These files provide the routine 'pmrrr'. Its purpose is to compute all 
or a subset of eigenvalues and optionally eigenvectors of a symmetric 
tridiagonal matrix. 'pmrrr' is based on the algorithm of Multiple 
Relatively Robust Representations (MRRR). The routine thereby targets 
hybrid architectures consisting of multiple SMP nodes. It also runs in 
fully distributed mode, with each node having only one processor, and 
fully SMP mode, in which case no message passing is required. The 
implementation is based on LAPACK's routine 'dstemr'.


Building the archive libpmrrr.a:
--------------------------------
1) In the folder INSTALL, depending on the compiler used, 
   replace the file 'make.inc' (which assumes GNU gcc) 
   with the corresponding file. 
   For example if you want to use Intel's compilers: 
   $cp ./INSTALL/make.inc.intel ./make.inc
   If you do not find a file matching your compiler, then use any of 
   the files as a basis to edit.
2) Edit the file 'make.inc' according to your system.
3) Run make.


Using libmrrr.a:
----------------
The folder EXAMPLES contains examples of how to use the routine 
'pmrrr' in C and Fortran code. Edit the 'Makefile' in these folders if 
you do not use the GNU compilers and run 'make' to compile the 
examples.

In general, the code that calls 'pmrrr' needs to be linked to the 
library 'libpmrrr.a', which is created in the LIB folder. 
Additionally, it has to be linked to the libraries on which it 
depends (see example files).

Below are given the C and Fortran prototypes of the function 'pmrrr'.
For more information please see 'INCLUDE/pmrrr.h'.


######################################################################
# C function prototype:                                              #
###################################################################### 
# int pmrrr(char *jobz, char *range, int *n, double  *D,             #
#           double *E, double *vl, double *vu, int *il, int *iu,     #
#           int *tryrac, MPI_Comm comm, int *nz, int *offset,        #
#           double *W, double *Z, int *ldz, int *Zsupp);             #
#                                                                    #
# Arguments:                                                         #
# ----------                                                         #
#                                                                    #
# INPUTS:                                                            #
# -------                                                            #
# jobz              "N" or "n" - compute only eigenvalues            #
#                   "V" or "v" - compute also eigenvectors           #
#                   "C" or "c" - count the maximal number of         #
#                                locally computed eigenvectors       #
# range             "A" or "a" - all                                 #
#                   "V" or "v" - by interval: (VL,VU]                #
#                   "I" or "i" - by index:     IL-IU                 #
# n                 matrix size                                      #
# ldz               must be set on input to the leading dimension    #
#                   of of eigenvector matrix Z; this is often equal  #
#                   to matrix size n (not changed on output)         #
#                                                                    #
# INPUT + OUTPUT:                                                    #
# ---------------                                                    #
# D (double[n])     Diagonal elements of tridiagonal T.              #
#                   (On output the array will be overwritten).       #
# E (double[n])     Off-diagonal elements of tridiagonal T.          #
#                   First n-1 elements contain off-diagonals,        #
#                   the last element can have an abitrary value.     #
#                   (On output the array will be overwritten.)       #
# vl                If range="V", lower bound of interval            #
#                   (vl,vu], on output refined.                      #
#                   If range="A" or "I" not referenced as input.     #
#                   On output the interval (vl,vu] contains ALL      #
#                   the computed eigenvalues.                        #
# vu                If range="V", upper bound of interval            #
#                   (vl,vu], on output refined.                      #
#                   If range="A" or "I" not referenced as input.     #
#                   On output the interval (vl,vu] contains ALL      #
#                   the computed eigenvalues.                        #
# il                If range="I", lower index (1-based indexing) of  #
#                   the subset 'il' to 'iu'.                         #
#                   If range="A" or "V" not referenced as input.     #
#                   On output the eigenvalues with index il to iu    #
#                   are computed by ALL processes.                   #
# iu                If range="I", upper index (1-based indexing) of  #
#                   the subset 'il' to 'iu'.                         #
#                   If range="A" or "V" not referenced as input.     #
#                   On output the eigenvalues with index il to iu    # 
#                   are computed by ALL processes.                   #
# tryrac            0 - do not try to achieve high relative accuracy.#
#                   1 - relative accuracy will be attempted;         #
#                       on output it is set to zero if high relative #
#                       accuracy is not achieved.                    #
# comm              MPI communicator; commonly: MPI_COMM_WORLD.      #
#                                                                    #
# OUTPUT:                                                            #
# -------                                                            #
# nz                Number of eigenvalues and eigenvectors computed  #
#                   locally.                                         #
#                   If jobz="C", 'nz' will be set to the maximal     #
#                   number of locally computed eigenvectors such     #
#                   that double[n*nz] will provide enough memory     #
#                   for the local eigenvectors;  this is only        #
#                   important in case of range="V" since             #
#                   '#eigenpairs' are not known in advance           #
# offset            Index, relative to the computed eigenvalues, of  #
#                   the smallest eigenvalue computed locally         #
#                   (0-based indexing).                              #
# W (double[n])     Locally computed eigenvalues;                    #
#                   The first nz entries contain the eigenvalues     #
#                   computed locally; the first entry contains the   #
#                   'offset + 1'-th eigenvalue computed and the      #
#                   'offset + il'-th eigenvalue of the input matrix  #
#                   (1-based indexing in both cases).                #
#                   In some situations it is desirable to have all   #
#                   computed eigenvalues in W, instead of only       #
#                   those computed locally. In this case the         #
#                   routine 'PMR_comm_eigvals' can be called right   #
#                   after 'pmrrr' returns (see below).               #
# Z                 Locally computed eigenvectors.                   #
# (double[n*nz])    Enough space must be provided to store the       #
#                   vectors. 'nz' should be bigger or equal          #
#                   to ceil('#eigenpairs'/'#processes'), where       #
#                   '#eigenpairs' is 'n' in case of range="A" and    #
#                  'iu-il+1' in case of range="I". Alternatively,    #
#                   and for range="V" 'nz' can be obtained           #
#                   by running the routine with jobz="C".            #
# Zsupp             Support of eigenvectors, which is given by       #
# (double[2*n])     Z[2*i] to Z[2*i+1] for the i-th eigenvector      #
#                   stored locally (1-based indexing in both         #
#                   cases).                                          #
#                                                                    #
# RETURN VALUE:                                                      #
# -------------                                                      #
#                 0 - success                                        #
#                 1 - wrong input parameter                          #
#                 2 - misc errors                                    #
#                                                                    #
######################################################################

