PDGEBD2(3)    ScaLAPACK routine of NEC Numeric Library Collection   PDGEBD2(3)



NAME
       PDGEBD2  -  reduce  a real general M-by-N distributed matrix sub( A ) =
       A(IA:IA+M-1,JA:JA+N-1) to upper  or  lower  bidiagonal  form  B  by  an
       orthogonal transformation

SYNOPSIS
       SUBROUTINE PDGEBD2( M,  N,  A,  IA,  JA, DESCA, D, E, TAUQ, TAUP, WORK,
                           LWORK, INFO )

           INTEGER         IA, INFO, JA, LWORK, M, N

           INTEGER         DESCA( * )

           DOUBLE          PRECISION A( * ), D( * ), E( * ), TAUP( * ),  TAUQ(
                           * ), WORK( * )

PURPOSE
       PDGEBD2  reduces  a  real  general M-by-N distributed matrix sub( A ) =
       A(IA:IA+M-1,JA:JA+N-1) to upper  or  lower  bidiagonal  form  B  by  an
       orthogonal  transformation:  Q'  *  sub(  A ) * P = B.  If M >= N, B is
       upper bidiagonal; if M < N, B is lower bidiagonal.


       Notes
       =====

       Each global data object is described by an associated description  vec-
       tor.  This vector stores the information required to establish the map-
       ping between an object element and its corresponding process and memory
       location.

       Let  A  be  a generic term for any 2D block cyclicly distributed array.
       Such a global array has an associated description vector DESCA.  In the
       following  comments,  the  character _ should be read as "of the global
       array".

       NOTATION        STORED IN      EXPLANATION
       --------------- -------------- --------------------------------------
       DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                                      DTYPE_A = 1.
       CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                                      the BLACS process grid A is distribu-
                                      ted over. The context itself is glo-
                                      bal, but the handle (the integer
                                      value) may vary.
       M_A    (global) DESCA( M_ )    The number of rows in the global
                                      array A.
       N_A    (global) DESCA( N_ )    The number of columns in the global
                                      array A.
       MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                                      the rows of the array.
       NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                                      the columns of the array.
       RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                                      row  of  the  array  A  is  distributed.
       CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                                      first column of the array A is
                                      distributed.
       LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
                                      array.  LLD_A >= MAX(1,LOCr(M_A)).

       Let  K  be  the  number of rows or columns of a distributed matrix, and
       assume that its process grid has dimension p x q.
       LOCr( K ) denotes the number of elements of  K  that  a  process  would
       receive  if K were distributed over the p processes of its process col-
       umn.
       Similarly, LOCc( K ) denotes the number of elements of K that a process
       would receive if K were distributed over the q processes of its process
       row.
       The values of LOCr() and LOCc() may be determined via  a  call  to  the
       ScaLAPACK tool function, NUMROC:
               LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
               LOCc(  N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).  An upper
       bound for these quantities may be computed by:
               LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
               LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A


ARGUMENTS
       M       (global input) INTEGER
               The number of rows to be operated on, i.e. the number  of  rows
               of the distributed submatrix sub( A ). M >= 0.

       N       (global input) INTEGER
               The  number  of  columns  to be operated on, i.e. the number of
               columns of the distributed submatrix sub( A ). N >= 0.

       A       (local input/local output) DOUBLE PRECISION pointer into the
               local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).  On
               entry, this array contains the local pieces of the general dis-
               tributed matrix sub( A ). On exit, if M >= N, the diagonal  and
               the  first  superdiagonal  of sub( A ) are overwritten with the
               upper bidiagonal matrix B; the  elements  below  the  diagonal,
               with  the  array  TAUQ,  represent the orthogonal matrix Q as a
               product of elementary reflectors, and the  elements  above  the
               first superdiagonal, with the array TAUP, represent the orthog-
               onal matrix P as a product of elementary reflectors. If M <  N,
               the diagonal and the first subdiagonal are overwritten with the
               lower bidiagonal matrix B; the elements below the first  subdi-
               agonal,  with the array TAUQ, represent the orthogonal matrix Q
               as a product of elementary reflectors, and the  elements  above
               the  diagonal,  with  the  array TAUP, represent the orthogonal
               matrix P as a product of  elementary  reflectors.  See  Further
               Details.   IA       (global input) INTEGER The row index in the
               global array A indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array  A  indicating  the  first
               column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       D       (local output) DOUBLE PRECISION array, dimension
               LOCc(JA+MIN(M,N)-1)  if  M >= N; LOCr(IA+MIN(M,N)-1) otherwise.
               The distributed diagonal elements of the bidiagonal  matrix  B:
               D(i) = A(i,i). D is tied to the distributed matrix A.

