so that one can choose the dzpuis of the result (default=2).
* -- Change of method name: laplacien --> laplacian.
*
* Revision 1.28 2003/10/29 11:00:42 e_gourgoulhon
* Virtual functions dec_dzpuis and inc_dzpuis have now an integer argument to
* specify by which amount dzpuis is to be increased.
* Accordingly virtual methods dec2_dzpuis and inc2_dzpuis have been suppressed.
*
* Revision 1.27 2003/10/20 14:26:02 j_novak
* New assignement operators.
*
* Revision 1.26 2003/10/19 19:46:33 e_gourgoulhon
* -- Method spectral_display now virtual (from Tensor), list of argument
* changed.
*
* Revision 1.25 2003/10/17 13:46:14 j_novak
* The argument is now between 1 and 3 (instead of 0->2)
*
* Revision 1.24 2003/10/16 15:23:41 e_gourgoulhon
* Name of method div_r_ced() changed to div_r_inc2().
* Name of method div_rsint_ced() changed to div_rsint_inc2().
*
* Revision 1.23 2003/10/15 21:10:11 e_gourgoulhon
* Added method poisson_angu().
*
* Revision 1.22 2003/10/15 16:03:35 j_novak
* Added the angular Laplace operator for Scalar.
*
* Revision 1.21 2003/10/15 10:29:05 e_gourgoulhon
* Added new members p_dsdt and p_stdsdp.
* Added new methods dsdt(), stdsdp() and div_tant().
*
* Revision 1.20 2003/10/13 13:52:39 j_novak
* Better managment of derived quantities.
*
* Revision 1.19 2003/10/10 15:57:27 j_novak
* Added the state one (ETATUN) to the class Scalar
*
* Revision 1.18 2003/10/08 14:24:08 j_novak
* replaced mult_r_zec with mult_r_ced
*
* Revision 1.17 2003/10/06 16:16:02 j_novak
* New constructor from a Tensor.
*
* Revision 1.16 2003/10/06 13:58:45 j_novak
* The memory management has been improved.
* Implementation of the covariant derivative with respect to the exact Tensor
* type.
*
* Revision 1.15 2003/10/05 21:06:31 e_gourgoulhon
* - Added new methods div_r_ced() and div_rsint_ced().
* - Added new virtual method std_spectral_base()
* - Removed method std_spectral_base_scal()
*
* Revision 1.14 2003/10/01 13:02:58 e_gourgoulhon
* Suppressed the constructor from Map* .
*
* Revision 1.13 2003/09/29 12:52:56 j_novak
* Methods for changing the triad are implemented.
*
* Revision 1.12 2003/09/25 09:33:36 j_novak
* Added methods for integral calculation and various manipulations
*
* Revision 1.11 2003/09/25 09:11:21 e_gourgoulhon
* Added functions for radial operations (divr, etc...)
*
* Revision 1.10 2003/09/25 08:55:23 e_gourgoulhon
* Added members raccord*.
*
* Revision 1.9 2003/09/25 08:50:11 j_novak
* Added the members import
*
* Revision 1.8 2003/09/25 08:13:51 j_novak
* Added method for calculating derivatives
*
* Revision 1.7 2003/09/25 07:59:26 e_gourgoulhon
* Added prototypes for PDE resolutions.
*
* Revision 1.6 2003/09/25 07:17:58 j_novak
* Method asymptot implemented.
*
* Revision 1.5 2003/09/24 20:53:38 e_gourgoulhon
* Added -- constructor by conversion from a Cmp
* -- assignment from Cmp
*
* Revision 1.4 2003/09/24 15:10:54 j_novak
* Suppression of the etat flag in class Tensor (still present in Scalar)
*
* Revision 1.3 2003/09/24 12:01:44 j_novak
* Added friend functions for math.
*
* Revision 1.2 2003/09/24 10:22:01 e_gourgoulhon
* still in progress...
*
* Revision 1.1 2003/09/22 12:50:47 e_gourgoulhon
* First version: not ready yet!
*
*
* $Header: /cvsroot/Lorene/C++/Include/scalar.h,v 1.94 2015/12/18 15:52:51 j_novak Exp $
*
*/
// Headers Lorene
#include "valeur.h"
#include "tensor.h"
namespace Lorene {
class Param ;
class Cmp ;
class Param_elliptic ;
/**
* Tensor field of valence 0 (or component of a tensorial field).
* \ingroup (tensor)
*
*
*/
class Scalar : public Tensor {
// Data :
// -----
protected:
/** The logical state \c ETATNONDEF (undefined), \c ETATZERO (null),
* \c ETATUN (one), or \c ETATQCQ (ordinary).
*/
int etat ;
/**
* Power of \e r by which the quantity represented by \c this
* must be divided in the compactified external domain (CED) in order
* to get the correct physical values
*/
int dzpuis ;
Valeur va ; ///< The numerical value of the \c Scalar
// Derived data :
// ------------
protected:
/// Pointer on \f$\partial/\partial r\f$ of \c *this (0x0 if not up to date)
mutable Scalar* p_dsdr ;
/** Pointer on \f$1/r \partial/\partial \theta\f$ of \c *this
* (0x0 if not up to date)
*/
mutable Scalar* p_srdsdt ;
/** Pointer on \f$1/(r\sin\theta) \partial/\partial \phi\f$ of \c *this
* (0x0 if not up to date)
*/
mutable Scalar* p_srstdsdp ;
/// Pointer on \f$\partial/\partial \theta\f$ of \c *this (0x0 if not up to date)
mutable Scalar* p_dsdt ;
/** Pointer on \f$1/\sin\theta \partial/\partial \phi\f$ of \c *this
* (0x0 if not up to date)
*/
mutable Scalar* p_stdsdp ;
/** Pointer on \f$\partial/\partial x\f$ of \c *this ,
* where \f$x=r\sin\theta \cos\phi\f$ (0x0 if not up to date)
*/
mutable Scalar* p_dsdx ;
/** Pointer on \f$\partial/\partial y\f$ of \c *this ,
* where \f$y=r\sin\theta \sin\phi\f$(0x0 if not up to date)
*/
mutable Scalar* p_dsdy ;
/** Pointer on \f$\partial/\partial z\f$ of \c *this ,
* where \f$z=r\cos\theta\f$ (0x0 if not up to date)
*/
mutable Scalar* p_dsdz ;
/** Pointer on the Laplacian of \c *this (0x0 if not up to date)
*/
mutable Scalar* p_lap ;
/** Pointer on the Laplacian of \c *this (0x0 if not up to date)
*/
mutable Scalar* p_lapang ;
/// Pointer on \f$\partial/\partial radial \f$ of \c *this
mutable Scalar* p_dsdradial ;
/// Pointer on \f$\partial/\partial \rho \f$ of \c *this
mutable Scalar* p_dsdrho ;
/** Power of \e r by which the last computed Laplacian has been
* multiplied in the compactified external domain.
*/
mutable int ind_lap ;
/** Pointer on the space integral of \c *this (values in each
* domain) (0x0 if not up to date)
*/
mutable Tbl* p_integ ;
// Constructors - Destructor
// -------------------------
public:
explicit Scalar(const Map& mpi) ; ///< Constructor from mapping
/// Constructor from a Tensor (must be of valence 0)
Scalar(const Tensor& a) ;
Scalar(const Scalar& a) ; ///< Copy constructor
Scalar(const Cmp& a) ; ///< Constructor by conversion of a Cmp
/// Constructor from a file (see \c sauve(FILE*) )
Scalar(const Map&, const Mg3d&, FILE* ) ;
virtual ~Scalar() ; ///< Destructor
// Memory management
// -----------------
protected:
void del_t() ; ///< Logical destructor
virtual void del_deriv() const; ///< Logical destructor of the derivatives
void set_der_0x0() const; ///< Sets the pointers for derivatives to 0x0
public:
/**
* Sets the logical state to \c ETATNONDEF (undefined).
