=====================
                  RADIAL++ README FILE
                  =====================
                            
                     Alpha -- Version
                      Version 0.1.2

==============================================================================
 Radial++
==============================================================================

Radial++ is a library of classes and templates in C++ for the numerical solution 
of partial differential equations (no weak solutions) and the interpolation
of scattered data by Radial Basis Functions (RBFs).

Radial++ works with generalized RBFs, like thin-plate splines, power splines,
Gaussian, multiquadrics and inverse multiquadrics. This means that you can
configurate the RBF to work with any degreee of the associated polynomial
to the RBF and also, give the free exponent in the generalized RBF.

In particular, we can interpolate data in any dimension, and also, the library
provide several examples for numerical solution of partial differential
equations by using the unsymmetric collocation method.


The RBFs in Radial++ are:

      RBF                          C++
    -----------------------------------
    Multiquadric                   MQ
    Thin-Plate Splines             TPS
    Radial Powers                  POT
    Gaussian                       GAU
    Inverse Multiquadric           IMQ
   
    C++                                      beta        
    ----------------------------------------------------
    MQ         phi(x) = (r^2+c^2)^(beta/2)   1,3,5,7,..
    IMQ        phi(x) = (r^2+c^2)^(-beta/2)  1,3,5,7,..
    GAU        phi(x) = exp(-r^2*c^2)        ---------
    TPS        phi(x) = r^beta log(r)        2,4,6,8,...
    POT        phi(x) = r^beta               1,3,5,7,...  

    r = || x ||_2  Euclidean distance
     
    c = shape parameter
    
    C++    default beta value
    -------------------------
    MQ            1
    IMQ           1
    GAU           -
    TPS           2
    POT           1      
    
    The degree of the polynomial depends of the choosed RBF, the
    default values are:
    
     C++          m
    -------------------------
    MQ            ceil(beta/2)     see theorem 8.15 page 109 in book Scattered Data Approximation          
    IMQ           ---              see theorem 6.13 page 76  in book Scattered Data Approximation
    GAU           ---              see theorem 6.10 page 74  in book Scattered Data Approximation
    TPS           1+beta/2         see theorem 8.17 page 112 in book Scattered Data Approximation
    POT           ceil(beta/2)     see theorem 8.16 page 111 in book Scattered Data Approximation
    
    Recall, the total degree of the polynomial is m-1.
    Also, you can configurate this value to obtain more polynomial precission.
    Each RBF can be configurated with beta and m.


Radial use the library called Seldon to perform matrix and vector operations,
Seldon is included, so, you dont need to download Seldon.  It is convenient to
say that the best performance is obtained with the interface to BLAS and LAPACK,
see Seldon. Radial++ can work without the interface of BLAS and LAPACK. In 
particular, all the provided examples run without any external pre-installed 
library.

This README file describes how to install the libray called RADIAL++.

==============================================================================
 The Compiler
==============================================================================

Radial++ is supposed to be fully compliant with the C++ standard. 
Therefore, it can be compiled by GNU GCC (>=3.0; tested with version 3.4, 4.4). 
No tests were conducted with proprietary compilers under Unix, but the 
compliance with the C++ standard should ensure portability.  Please tell 
me if you have any problem running the test problems.

==============================================================================
 Installation
==============================================================================
The library called Radial++ is ready to run "out the box", it is not neccesary
to have a previous library installed in your machine. All the necessary files
are contained in Radial++.

To compile the examples only modify the file:

  examples/Makefile.in 

in particular the "tag" <RBF> which correspond to the path where the library 
was unzipped.

!!This is the only modification that you need to do to run all the examples.


If additionn, optionally, in the file examples/Makefile.in you can modify the
compiler selected (CXX==?) and the flags in the compilation (CXXFLAGS=?)

==============================================================================
 Running Examples
==============================================================================
Before to continue, check that  you are using the correct Makefile.in  for your 
system.

The examples are divided into two categories:

examples/interpolation
examples/pde

each folder has several folders

examples/interpolation/1d
         interpolation/2d
         interpolation/3d
         interpolation/nd  <- scattered data interpolation in any dimension with any RBF 

examples/pde/advection
         pde/allen-cahn
         pde/burger
         pde/circle        <- Poisson in the circle with analytical solution
         pde/helmholtz
         pde/poisson
         pde/laplace
         
         
Inside of each folder you only need to write in the shell:  

make help

and follow the given instructions.

All the examples contain ascii outputs with the expected running outputs.


==============================================================================
 Author
==============================================================================
This work was proposed by the professor Pedro Gonzalez-Casanova, and provide me
the theoretical background. See  http://www.dci.dgsca.unam.mx/pderbf

Jose Antonio Munoz-Gomez
jantoniomex@gmail.com
University of Guadalajara
Mexico
2  Agosto  2009