       E       (local output) DOUBLE PRECISION array, dimension
               LOCr(IA+MIN(M,N)-1)  if  M >= N; LOCc(JA+MIN(M,N)-2) otherwise.
               The distributed off-diagonal elements of  the  bidiagonal  dis-
               tributed  matrix  B:  if  m  >=  n,  E(i)  =  A(i,i+1)  for i =
               1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.   E
               is tied to the distributed matrix A.

       TAUQ    (local output) DOUBLE PRECISION array dimension
               LOCc(JA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary
               reflectors which represent the orthogonal  matrix  Q.  TAUQ  is
               tied  to  the  distributed matrix A. See Further Details.  TAUP
               (local    output)    DOUBLE    PRECISION    array,    dimension
               LOCr(IA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary
               reflectors which represent the orthogonal  matrix  P.  TAUP  is
               tied  to  the  distributed matrix A. See Further Details.  WORK
               (local workspace/local output) DOUBLE PRECISION  array,  dimen-
               sion  (LWORK)  On exit, WORK(1) returns the minimal and optimal
               LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must
               be at least LWORK >= MAX( MpA0, NqA0 )

               where  NB  =  MB_A  =  NB_A,  IROFFA  = MOD( IA-1, NB ) IAROW =
               INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL =  INDXG2P(  JA,
               NB, MYCOL, CSRC_A, NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW,
               IAROW, NPROW ), NqA0 =  NUMROC(  N+IROFFA,  NB,  MYCOL,  IACOL,
               NPCOL ).

               INDXG2P  and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL,
               NPROW and NPCOL can be determined  by  calling  the  subroutine
               BLACS_GRIDINFO.

               If LWORK = -1, then LWORK is global input and a workspace query
               is assumed; the routine only calculates the minimum and optimal
               size  for  all work arrays. Each of these values is returned in
               the first entry of the corresponding work array, and  no  error
               message is issued by PXERBLA.

       INFO    (local output) INTEGER
               = 0:  successful exit
               <  0:   If the i-th argument is an array and the j-entry had an
               illegal value, then INFO = -(i*100+j), if the i-th argument  is
               a scalar and had an illegal value, then INFO = -i.

FURTHER DETAILS
       The  matrices Q and P are represented as products of elementary reflec-
       tors:

       If m >= n,

          Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

       Each H(i) and G(i) has the form:

          H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

       where tauq and taup are real scalars, and v and  u  are  real  vectors;
       v(1:i-1)   =   0,  v(i)  =  1,  and  v(i+1:m)  is  stored  on  exit  in
       A(ia+i:ia+m-1,ja+i-1);
       u(1:i)  =  0,  u(i+1)  =  1,  and  u(i+2:n)  is  stored  on   exit   in
       A(ia+i-1,ja+i+1:ja+n-1);
       tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

       If m < n,

          Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

       Each H(i) and G(i) has the form:

          H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

       where  tauq  and  taup  are real scalars, and v and u are real vectors;
       v(1:i)  =  0,  v(i+1)  =  1,  and  v(i+2:m)  is  stored  on   exit   in
       A(ia+i+1:ia+m-1,ja+i-1);
       u(1:i-1)   =   0,  u(i)  =  1,  and  u(i+1:n)  is  stored  on  exit  in
       A(ia+i-1,ja+i:ja+n-1);
       tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).

       The contents of sub( A ) on exit are illustrated by the following exam-
       ples:

       m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

         (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
         (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
         (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
         (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
         (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
         (  v1  v2  v3  v4  v5 )

       where  d  and  e  denote  diagonal  and  off-diagonal elements of B, vi
       denotes an element of the vector defining H(i), and ui  an  element  of
       the vector defining G(i).


ALIGNMENT REQUIREMENTS
       The  distributed  submatrix sub( A ) must verify some alignment proper-
       ties, namely the following expressions should be true:
                       ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )




ScaLAPACK routine               31 October 2017                     PDGEBD2(3)