* Calls the logical destructor of the \c Valeur \c va
* deallocates the memory occupied by all the derivatives.
*/
virtual void set_etat_nondef() ;
/**
* Sets the logical state to \c ETATZERO (zero).
* Calls the logical destructor of the \c Valeur \c va and
* deallocates the memory occupied by all the derivatives.
*/
virtual void set_etat_zero() ;
/**
* Sets the logical state to \c ETATQCQ (ordinary state).
* If the state is already \c ETATQCQ , this function does nothing.
* Otherwise, it calls the logical destructor of the \c Valeur \c va and
* deallocates the memory occupied by all the derivatives.
*/
virtual void set_etat_qcq() ;
/**
* Sets the logical state to \c ETATUN (one).
* Fills the \c Valeur \c va with ones and
* deallocates the memory occupied by all the derivatives.
*/
void set_etat_one() ;
/**
* Sets the logical state to \c ETATQCQ (ordinary state)
* and performs the memory allocation of all the
* elements, down to the \c double arrays of the \c Tbl s.
* This function performs in fact recursive calls to \c set_etat_qcq()
* on each element of the chain \c Scalar ->
* \c Valeur -> \c Mtbl -> \c Tbl .
*/
virtual void allocate_all() ;
/**
* Sets the \c Scalar to zero in a hard way.
* 1/ Sets the logical state to \c ETATQCQ , i.e. to an ordinary state.
* 2/ Fills the \c Valeur \c va with zeros.
* NB: this function must be used for debugging purposes only.
* For other operations, the functions \c set_etat_zero()
* or \c annule(int,int) must be perferred.
*/
void annule_hard() ;
// Extraction of information
// -------------------------
public:
/** Returns the logical state \c ETATNONDEF (undefined),
* \c ETATZERO (null) or \c ETATQCQ (ordinary).
*/
int get_etat() const {return etat;} ;
/// Returns \c dzpuis
int get_dzpuis() const {return dzpuis;} ;
/** Returns \c true if the last domain is compactified and
* \c *this is not zero in this domain
*/
bool dz_nonzero() const ;
/** Returns \c false if the last domain is compactified
* and \c *this is not zero in this domain and \c dzpuis
* is not equal to \c dzi , otherwise return true.
*/
bool check_dzpuis(int dzi) const ;
/** Looks for NaNs (not a number) in the scalar field.
* If at least one NaN is found, it returns \c true. If the flag \c verbose
* is set to \c true, it outputs to the standard output the indices where NaNs
* have been found.
*/
bool is_nan(bool verbose=false) const ;
// Assignment
// -----------
public:
/// Assignment to another \c Scalar defined on the same mapping
void operator=(const Scalar& a) ;
/// Assignment to a \c Tensor (of valence 0)
virtual void operator=(const Tensor& a) ;
void operator=(const Cmp& a) ; ///< Assignment to a \c Cmp
void operator=(const Valeur& a) ; ///< Assignment to a \c Valeur
void operator=(const Mtbl& a) ; ///< Assignment to a \c Mtbl
void operator=(double ) ; ///< Assignment to a \c double
void operator=(int ) ; ///< Assignment to an \c int
// Conversion oprators
//---------------------
operator Cmp() const ; ///< Conversion to a \c Cmp
// Access to individual elements
// -----------------------------
public:
/// Returns \c va (read only version)
const Valeur& get_spectral_va() const {return va;} ;
/// Returns \c va (read/write version)
Valeur& set_spectral_va() {return va;} ;
/** Read/write of the value in a given domain.
* This method should be used only to set the value in a given
* domain (it performs a call to \c del_deriv); for reading the
* value in a domain without changing it, the method \c domain(int )
* is preferable.
*
* @param l [input] domain index
* @return writable \c Tbl containing the value of the field in domain \c l .
*/
Tbl& set_domain(int l) {
assert(etat == ETATQCQ) ;
del_deriv() ;
return va.set(l) ;
};
/** Read-only of the value in a given domain.
* @param l [input] domain index
* @return \c Tbl containing the value of the field in domain \c l .
*/
const Tbl& domain(int l) const {
assert( (etat == ETATQCQ) || (etat == ETATUN) ) ;
return va(l) ;
};
/** Returns the value of the field at a specified grid point.
* @param l [input] domain index
* @param k [input] \f$\phi\f$ index
* @param j [input] \f$\theta\f$ index
* @param i [input] \e r (\f$\xi\f$) index
*/
double val_grid_point(int l, int k, int j, int i) const {
assert(etat != ETATNONDEF) ;
if (etat == ETATZERO) {
double zero = 0. ;
return zero ;
}
else {
if (etat == ETATUN) {
double one = 1. ;
return one ;
}
else{
return va(l, k, j, i) ;
}
}
};
/** Computes the value of the field at an
* arbitrary point \f$(r, \theta, \phi)\f$, by means of the spectral
* expansion.
* NB: if \f$(r, \theta, \phi)\f$ is a point of the spectral grid,
* the method \c val_grid_point is to be preferred,
* being much more efficient.
* @param r [input] value of the coordinate \e r
* @param theta [input] value of the coordinate \f$\theta\f$
* @param phi [input] value of the coordinate \f$\phi\f$
* @return value at the point \f$(r, \theta, \phi)\f$
* of the field represented by \c *this . NB: in the compactified
* external domain, the returned value is the actual value of the field,
* i.e. the stored value divided by \f$ r^{\rm dzpuis} \f$.
*/
double val_point(double r, double theta, double phi) const ;
/** Setting the value of the field at a given grid point.
* CAUTION: to gain in efficiency (especially when this method is
* invoqued inside a loop), the method \c del_deriv() (to delete
* the derived members) is not called by \c set_grid_point .
* It must thus be invoqued by the user, after all the calls
* to \c set_grid_point have been performed.
*
* @param l [input] domain index
* @param k [input] \f$\phi\f$ index
* @param j [input] \f$\theta\f$ index
* @param i [input] \e r (\f$\xi\f$) index
* @return writable value of the field at the specified grid point
*/
double& set_grid_point(int l, int k, int j, int i) {
assert(etat == ETATQCQ) ;
return va.set(l, k, j, i) ;
};
/**
* Sets the \c Scalar to zero in several domains.
* @param l_min [input] The \c Scalar will be set (logically) to zero
* in the domains whose indices are in the range
* \c [l_min,l_max] .
* @param l_max [input] see the comments for \c l_min .
*
* Note that \c annule(0,va.mg->get_nzone()-1) is equivalent to
* \c set_etat_zero() .
*/
virtual void annule(int l_min, int l_max) ;
/** Sets the value of the \c Scalar at the inner boundary of a given
* domain.
* @param l [input] domain index
* @param x [input] (constant) value at the inner boundary of domain no. \c l
*/
void set_inner_boundary(int l, double x) ;
/** Sets the value of the \c Scalar at the outer boundary of a given
* domain.
* @param l [input] domain index
* @param x [input] (constant) value at the outer boundary of domain no. \c l
*/
void set_outer_boundary(int l, double x) ;
/**
* Gives the spectrum in terms of multipolar modes \e l .
* @return a \c Tbl of size (nzone, lmax), where lmax is the
* maximal multipolar momentum over all domains. The \e l -th
* element contains the L1 norm of the \e l -th multipole
* (\e i.e. a sum over all \e m of the norms (coefficient space)
* of the component of a given \f$Y_l^m\f$.
*/
Tbl multipole_spectrum () const ;
/** Returns the \c Tbl containing the values of angular coefficients
* at the outer boundary.
