Helmholtz particle FMMs in R^2.
===============================

hfmm2d: charge and dipstr are complex valued, x \in R^2

\phi(x_i) = (ima/4) \sum_{j\ne i}   charge_j H^{(1)}_0(k |x_i-x_j| )
            + dipstr_j (dipvec_j \dot \grad_j H^{(1)}_0(k |x_i-x_j| )

or, more precisely,

\phi(x_i) = (ima/4) \sum_{j\ne i}   charge_j H^{(1)}_0(k |x_i-x_j| )
+ dipstr_j (dipvec_j \dot (x_i-x_j) ) H^{(1)}_1(k |x_i-x_j|) *k/|x_i-x_j|


Laplace particle FMMs in R^2.
=============================

lfmm2d: charge and dipstr are complex valued, x \in R^2

\phi(x_i) = \sum_{j\ne i}   charge_j \log |x_i-x_j|  
                   + dipstr_j (dipvec_j \dot \grad_j log|x_i-x_j|)

or, more precisely, 

\phi(x_i) = \sum_{j\ne i}   charge_j \log |x_i-x_j|  
                   + dipstr_j (dipvec_j \dot (x_i-x_j)) * (-1/|x_i-x_j|^2)


rfmm2d: charge and dipstr are real valued, x \in R^2

\phi(x_i) = \sum_{j\ne i}   charge_j \log |x_i-x_j| 
                   + dipstr_j (dipvec_j \dot \grad_j log|x_i-x_j|)

or, more precisely,

\phi(x_i) = \sum_{j\ne i}   charge_j \log |x_i-x_j|  
                   + dipstr_j (dipvec_j \dot (x_i-x_j)) * (-1/|x_i-x_j|^2)


zfmm2d: dipstr are complex valued, z are complex numbers, Cauchy sums

\phi(z_i) = \sum_{j\ne i} dipstr_j \frac{1}{z_i-z_j}

       In this routine, we define the gradient as the first
       derivative with respect to z, and the hessian as the second
       derivative with respect to z.

\gradient \phi(z_i) = \frac{\partial \phi(z_i)}{\partial z}
\hessian  \phi(z_i) = \frac{\partial^2 \phi(z_i)}{\partial z^2}


cfmm2d: charge and dipstr are complex valued, z are complex numbers, 
        generalized Cauchy sums.

       Note, that this is a highly specialized routine, please use
       lfmm2d, rfmm2d and zfmm2d instead. Note also, that the complex
       valued logarithm is a multi-valued function, therefore, the
       potential values returned by cfmm2d have to be interpreted
       carefully, if charges are specified.  For example, only the
       real part of potential is meaningful for real valued charges.
       The gradients and hessians are valid for arbitrary complex
       charges.

\phi(z_i) = \sum_{j\ne i} charge_j \log(z_i-z_j) + dipstr_j \frac{1}{z_i-z_j}

       In this routine, we define the gradient as the first
       derivative with respect to z, and the hessian as the second
       derivative with respect to z.

\gradient \phi(z_i) = \frac{\partial \phi(z_i)}{\partial z}
\hessian  \phi(z_i) = \frac{\partial^2 \phi(z_i)}{\partial z^2}



Warning, the direct evaluation routines are different for zfmm2d, cfmm2d,
rfmm2d and lfmm2d, respectively.

zfmm2d - cpotgrad2d_sdp  - zfmm2dpart.f

cfmm2d - cpotgrad2d_sdp  - cfmm2dpart.f

rfmm2d - rpotgrad2d_sdp  - rfmm2dpart.f

lfmm2d - rcpotgrad2d_sdp - lfmm2dpart.f


The lfmm2d is the closest analog for hfmm2d, i.e. it implements
complex valued charges and dipoles. rfmm2d is the real version of
lfmm2d, it is slightly faster (2x) but can handle only real valued charges
and dipoles. zfmm2d does the Cauchy sums of complex valued dipoles.

Both rfmm2d and lfmm2d will internally form real charges and complex
dipoles (lfmm2d does it twice), after that the actual work is
performed by cfmm2d.

cfmm2d does the Cauchy like sums of complex valued
charges and dipoles - dipole vectors have no meaning in cfmm2d.

==============================================================================

Sign conventions for Helmholtz and Laplace potentials:

> f := HankelH1(0,k*r);
                                            2    2 1/2
                       f := HankelH1(0, k (x  + y )   )

> diff(f,x);           
                                         2    2 1/2
                         HankelH1(1, k (x  + y )   ) k x
                       - -------------------------------
                                    2    2 1/2
                                  (x  + y )

> f := log(r);         
                                          2    2
                             f := 1/2 ln(x  + y )

> diff(f,x);           
                                       x
                                    -------
                                     2    2
                                    x  + y

Helmholtz dipole potential:

   (dipvec(1)*xdiff+dipvec(2)*ydiff) * (h1*wavek/r) * (ima/4)

Laplace dipole potential:

   (dipvec(1)*xdiff+dipvec(2)*ydiff) * (-1/r**2)


Note the sign flip in the Laplace dipole potential, this is due to our Green's
function choice as log(r). 

==============================================================================

hfmm2dpart.f  Helmholtz FMM for particles 
                     (potential, field, and hessian)

lfmm2dpart.f  Complex valued Laplace FMM for particles 
                     (potential, field, and hessian)
		     hfmm2dparthess compatible

rfmm2dpart.f  Real valued Laplace FMM for particles 
                     (potential, field, and hessian)

zfmm2dpart.f  Cauchy FMM for particles (Cauchy sums) 
                     (potential, field, and hessian)
                     evaluate potz and potzz only
		     (dz = dx + i*dy = 0 => dy = i*dx)

cfmm2dpart.f  Generalized Cauchy FMM for particles (Cauchy like sums) 
                     (potential, field, and hessian)
                     evaluate potz and potzz only
		     (dz = dx + i*dy = 0 => dy = i*dx)


d2tstrcr.f       FMM tree routines, for self and target interactions 
d2tstrcr_omp.f       FMM tree routines, for self and target interactions 
                     fortran 90 version, with better OpenMP scaling
d2mtreeplot.f    FMM tree plotting routines

*_dr.f         driver files
*.make         make files