ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems. ARPACK should be readily packaged by most Linux distributions. Don't forget to install a development version of the library.
Since ARPACK has not been maintained by Rice university for some years, Debian, Octave and Scilab developers gathered their own modifications and bug fixes in the library ARPACK-NG. Furthermore, ARPACK-NG equips ARPACK with a CMake build system.
Below is a short summary of instructions on how to compile and install ARPACK or ARPACK-NG by hand (for the case you wish to do so).
After you downloaded the Fortran version of ARPACK and the patch, unzip the files you got. That will create a directory named ARPACK. If you need further instructions please read the README file or the instructions. We will explain here in a few steps what has to be done to be able to compile ARPACK.
ARmake.incmake lib in the current directory
      to build the standard library libarpack_$(PLAT).a
      
      Note: For compilation of ARPACK we emphasize
      adding the compiler flag -fPIC. This is a definite
      requirement if we are compiling deal.II with
      shared libraries (which is the default). If we had preferred to be
      compiling deal.II without shared libraries,
      that's ok too; in that case we would do exactly the same thing
      as described above, but this time omitting
      the -fPIC flag from the scheme.
    
      Try to run one of the examples and compare the output.
      How the output should look like is stated in the README
      that can be found in the EXAMPLES directory.
    
If that output you produced looks like it should you can proceed to compile deal.II with ARPACK.
First clone the ARPACK-NG repository. The following commands will set up an appropriate configuration:
        cd arpack-ng
        mkdir build
        cd build
        cmake                                       \
        -DEXAMPLES=ON                               \
        -DMPI=ON                                    \
        -DBUILD_SHARED_LIBS=ON                      \
        -DCMAKE_INSTALL_PREFIX:PATH=$HOME/ARPACK-NG \
        ../
        make
      
      You will need to adjust the path into which you want to install ARPACK-NG
      in the CMAKE_INSTALL_PREFIX line. If you do not have MPI installed, you
      should use -DMPI=OFF instead.
    
    Afterwards run
        make check
        make test
        make install
      
      to test that the build of ARPACK-NG works correctly
      and to install the library finally.
    
    Support for ARPACK will be enabled automatically if a system wide installation of ARPACK can be found. To use a self compiled version, specify
    -DARPACK_DIR=/path/to/arpack
      
      when invoking cmake.
    
    You can override the autodetection by manually setting
    -DDEAL_II_WITH_ARPACK=OFF|ON
      .