* @param l_dom [input] domain index
* @param leave_ylm [input] flag to decide whether the coefficients
* are expressed in spherical harmonics or Fourier base
*/
Tbl tbl_out_bound(int l_dom, bool leave_ylm = false) ;
/** Returns the \c Tbl containing the values of angular coefficients
* at the inner boundary.
* @param l_dom [input] domain index
* @param leave_ylm [input] flag to decide whether the coefficients
* are expressed in spherical harmonics or Fourier base
*/
Tbl tbl_in_bound(int n, bool leave_ylm = false) ;
/** Returns the \c Scalar containing the values of angular coefficients
* at the outer boundary.
* @param l_dom [input] domain index
* @param leave_ylm [input] flag to decide whether the coefficients
* are expressed in spherical harmonics or Fourier base
*/
Scalar scalar_out_bound(int n, bool leave_ylm = false) ;
// Differential operators and others
// ---------------------------------
public:
/** Returns \f$\partial / \partial r\f$ of \c *this .
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdr() const ;
/** Returns \f$1/r \partial / \partial \theta\f$ of \c *this .
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& srdsdt() const ;
/** Returns \f$1/(r\sin\theta) \partial / \partial \phi\f$ of \c *this .
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& srstdsdp() const ;
/** Returns \f$\partial / \partial \theta\f$ of \c *this .
*/
const Scalar& dsdt() const ;
/** Returns \f$\partial / \partial r\f$ of \c *this if the mapping is
* affine (class \c Map_af) and \f$\partial / \partial \ln r\f$
* of \c *this if the mapping
* is logarithmic (class \c Map_log).
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdradial() const ;
/** Returns \f$\partial / \partial \rho \f$ of \c *this .
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdrho() const ;
/** Returns \f$1/\sin\theta \partial / \partial \phi\f$ of \c *this .
*/
const Scalar& stdsdp() const ;
/** Returns \f$\partial/\partial x\f$ of \c *this ,
* where \f$x=r\sin\theta \cos\phi\f$.
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdx() const ;
/** Returns \f$\partial/\partial y\f$ of \c *this ,
* where \f$y=r\sin\theta \sin\phi\f$.
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdy() const ;
/** Returns \f$\partial/\partial z\f$ of \c *this ,
* where \f$z=r\cos\theta\f$.
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
*/
const Scalar& dsdz() const ;
/** Returns \f$\partial/\partial x_i\f$ of \c *this ,
* where \f$x_i = (x, y, z)\f$.
* If \c dzpuis is zero, then the returned \c Scalar has
* \c dzpuis = 2. It is increased by 1 otherwise.
* @param i [input] i=1 for \e x , i=2 for \e y , i=3 for \e z .
*/
const Scalar& deriv(int i) const ;
/** Returns the gradient (1-form = covariant vector) of \c *this
* @param gam metric components only used to get the triad with
* respect to which the components of the result are defined
*/
const Vector& derive_cov(const Metric& gam) const ;
/** Returns the "contravariant" derivative of \c *this with respect
* to some metric \f$\gamma\f$, by raising the index of the
* gradient (cf. method \c derive_cov() ) with
* \f$\gamma\f$.
*/
const Vector& derive_con(const Metric& gam) const ;
/// Computes the derivative of \c this along a vector field \c v
Scalar derive_lie(const Vector& v) const ;
/** Returns the Laplacian of \c *this
* @param ced_mult_r [input] Determines the quantity computed in
* the compactified external domain (CED)
* (\e u in the field represented by \c *this ) :
* \li ced_mult_r = 0 : \f$\Delta u\f$
* \li ced_mult_r = 2 : \f$r^2 \, \Delta u\f$
* \li ced_mult_r = 4 (default) : \f$r^4 \, \Delta u\f$
*/
const Scalar& laplacian(int ced_mult_r = 4) const ;
/** Returns the angular Laplacian \f$\Delta_{\theta\varphi}\f$ of \c *this ,
* where \f$\Delta_{\theta\varphi} f = \frac{\partial^2 f}
* {\partial \theta^2} + \frac{1}{\tan \theta} \frac{\partial f}
* {\partial \theta} +\frac{1}{\sin^2 \theta}\frac{\partial^2 f}
* {\partial \varphi^2}\f$
*
*/
const Scalar& lapang() const ;
/// Division by \e r everywhere; \c dzpuis is not changed.
void div_r() ;
/** Division by \e r everywhere but with the output flag \c dzpuis
* set to \c ced_mult_r .
* @param ced_mult_r [input] value of \c dzpuis of the result.
*/
void div_r_dzpuis(int ced_mult_r) ;
/** Division by \e r in the compactified external domain (CED), the
* \c dzpuis flag is not changed.
*/
void div_r_ced() ;
/// Multiplication by \e r everywhere; \c dzpuis is not changed.
void mult_r() ;
/** Multiplication by \e r everywhere but with the output flag \c dzpuis
* set to \c ced_mult_r .
* @param ced_mult_r [input] value of \c dzpuis of the result.
*/
void mult_r_dzpuis(int ced_mult_r) ;
/** Multiplication by \e r in the compactified external domain (CED), the
* \c dzpuis flag is not changed.
*/
void mult_r_ced() ;
/// Multiplication by \f$r\sin\theta\f$ everywhere; \c dzpuis is not changed.
void mult_rsint() ;
/** Multiplication by \f$r\sin\theta\f$ but with the output flag \c dzpuis
* set to \c ced_mult_r .
* @param ced_mult_r [input] value of \c dzpuis of the result.
*/
void mult_rsint_dzpuis(int ced_mult_r) ;
/// Division by \f$r\sin\theta\f$ everywhere; \c dzpuis is not changed.
void div_rsint() ;
/** Division by \f$r\sin\theta\f$ but with the output flag \c dzpuis
* set to \c ced_mult_r .
* @param ced_mult_r [input] value of \c dzpuis of the result.
*/
void div_rsint_dzpuis(int ced_mult_r) ;
void mult_cost() ; ///< Multiplication by \f$\cos\theta\f$
void div_cost() ; ///< Division by \f$\cos\theta\f$
void mult_sint() ; ///< Multiplication by \f$\sin\theta\f$
void div_sint() ; ///< Division by \f$\sin\theta\f$
void div_tant() ; ///< Division by \f$\tan\theta\f$
/** Computes the radial primitive which vanishes for \f$r\to \infty\f$.
* i.e. the function
* \f$ F(r,\theta,\varphi) = \int_r^\infty f(r',\theta,\varphi) \, dr' \f$
* where \e f is the function represented by \c *this
* (and must have a \c dzpuis = 2).
* @param null_infty if true (default), the primitive is null
* at infinity (or on the grid boundary). \e F is null at the
* center otherwise
* @return function \e F
*/
Scalar primr(bool null_infty = true) const ;
/** Computes the integral over all space of \c *this .
* The computed quantity is (\e u being the field represented by
* \c *this )
* \f$\int u \, r^2 \sin\theta \, dr\, d\theta \, d\phi\f$.
* Note that in the compactified external domain (CED), \c dzpuis
* must be 4 for the computation to take place.
*/
double integrale() const ;
/** Computes the integral in each domain of \c *this .
* The computed quantity is (\e u being the field represented by
* \c *this )
* \f$\int u \, r^2 \sin\theta \, dr\, d\theta \, d\phi\f$
* in each domain. The result is returned a \c Tbl on the
* various domains.
* Note that in the compactified external domain (CED), \c dzpuis
* must be 4 for the computation to take place.
*/
const Tbl& integrale_domains() const ;
/** Decreases by \c dec units the value of \c dzpuis and
* changes accordingly the values of the \c Scalar in the
* compactified external domain (CED).
*/
virtual void dec_dzpuis(int dec = 1) ;
/** Increases by \c inc units the value of \c dzpuis and
* changes accordingly the values of the \c Scalar in the
* compactified external domain (CED).
