This directory contains the source codes of the sample Fortran90 programs
  of the Filter Diagonalization Method to solve symmetric eigenvalue problems.

  0) Prepare the gnu-fortran90 "gfortran", and Unit "xgraph" command
     for the simple and easy graph plot.

  1) First, in the sub-directory "./Filter", make the object archive 
     library "lib.a" by "make" command.
     The files in this subdirectry are the library routines to generate 
     the abscissas for the shifts of resolvents and the coefficients 
     of the linear combination of resolvents for the typical kind of
     filters (Butterworth, Chebyshev, Inverse Chebyshev, elliptic).

  2) The sub-directroy "./Filter-Test" is the directory to test the 
     library routines in the sub-directory "./Filter".
     After the "make" in the directry is finished, the tests of the 
     filter library can be made. Please read the file README in the 
     file "./Filter-Test".
    
  3) The files of this current directry are the test programs of
     the filter-diagonalization-method for the real symmetric eigenproblems.
     After the "make" is made by
          % make
     four executables "main_std.out", "main_std_mp.out", "main_gen.out"
     and "main_gen_mp.out" are created. The executable whose name 
     with _mp are the programs with the multi-threaded parallellizations 
     by OpenMP. The number of threads to use can be controlled by 
     setting the environment variable "OMP_NUM_THREADS". For example,
     using the borne shell by
           % OMP_NUM_THREADS=3; export OMP_NUM_THREADS;
     (Large memory space is required as the number of threads in use.)

  4) The programs "main_std.out" and "main_std_mp.out" solves the real
     symmetric standard eigenproblem $Ax=\lambda x$. 
       The programs "main_gen.out" and "main_gen_mp.out" solves the 
     generalized real symmetric definite eigenproblem $Ax=\lambda Bx$,
     where $A$,$B$ are real symmetric matrices and $B$ is positive definite.
     The current sample programs are for the case both $A$ and $B$ are 
     banded matrices.

     The input data can be supplied from file by the redirect. For example,
         %  main_gen  < Input
       or
         %  main_gen_mp  < Input
     Then the program calculates the approximated eigenpairs and writes 
     some files to show the calculation process and the result.

  5) After the run is finished, the shell script "show.sh" can be used 
     to graphically view the result of execution. It shows two graphs
     using the "xgraph" command for Unix. The first graph shows the 
     distribution of the singular values of the set of output vectors 
     from the filter.
       After the first graph is shown, close it (Control-C will do) then
     the second graph will be shown which plots the value of the approximated
     eigenvalues of the eigenpair in the horizontal axis, and in the vertical
     axis the values of $\Delta=||r||_{B^{-1}}=(r,B^{-1}r)$ which is the 
     bound of error of the eigenvalue for the approximated eigenpair, 
     here $r$ denotes the residual of the eigenpair $r=(A-\lambda B)x$ 
     for the approximated eigenpair $(\lambda, x)$. 
     The curve "IT0" is the solutions of the filter diagonalization method.
     The curve "IT1" is the improved result by the application of the 
     Rayleigh inverse-iteration once, and for the curve "IT2" twice.

  5') Even if the "xgraph" command is not available on your system, 
      the shell script "show.sh" before it fails to find the "xgraph" command,
      two data files "sigma.data" and  "delta.data" are already generated, 
      which are to be plotted by "gnuplot" command.
      By the use of gnuplot command files "sigma.gnuplot" and "delta.gnuplot",
      we do
           % gnuplot < sigma.gnuplot
           % gnuplot < delta.gnuplot
      which generates the eps files "sigma.eps" and "delta.eps".
        The eps files can be view by "ghostscript" or they can be included 
      for the LaTeX graphics environment.
      (If you modify the gnuplot command files "sigma.gnuplot" and 
      "delta.gnuplot", the graphs can be directly shown on the terminal 
      by the gnuplot command.)

  6) The samples of the execution of the programs are stored in the
     sub-directory "./RUN.SAMPLES".