######################################################################
# Fortran prototype:                                                 #
######################################################################
# PMRRR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, TRYRAC,                #
#       MPI_COMM, NZ, OFFSET, W, Z, LDZ, ZSUPP, INFO)                #
#                                                                    #
# CHARACTER        JOBZ, RANGE                                       #
# INTEGER          N, IL, IU, TRYRAC, MPI_COMM, NZ, OFFSET, LDZ,     # 
#                  ZSUPP(*), INFO                                    #
# DOUBLE PRECISION D(*), D(*), VL, VU, W(*), Z(*,*)                  #
######################################################################


The function is called by every process of the MPI communicator 
specified as an argument. Each process is assumed to have the diagonal 
and sub-diagonal of the symmetric tridiagonal input matrix. It is 
thereby important that MPI is always initialized by 
'MPI_Init_thread' (as opposed to 'MPI_Init'!). If multithreading is 
desired, the thread level support MPI_THREAD_MULTIPLE has to be 
requested in 'MPI_Init_thread'. If no multithreading is used, the 
thread support 'MPI_THREAD_SINGLE' can be requested. Notice that 
multithreading is disabled if the implementation of the MPI library 
does not support MPI_THREAD_MULTIPLE.
The number of threads created in each process can be specified via the 
environment variable PMR_NUM_THREADS. In case this variable is not 
defined, then the routine will create as many threads as specified by 
the variable DEFAULT_NUM_THREADS (which is set in 'pmrrr.h'). 
Note: Do not forget to tell mpiexec/mpirun about the environment 
variable PMR_NUM_THREADS if you use it. This is typically done via the 
option '-env' or '-x'.


Some additinal comments:
------------------------

COMMUNICATING COMPUTED EIGENVALUES:
Upon completion of routine 'pmrrr', each process has a subset of the 
eigenvalues, stored in the array W (see 'pmrrr.h'). In some situations
it is desirable to let all processes have all the computed eigenvalues. 
To this end, an additional routine 'PMR_comm_eigvals' is provided. It 
can be called right after 'pmrrr' returned. Upon completion, each 
process has all the computed eigenvalues, stored in W.
The C and Fortran prototypes of the function 'PMR_comm_eigvals' are 
given below. For more information please see 'pmrrr.h'.

######################################################################
# C function prototype:                                              #
###################################################################### 
# int PMR_comm_eigvals(MPI_Comm comm, int *nz, int *offset,          # 
#                      double *W);                                   #
######################################################################

######################################################################
# Fortran prototype:                                                 #
######################################################################
# PMR_COMM_EIGVALS(MPI_COMM, NZ, OFFSET, W, INFO)                    #
#                                                                    #
# INTEGER          MPI_COMM, NZ, OFFSET, INFO                        #
# DOUBLE PRECISION W(*)                                              #
######################################################################


COMMENTS AND BUGS:
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