*/
virtual void inc_dzpuis(int inc = 1) ;
/** Sets a new vectorial basis (triad) of decomposition and modifies
* the components accordingly.
*/
virtual void change_triad(const Base_vect& new_triad) ;
/**
* Sets the \c n lasts coefficients in \e r to 0 in the
* external domain.
*/
void filtre (int n) ;
/**
* Sets the \c n lasts coefficients in \e r to 0 in all
* domains.
*/
void filtre_r (int* nn) ;
/**
* Sets the \c n last coefficients in \e r to 0 in the domain \c nzone .
*/
void filtre_r (int n, int nzone) ;
/**
* Applies an exponential filter to the spectral coefficients in the radial direction.
* The filter is of the type: \f$ \forall n\leq N,\, b_n = \sigma(n/N ) a_n\f$, with
* \f$ \sigma(x) = \exp\left( -\ln (10^\alpha ) x^{2p} \right) \f$ and \e N the number
* of radial coefficients.
* @param lzmin, lzmax [input] the indices of the domain where the filter is applied
* (in [\c lzmin , \c lzmax ])
* @param p [input] the order of the filter
* @param alpha [input] \f$\alpha\f$ appearing in the above formula.
*/
// virtual void exponential_filter_r(int lzmin, int lzmax, int p,
// double alpha= -16.) ;
virtual void exponential_filter_r(int lzmin, int lzmax, int p,
double alpha= -16.) ;
/**
* Applies an exponential filter to the spectral coefficients in the radial direction.
* The filter is of the type: \f$ \forall n\leq N,\, b_n = \sigma(n/N ) a_n\f$, with
* \f$ \sigma(x) = \exp\left( \alpha x^{p} \right) \f$ and \e N the number
* of radial coefficients.
* @param lzmin, lzmax [input] the indices of the domain where the filter is applied
* (in [\c lzmin , \c lzmax ])
* @param p [input] the order of the filter
* @param alpha [input] \f$\alpha\f$ appearing in the above formula.
*/
void sarra_filter_r(int lzmin, int lzmax, double p,
double alpha= -1E-16) ;
/**
* Applies an exponential filter in radial direction in all domains.
* (see \c Scalar:exponential_filter_r ). Note that this may cause
* regularity problems at the origin if applied in a nucleus.
*/
void exp_filter_r_all_domains(Scalar &ss, int p, double alpha=-16.) ;
/**
* Applies an exponential filter in radial direction in all domains
* for the case where p is a double
* (see \c Scalar:sarra_filter_r ). Note that this may cause
* regularity problems at the origin if applied in a nucleus.
*/
void sarra_filter_r_all_domains(double p, double alpha=1E-16) ;
/**
* Applies an exponential filter to the spectral coefficients in the angular directions.
* The filter is of the type:
* \f$ \forall \ell \leq \ell_{\rm max},\, \forall m,\, b_{\ell m} = \sigma(\ell/\ell_{\rm max} ) a_{\ell m}\f$, with
* \f$ \sigma(x) \f$ defined for \c Scalar::exponential_filter_r and
* \f$\ell_{\rm max}\f$ the number of spherical harmonics used.
*/
virtual void exponential_filter_ylm(int lzmin, int lzmax, int p,
double alpha= -16.) ;
/**
* Sets all the multipolar components between \c l_min and \c l_max
* to zero. This is done for [ \c l_min , \c l_max ] and all relevant
* \c m in the spherical harmonics expansion basis. If \c ylm_output
* is set to \c true , then the spectral expansion basis of \c this is
* left to be that of spherical harmonics.
*/
void annule_l (int l_min, int l_max, bool ylm_output= false ) ;
/**
* Sets the \c n lasts coefficients in \f$\Phi\f$ to 0 in the
* domain \c zone .
*/
void filtre_phi (int n, int zone) ;
/**
* Sets the \c n lasts coefficients in \f$\theta\f$ to 0 in the
* domain \c nz1 to \c nz2 when expressed in spherical harmonics.
*/
void filtre_tp(int nn, int nz1, int nz2) ;
/**
* Substracts all the components behaving like \f$r^{-n}\f$ in the external
* domain, with \e n strictly lower than \c puis , so that \c *this
* decreases at least like \f$r^{\tt puis} \f$ at infinity.
*/
void fixe_decroissance (int puis) ;
/** Performs a \f$C^k\f$ matching of the last non-compactified shell with
* a decaying function \f$\sum_{j=0}^k {\alpha_j \over r^{\ell+n+j}}\f$ where
* \f$\ell\f$ is the spherical harmonic index and \e n is some
* specifiable parameter.
*/
void smooth_decay(int k, int n) ;
/**
* Performs the \f$C^n\f$ matching of the nucleus with respect to the
* first shell.
*/
void raccord(int n) ;
/**
* Performs the \f$C^1\f$ matching of the external domain with respect to
* the last shell using function like \f$\frac{1}{r^i}\f$ with
* \f${\tt puis} \leq i \leq {\tt puis+nbre}\f$ for each spherical harmonics
* with \f$l \leq {\tt lmax}\f$.
*/
void raccord_c1_zec(int puis, int nbre, int lmax) ;
/**
* Matching of the external domain with the outermost shell
*/
void raccord_externe(int puis, int nbre, int lmax) ;
/**
* Method for matching accross domains and imposing boundary condition.
* Matching of the field represented by \c this accross domains and imposition of the
* boundary condition using the tau method.
* @param par_bc [input] \c Param to control the boundary conditions
* par_bc must contain (at a minimum)
* a modifiable Tbl which specifies a physical boundary
* two integers, one specifying the domain that has the boundary
* the other specifying the number of conditions
* 1 -> Dirichlet
* 2 -> Robin (which may reduce to von Neumann, see below)
* two doubles, specifying the Robin BC parameters. If the first is zero, we see that
* Robin will reduce to von Neumann
* @param par_mat [input/output] optional \c Param in which the matching matrices are
* stored (together with their LU decomposition).
*/
void match_tau(Param& par_bc, Param* par_mat=0x0) ;
/**
* Method for matching accross domains and imposing boundary condition.
* Matching of the field represented by \c this accross domains and imposition of the
* boundary condition using the tau method.
* @param par_bc [input] \c Param to control the boundary conditions
* @param par_mat [input/output] optional \c Param in which the matching matrices are
* stored (together with their LU decomposition).
*/
void match_tau_shell(Param& par_bc, Param* par_mat=0x0) ;
/**
* Method for matching accross domains and imposing boundary condition.
* Matching of the field represented by \c this accross domains and imposition of the
* boundary condition using the collocation method.
* @param par_bc [input] \c Param to control the boundary conditions
* @param par_mat [input/output] optional \c Param in which the matching matrices are
* stored (together with their LU decomposition).
*/
void match_collocation(Param& par_bc, Param* par_mat=0x0) ;
// Outputs
// -------
public:
virtual void sauve(FILE *) const ; ///< Save in a file
/** Displays the spectral coefficients and the associated
* basis functions. This function shows only the values greater than a
* given threshold.
* @param comment comment to be printed at top of the display
* (default: 0x0 = nothing printed)
* @param threshold [input] Value above which a coefficient is printed
* (default: 1.e-7)
* @param precision [input] Number of printed digits (default: 4)
* @param ostr [input] Output stream used for the printing (default: cout)
*/
virtual void spectral_display(const char* comment = 0x0,
double threshold = 1.e-7, int precision = 4,
ostream& ostr = cout) const ;
/// Display
friend ostream& operator<<(ostream& , const Scalar & ) ;
/** 3D visualization via a plane section.
* Prepares files for visualization by OpenDX of the values of the field in
* a plane x=const, y=const or z=const
*
* @param section_type [input] defines the type of section :
* \li 'x' for a plane x = a with a = const (parameter \c aa )
* \li 'y' for a plane y = a with a = const (parameter \c aa )
* \li 'z' for a plane z = a with a = const (parameter \c aa )
* @param aa [input] constant a defining the section plane
* @param umin [input] defines with \c umax the range of the plane coordinate u
* @param umax [input] defines with \c umin the range of the plane coordinate u
* @param vmin [input] defines with \c vmax the range of the plane coordinate v
* @param vmax [input] defines with \c vmin the range of the plane coordinate v
* @param title [input] title for the graph (for OpenDX legend)
* @param filename [input] name for the file which will be the input for
* OpenDX; the default 0x0 is transformed into "scalar_section"
* @param start_dx [input] determines whether OpenDX must be launched (as a
* subprocess) to view the field; if set to \c false , only input files
* for future usage of OpenDX are created
* @param nu [input] number of points in the u direction (uniform sampling)
* @param nv [input] number of points in the v direction (uniform sampling)
*
*/
void visu_section(const char section_type, double aa, double umin, double umax, double vmin,
double vmax, const char* title = 0x0, const char* filename = 0x0,
bool start_dx = true, int nu = 200, int nv = 200) const ;
/** 3D visualization via a plane section.
* Prepares files for visualization by OpenDX of the values of the field in
* any given plane.
*
* @param plane [input] : 2D \c Tbl defining the section plane: \c plane
* must of dimension 3x3 with the following content:
* \li \c plane(0,i) : absolute Cartesian coordinates (xa0,ya0,za0) of some
* point in the plane considered as the origin for the plane coordinates
* (u,v): \c plane(0,0) = xa0 , \c plane(0,1) = ya0 ,
* \li \c plane(0,2) = za0
* \li \c plane(1,i) : components w.r.t. absolute Cartesian coordinates
* of the u-coordinate unit vector in the section plane
* \li \c plane(2,i) : components w.r.t. absolute Cartesian coordinates
* of the v-coordinate unit vector in the section plane
* @param umin [input] defines with \c umax the range of the plane coordinate u
* @param umax [input] defines with \c umin the range of the plane coordinate u
* @param vmin [input] defines with \c vmax the range of the plane coordinate v
* @param vmax [input] defines with \c vmin the range of the plane coordinate v
* @param title [input] title for the graph (for OpenDX legend)
* @param filename [input] name for the file which will be the input for
* OpenDX; the default 0x0 is transformed into "scalar_section"
* @param start_dx [input] determines whether OpenDX must be launched (as a
* subprocess) to view the field; if set to \c false , only input files
* for future usage of OpenDX are created
* @param nu [input] number of points in the u direction (uniform sampling)
* @param nv [input] number of points in the v direction (uniform sampling)
*
*/
void visu_section(const Tbl& plane, double umin, double umax, double vmin,
double vmax, const char* title = 0x0, const char* filename = 0x0,
bool start_dx = true, int nu = 200, int nv = 200) const ;
/** 3D visualization via time evolving plane section (animation).
* Prepares files for visualization by OpenDX of the values of the field in
* a plane x=const, y=const or z=const at successive time steps
*
* @param section_type [input] defines the type of section :
* \li 'x' for a plane x = a with a = const (parameter \c aa )
* \li 'y' for a plane y = a with a = const (parameter \c aa )
* \li 'z' for a plane z = a with a = const (parameter \c aa )
* @param aa [input] constant a defining the section plane
* @param umin [input] defines with \c umax the range of the plane coordinate u
* @param umax [input] defines with \c umin the range of the plane coordinate u
* @param vmin [input] defines with \c vmax the range of the plane coordinate v
* @param vmax [input] defines with \c vmin the range of the plane coordinate v
* @param jtime [input] time step label
* @param ttime [input] time t corresponding to \c jtime
* @param jgraph [input] number of time steps between two graphs: the graph
* will be generated only if \c jtime is a multiple of \c jgraph
* @param title [input] title for the graph (for OpenDX legend)
* @param filename_root [input] beginning of the names for the files which will
* be the input for OpenDX (the end of names will be automatically generated
* from the time steps); the default 0x0 is transformed into "anim"
* @param start_dx [input] determines whether OpenDX must be launched (as a
* subprocess) to view the field; if set to \c false , only input files
* for future usage of OpenDX are created
* @param nu [input] number of points in the u direction (uniform sampling)
* @param nv [input] number of points in the v direction (uniform sampling)
*
*/
void visu_section_anim(const char section_type, double aa, double umin,
double umax, double vmin, double vmax, int jtime, double ttime,
int jgraph = 1, const char* title = 0x0, const char* filename_root = 0x0,
bool start_dx = false, int nu = 200, int nv = 200) const ;
/** 3D visualization (volume rendering) via OpenDX.
* Prepares files for visualization by OpenDX of the values of the field in
* some rectangular box.
*
* @param xmin [input] defines with \c xmax the x range of the visualization box
* @param xmax [input] defines with \c xmin the x range of the visualization box
* @param ymin [input] defines with \c ymax the y range of the visualization box
* @param ymax [input] defines with \c ymin the y range of the visualization box
* @param zmin [input] defines with \c zmax the z range of the visualization box
* @param zmax [input] defines with \c zmin the z range of the visualization box
* @param title [input] title for the graph (for OpenDX legend)
* @param filename [input] name for the file which will be the input for
* OpenDX; the default 0x0 is transformed into "scalar_box"
* @param start_dx [input] determines whether OpenDX must be launched (as a
* subprocess) to view the field; if set to \c false , only input files
* for future usage of OpenDX are created
* @param nx [input] number of points in the x direction (uniform sampling)
* @param ny [input] number of points in the y direction (uniform sampling)
* @param nz [input] number of points in the z direction (uniform sampling)
*
*/
void visu_box(double xmin, double xmax, double ymin, double ymax,
double zmin, double zmax, const char* title0 = 0x0,
const char* filename0 = 0x0, bool start_dx = true, int nx = 40, int ny = 40,
int nz = 40) const ;
// Member arithmetics
// ------------------
public:
void operator+=(const Scalar &) ; ///< += Scalar
void operator-=(const Scalar &) ; ///< -= Scalar
void operator*=(const Scalar &) ; ///< *= Scalar
// Manipulation of spectral bases
// ------------------------------
/** Sets the spectral bases of the \c Valeur \c va to the standard ones
* for a scalar field
*/
virtual void std_spectral_base() ;
/** Sets the spectral bases of the \c Valeur \c va to the standard odd ones
* for a scalar field
*/
virtual void std_spectral_base_odd() ;
/** Sets the spectral bases of the \c Valeur \c va
*/
void set_spectral_base(const Base_val& ) ;
/// Returns the spectral bases of the \c Valeur \c va
const Base_val& get_spectral_base( ) const {return va.base ;} ;
/** Modifies the \c dzpuis flag.
* NB: this method does not change the field values stored in
* the compactified external domain (use methods \c dec_dzpuis() ,
* etc... for this purpose).
*/
void set_dzpuis(int ) ;
/** Asymptotic expansion at r = infinity.
*
* Determines the coefficients \f$a_k(\theta, \phi)\f$ of the expansion
* \f[
* \sum_{k=0}^n {a_k(\theta, \phi) \over r^k}
* \f]
* of \c *this when \f$r \rightarrow \infty\f$.
*
* @param n order of the expansion
* @param flag : output
* @return Array of \c n +1 \c Valeur s on \c mg->angu
* describing the coefficients \f$a_k(\theta, \phi)\f$.
* This array is allocated by the routine.
*
*/
Valeur** asymptot(int n, const int flag = 0) const ;
// PDE resolution
// --------------
public:
/** Solves the scalar Poisson equation with \c *this as a source.
* The source \f$\sigma\f$ of the equation \f$\Delta u = \sigma\f$ is
* represented by the \c Scalar \c *this .
* Note that \c dzpuis must be equal to 2 or 4, i.e. that the
* quantity stored in \c *this is in fact \f$r^2 \sigma\f$ or
* \f$r^4 \sigma\f$ in the compactified external domain.
* The solution \e u with the boundary condition \e u =0 at spatial
* infinity is the returned \c Scalar.
*/
Scalar poisson() const ;
/** Solves the scalar Poisson equation with \c *this as a source
* (version with parameters to control the resolution).
* The source \f$\sigma\f$ of the equation \f$\Delta u = \sigma\f$ is
* represented by the \c Scalar \c *this .
* Note that \c dzpuis must be equal to 2 or 4, i.e. that the
* quantity stored in \c *this is in fact \f$r^2 \sigma\f$ or
* \f$r^4 \sigma\f$ in the compactified external domain.
* @param par [input/output] possible parameters
* @param uu [input/output] solution \e u with the boundary condition
* \e u =0 at spatial infinity.
*/
void poisson(Param& par, Scalar& uu) const ;
/** Solves the scalar Poisson equation with \c *this as a source using a real Tau method
* The source \f$\sigma\f$ of the equation \f$\Delta u = \sigma\f$ is
* represented by the \c Scalar \c *this .
* Note that \c dzpuis must be equal to 2, 3 or 4, i.e. that the
* quantity stored in \c *this is in fact \f$r^2 \sigma\f$, \f$r^3 \sigma\f$ or
* \f$r^4 \sigma\f$ in the compactified external domain.
* The solution \e u with the boundary condition \e u =0 at spatial
* infinity is the returned \c Scalar.
*/
Scalar poisson_tau() const ;
/** Solves the scalar Poisson equation with \c *this as a source using a real Tau method
* (version with parameters to control the resolution)
* The source \f$\sigma\f$ of the equation \f$\Delta u = \sigma\f$ is
* represented by the \c Scalar \c *this .
* Note that \c dzpuis must be equal to 2, 3 or 4, i.e. that the
* quantity stored in \c *this is in fact \f$r^2 \sigma\f$, \f$r^3 \sigma\f$ or
* \f$r^4 \sigma\f$ in the compactified external domain.
* The solution \e u with the boundary condition \e u =0 at spatial
* infinity is the returned \c Scalar.
*/
void poisson_tau(Param& par, Scalar& uu) const ;
/**
* Is identicall to \c Scalar::poisson() . The regularity condition at the
* origin is replace by a boundary condition of the Dirichlet type.
*
* @param limite [input] : angular function. The boundary condition is
* given by \c limite[num] .
* @param num [input] : index of the boudary at which the condition is to
* be fullfilled.
*
* More precisely we impose the solution is equal to \c limite[num] at the
* boundary between the domains \c num and \c num+1 (the latter one being
* a shell).
*
*/
Scalar poisson_dirichlet (const Valeur& limite, int num) const ;
/**
* Idem as \c Scalar::poisson_dirichlet , the boundary condition being on
* the radial derivative of the solution.
*/
Scalar poisson_neumann (const Valeur&, int) const ;
/**
* Is identicall to \c Scalar::poisson() . The regularity condition at the
* origin is replace by a mixed boundary condition (Dirichlet + Neumann).
*
* @param limite [input] : angular function. The boundary condition is
* given by \c limite[num] .
* @param num [input] : index of the boudary at which the condition is to
* be fullfilled.
* @param fact_dir [input] : double in front of \f$\Psi\f$ (if \f$\Psi\f$
* is the variable solved).
* @param fact_neu [input] : double in front of the radial derivative
* of \f$\Psi\f$.
*
* More precisely we impose \f$ fact\_dir.\Psi + fact\_neu.\frac{\partial
* \Phi}{\partial r}\f$ is equal to the source at the
* boundary between the domains \c num and \c num+1 (the latter one being
* a shell).
*/
Scalar poisson_dir_neu (const Valeur& limite , int num,
double fact_dir, double fact_neu) const ;
/**
* Idem as \c Scalar::poisson_dirichlet , the boundary condition being on
* both the function and its radial derivative. The boundary condition
* at infinity is relaxed.
*/
Scalar poisson_frontiere_double (const Valeur&, const Valeur&, int) const ;
/** Solves the scalar Poisson equation with \c *this as a source
* (version with parameters to control the resolution).
* The source \f$\sigma\f$ of the equation \f$\Delta u = \sigma\f$ is
* represented by the \c Scalar \c *this .
* The regularized source
* \f$\sigma_{\rm regu} = \sigma - \sigma_{\rm div}\f$
* is constructed and solved.
* Note that \c dzpuis must be equal to 2 or 4, i.e. that the
* quantity stored in \c *this is in fact \f$r^2 \sigma\f$ or
* \f$r^4 \sigma\f$ in the compactified external domain.
* @param k_div [input] regularization degree of the procedure
* @param nzet [input] number of domains covering the star
* @param unsgam1 [input] parameter \f$1/(\gamma-1)\f$ where \f$\gamma\f$
* denotes the adiabatic index
* @param par [input/output] possible parameters
@param uu [input/output] solution
* @param uu_regu [output] solution of the regular part of
* the source.
* @param uu_div [output] solution of the diverging part of
* the source.
* @param duu_div [output] derivative of the diverging potential.
* @param source_regu [output] regularized source
* @param source_div [output] diverging part of the source
*/
void poisson_regular(int k_div, int nzet, double unsgam1, Param& par,
Scalar& uu, Scalar& uu_regu, Scalar& uu_div,
Tensor& duu_div,
Scalar& source_regu, Scalar& source_div) const ;
/** Checks if a Poisson equation with \c *this as a source
* has been correctly solved.
*
* @param uu [input] Solution \e u of the Poisson equation
* \f$\Delta u = \sigma\f$, \f$\sigma\f$ being
* represented by the \c Scalar \c *this .
*
* @param ostr [input/output] Output stream used for displaying
* \c err .
*
* @param detail [input] \li if \c true displays \c err(0,*) ,
* \c err(1,*) and \c err(2,*)
* \li if \c false (default), displays only
* the relative error \c err(0,*) .
*
* @return 2-D \c Tbl \c err decribing the errors in each
* domain:
* \li \c err(0,l) : Relative error in domain no. \c l ,
* defined as the maximum value of
* \f$|\Delta u - \sigma|\f$ in that domain divided by \e M ,
* where \e M is the maximum value of \f$|\sigma|\f$
* over all domains if \c dzpuis = 0 or \f$\sigma\f$ is
* zero in the compactified external domain (CED). If
* \c dzpuis != 0 and \f$\sigma\f$ does not vanish in the
* CED, the value of \e M used in the
* non-compactified domains is the maximum value over
* these domains, whereas the value of \e M used in the
* compactified external domain is the maximum value
* on that particular domain.
* \li \c err(1,l) : Maximum value of the absolute error
* \f$|\Delta u - \sigma|\f$ in domain no. \c l
* \li \c err(2,l) : Maximum value of \f$|\sigma|\f$ in domain
* no. \c l
*/
Tbl test_poisson(const Scalar& uu, ostream& ostr,
bool detail = false) const ;
/** Solves the (generalized) angular Poisson equation with \c *this
* as source.
* The generalized angular Poisson equation is
* \f$\Delta_{\theta\varphi} u + \lambda u = \sigma\f$,
* where \f$\Delta_{\theta\varphi} u := \frac{\partial^2 u}
* {\partial \theta^2} + \frac{1}{\tan \theta} \frac{\partial u}
* {\partial \theta} +\frac{1}{\sin^2 \theta}\frac{\partial^2 u}
* {\partial \varphi^2}\f$.
*
* @param lambda [input] coefficient \f$\lambda\f$ in the above equation
* (default value = 0)
* @return solution \e u .
*/
Scalar poisson_angu(double lambda =0) const ;
/** Performs one time-step integration (from \f$t=J \to J+1\f$) of the
* scalar d'Alembert equation with \c *this being the value of
* the function \e f at time \e J .
*
* Works only with an affine mapping (class \c Map_af ) and,
* if the last domain is a compactified one, it simply copies
* the value of the field in this last domain at the time-step \e J
* to the last domain of the returned solution.
* @param par [input/output] possible parameters to control the
* resolution of the d'Alembert equation:
* \li \c par.get_double(0) : [input] the time step \e dt ,
* \li \c par.get_int(0) : [input] the type of boundary conditions
* set at the outer boundary (0 : reflexion, 1 : Sommerfeld
* outgoing wave, valid only for \e l=0 components, 2 : Bayliss
* \& Turkel outgoing wave, valid for \e l=0, 1, 2 components)
* \li \c par.get_int_mod(0) : [input/output] set to 0 at first
* call, is used as a working flag after (must not be modified after
* first call)
* \li \c par.get_tensor_mod(0) : [input] (optional) if the wave
* equation is on a curved space-time, this is the potential in front
* of the Laplace operator. It has to be a Scalar and updated at
* every time-step (for a potential depending on time).
* Note: there are many other working objects attached to this
* \c Param , so one should not modify it.
* There should be exactly one \c Param for each wave equation to be
* solved.
* @param fJm1 [input] solution \f$f^{J-1}\f$ at time \e J-1
* @param source [input] source \f$\sigma\f$ of the d'Alembert equation
* \f$\diamond u = \sigma\f$.
* @return solution \f$f^{J+1}\f$ at time \e J+1
* with boundary conditions defined by \c par.get_int(0) .
*/
Scalar avance_dalembert(Param& par, const Scalar& fJm1, const Scalar& source)
const ;
/**
* Resolution of a general elliptic equation, putting zero at infinity.
* @param params [input] the operators and variables to be used.
**/
Scalar sol_elliptic(Param_elliptic& params) const ;
/**
* Resolution of a general elliptic equation, putting zero at infinity
* and with inner boundary conditions.
* @param params [input] the operators and variables to be used.
* @param bound [input] : the boundary condition
* @param fact_dir : 1 Dirchlet condition, 0 Neumann condition
* @param fact_neu : 0 Dirchlet condition, 1 Neumann condition
**/
Scalar sol_elliptic_boundary(Param_elliptic& params, const Mtbl_cf& bound,
double fact_dir, double fact_neu) const ;
/** Resolution of general elliptic equation, with inner boundary conditions as Scalars
* on mono-domain angulare grids
**/
Scalar sol_elliptic_boundary(Param_elliptic& params, const Scalar& bound,
double fact_dir, double fact_neu) const ;
/** Solves the scalar 2-dimensional elliptic equation
* with \c *this as a source.
* Note that \c dzpuis must be equal to 2, 3 or 4, i.e.
* The solution \e u with the boundary condition \e u =0 at spatial
* infinity is the returned \c Scalar.
*/
Scalar sol_elliptic_2d(Param_elliptic&) const ;
/** Solves a pseudo-1d elliptic equation
* with \c *this as a source.
*
*/
Scalar sol_elliptic_pseudo_1d(Param_elliptic&) const ;
/**
* Resolution of a general elliptic equation, putting a given value
* at the outermost
* shell and not solving in the compactified domain.
* @param params [input] the operators and variables to be used.
* @param val [input] value at the last shell.
**/
Scalar sol_elliptic_no_zec(Param_elliptic& params, double val = 0) const ;
/**
* Resolution of a general elliptic equation solving in the
* compactified domain and putting a given value at the inner boundary.
* @param params [input] the operators and variables to be used.
* @param val [input] value at the inner boundary of the external domain.
**/
Scalar sol_elliptic_only_zec(Param_elliptic& params, double val) const ;
/**
* General elliptic solver.
* The equation is not solved in the compactified domain and the
* matching is done with an homogeneous solution.
* @param params [input] the operators and variables to be used.
* @param coef [output] : coefficient of the oscillatory solution
* in the external domain.
* @param phases [output] : phases (i.e. choice of the homogeneous solution to match with).
**/
Scalar sol_elliptic_sin_zec(Param_elliptic& params, double* coefs, double* phases) const ;
/**
* Resolution of a general elliptic equation fixing the dericative at
* the origin and relaxing one continuity condition.
*
* @param val [input] value of the derivative.
* @param params [input] the operators and variables to be used.
**/
Scalar sol_elliptic_fixe_der_zero(double val,
Param_elliptic& params) const ;
/**
* Resolution of a divergence-like equation.
* The equation solved reads: \f$ \frac{\partial \phi}{\partial r} + \frac{n}{r}
* \phi = \sigma\f$, with \f$\phi\f$ the unknown and \f$\sigma\f$ the source
* represented by \c this.
*
* @param n [input] the coefficient in front of the 1/r term.
*
* @return the solution to the equation.
*/
Scalar sol_divergence(int n) const ;
// Import from other mapping
// -------------------------
/** Assignment to another \c Scalar defined on a different mapping.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param ci [input] \c Scalar to be imported.
*/
void import(const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping.
* Case where the \c Scalar is symmetric with respect to the plane y=0.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param ci [input] \c Scalar to be imported.
*/
void import_symy(const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping.
* Case where the \c Scalar is antisymmetric with respect to the
* plane y=0.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param ci [input] \c Scalar to be imported.
*/
void import_asymy(const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping.
* Case where the \c Scalar is symmetric with respect to the plane y=0.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_symy(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping.
* Case where the \c Scalar is antisymmetric with respect to the
* plane y=0.
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_asymy(int nzet, const Scalar& ci) ;
protected:
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings do not have a particular relative orientation.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_gal(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have aligned Cartesian axis.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_align(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have anti-aligned Cartesian axis (i.e.
* \f$x_1 = - x_2\f$, \f$y_1 = - y_2\f$, \f$z_1 = z_2\f$).
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_anti(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have aligned Cartesian axis.
* Case where the \c Scalar is symmetric with respect to the plane y=0.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_align_symy(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have anti-aligned Cartesian axis (i.e.
* \f$x_1 = - x_2\f$, \f$y_1 = - y_2\f$, \f$z_1 = z_2\f$).
* Case where the \c Scalar is symmetric with respect to the plane y=0.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_anti_symy(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have aligned Cartesian axis.
* Case where the \c Scalar is antisymmetric with respect to the
* plane y=0.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_align_asymy(int nzet, const Scalar& ci) ;
/** Assignment to another \c Scalar defined on a different mapping,
* when the two mappings have anti-aligned Cartesian axis (i.e.
* \f$x_1 = - x_2\f$, \f$y_1 = - y_2\f$, \f$z_1 = z_2\f$).
* Case where the \c Scalar is antisymmetric with respect to the
* plane y=0.
*
* This assignment is performed point to point by means of the
* spectral expansion of the original \c Scalar.
* @param nzet [input] Number of domains of the destination
* mapping (i.e. \c this->mp ) where the
* importation is performed: the assignment
* is done for the domains whose indices are
* between 0 and \c nzet-1 . In the other
* domains, \c *this is set to zero.
* @param ci [input] \c Scalar to be imported.
*/
void import_anti_asymy(int nzet, const Scalar& ci) ;
friend Scalar operator-(const Scalar& ) ;
friend Scalar operator+(const Scalar&, const Scalar &) ;
friend Scalar operator+(const Scalar&, const Mtbl&) ;
friend Scalar operator+(const Scalar&, double ) ;
friend Scalar operator-(const Scalar &, const Scalar &) ;
friend Scalar operator-(const Scalar&, const Mtbl&) ;
friend Scalar operator-(const Scalar&, double ) ;
friend Scalar operator*(const Scalar &, const Scalar &) ;
friend Scalar operator%(const Scalar &, const Scalar &) ;
friend Scalar operator|(const Scalar &, const Scalar &) ;
friend Scalar operator*(const Mtbl&, const Scalar &) ;
friend Scalar operator*(double, const Scalar &) ;
friend Scalar operator/(const Scalar &, const Scalar &) ;
friend Scalar operator/(const Scalar &, const Mtbl &) ;
friend Scalar operator/(const Mtbl &, const Scalar &) ;
friend Scalar operator/(const Scalar&, double ) ;
friend Scalar operator/(double, const Scalar &) ;
friend Scalar sin(const Scalar& ) ;
friend Scalar cos(const Scalar& ) ;
friend Scalar tan(const Scalar& ) ;
friend Scalar asin(const Scalar& ) ;
friend Scalar acos(const Scalar& ) ;
friend Scalar atan(const Scalar& ) ;
friend Scalar exp(const Scalar& ) ;
friend Scalar Heaviside(const Scalar& ) ;
friend Scalar log(const Scalar& ) ;
friend Scalar log10(const Scalar& ) ;
friend Scalar sqrt(const Scalar& ) ;
friend Scalar racine_cubique (const Scalar& ) ;
friend Scalar pow(const Scalar& , int ) ;
friend Scalar pow(const Scalar& , double ) ;
friend Scalar abs(const Scalar& ) ;
friend double totalmax(const Scalar& ) ;
friend double totalmin(const Scalar& ) ;
friend Tbl max(const Scalar& ) ;
friend Tbl min(const Scalar& ) ;
friend Tbl norme(const Scalar& ) ;
friend Tbl diffrel(const Scalar& a, const Scalar& b) ;
friend Tbl diffrelmax(const Scalar& a, const Scalar& b) ;
};
ostream& operator<<(ostream& , const Scalar & ) ;
// Prototypage de l'arithmetique
/**
* \defgroup sal_mat Scalar mathematics
* \ingroup (tensor)
* @{
*/
Scalar operator+(const Scalar& ) ; ///< + Scalar
Scalar operator-(const Scalar& ) ; ///< \c - Scalar
Scalar operator+(const Scalar&, const Scalar &) ; ///< Scalar + Scalar
Scalar operator+(const Scalar&, const Mtbl&) ; ///< Scalar + Mbtl
Scalar operator+(const Mtbl&, const Scalar&) ; ///< Mtbl + Scalar
Scalar operator+(const Scalar&, double ) ; ///< Scalar + double
Scalar operator+(double, const Scalar& ) ; ///< double + Scalar
Scalar operator+(const Scalar&, int ) ; ///< Scalar + int
Scalar operator+(int, const Scalar& ) ; ///< int + Scalar
Scalar operator-(const Scalar &, const Scalar &) ; ///< Scalar - Scalar
Scalar operator-(const Scalar&, const Mtbl&) ; ///< Scalar - Mbtl
Scalar operator-(const Mtbl&, const Scalar&) ; ///< Mtbl - Scalar
Scalar operator-(const Scalar&, double ) ; ///< Scalar - double
Scalar operator-(double, const Scalar& ) ; ///< double - Scalar
Scalar operator-(const Scalar&, int ) ; ///< Scalar - int
Scalar operator-(int, const Scalar& ) ; ///< int - Scalar
Scalar operator*(const Scalar &, const Scalar &) ; ///< Scalar * Scalar
/// Scalar * Scalar with desaliasing
Scalar operator%(const Scalar &, const Scalar &) ;
/// Scalar * Scalar with desaliasing only in \e r
Scalar operator|(const Scalar &, const Scalar &) ;
Scalar operator*(const Mtbl&, const Scalar&) ; ///< Mtbl * Scalar
Scalar operator*(const Scalar&, const Mtbl&) ; ///< Scalar * Mtbl
Scalar operator*(const Scalar&, double ) ; ///< Scalar * double
Scalar operator*(double, const Scalar &) ; ///< double * Scalar
Scalar operator*(const Scalar&, int ) ; ///< Scalar * int
Scalar operator*(int, const Scalar& ) ; ///< int * Scalar
Scalar operator/(const Scalar &, const Scalar &) ; ///< Scalar / Scalar
Scalar operator/(const Scalar&, double ) ; ///< Scalar / double
Scalar operator/(double, const Scalar &) ; ///< double / Scalar
Scalar operator/(const Scalar&, int ) ; ///< Scalar / int
Scalar operator/(int, const Scalar &) ; ///< int / Scalar
Scalar operator/(const Scalar &, const Mtbl&) ; ///< Scalar / Mtbl
Scalar operator/(const Mtbl&, const Scalar &) ; ///< Mtbl / Scalar
Scalar sin(const Scalar& ) ; ///< Sine
Scalar cos(const Scalar& ) ; ///< Cosine
Scalar tan(const Scalar& ) ; ///< Tangent
Scalar asin(const Scalar& ) ; ///< Arcsine
Scalar acos(const Scalar& ) ; ///< Arccosine
Scalar atan(const Scalar& ) ; ///< Arctangent
Scalar exp(const Scalar& ) ; ///< Exponential
Scalar Heaviside(const Scalar& ) ; ///< Heaviside function
Scalar log(const Scalar& ) ; ///< Neperian logarithm
Scalar log10(const Scalar& ) ; ///< Basis 10 logarithm
Scalar sqrt(const Scalar& ) ; ///< Square root
Scalar racine_cubique (const Scalar& ) ; ///< Cube root
Scalar pow(const Scalar& , int ) ; ///< Power \f${\tt Scalar}^{\tt int}\f$
Scalar pow(const Scalar& , double ) ; ///< Power \f${\tt Scalar}^{\tt double}\f$
Scalar abs(const Scalar& ) ; ///< Absolute value
/**
* Maximum values of a \c Scalar in each domain.
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are the set of the maximum values in each domain.
*/
double totalmax(const Scalar& ) ;
/**
* Minimum values of a \c Scalar in each domain.
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are the set of the minimum values in each domain.
*/
double totalmin(const Scalar& ) ;
/**
* Maximum values of a \c Scalar in each domain.
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are the set of the maximum values in each domain.
*/
Tbl max(const Scalar& ) ;
/**
* Minimum values of a \c Scalar in each domain.
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are the set of the minimum values in each domain.
*/
Tbl min(const Scalar& ) ;
/**
* Sums of the absolute values of all the values of the \c Scalar
* in each domain.
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are the set of the sums of the absolute values in each domain.
*/
Tbl norme(const Scalar& ) ;
/**
* Relative difference between two \c Scalar (norme version).
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are \c norme[a(l)-b(l)]/norme[b(l)] if \c b(l)!=0 and
* \c norme[a(l)-b(l)] if \c b(l)=0 , where \c a(l) and
* \c b(l) denote symbolically the values of \c a and \c b
* in domain no. \c l .
*/
Tbl diffrel(const Scalar& a, const Scalar& b) ;
/**
* Relative difference between two \c Scalar (max version).
* @return 1-D \c Tbl of size the number of domains, the elements of which
* are \c max[abs(a(l)-b(l))]/max[abs(b(l))] if \c b(l)!=0 and
* \c max[abs(a(l)-b(l))] if \c b(l)=0 , where \c a(l) and
* \c b(l) denote symbolically the values of \c a and \c b
* in domain no. \c l .
*/
Tbl diffrelmax(const Scalar& a, const Scalar& b) ;
/**
* Applies an exponential filter in angular directions in all domains.
* (see \c Scalar:exponential_filter_ylm ).
*/
void exp_filter_ylm_all_domains(Scalar& ss, int p, double alpha=-16.) ;
/** @} */
}
#endif
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