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NAME

       PZGEBRD   -   reduce   a   complex   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 unitary transformation

SYNOPSIS

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

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

           INTEGER         DESCA( * )

           DOUBLE          PRECISION D( * ), E( * )

           COMPLEX*16      A( * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE

       PZGEBRD  reduces  a   complex   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 unitary 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  vector.   This  vector
       stores the information required to establish the mapping 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 column.
       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) COMPLEX*16 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 distributed 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  unitary matrix Q as a product of elementary reflectors, and
               the elements above the first superdiagonal, with the  array  TAUP,  represent  the
               orthogonal  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 subdiagonal, with the array TAUQ, represent the unitary
               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 distributed 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) COMPLEX*16 array dimension
               LOCc(JA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary  reflectors  which
               represent  the  unitary  matrix  Q.  TAUQ is tied to the distributed matrix A. See
               Further   Details.    TAUP      (local   output)   COMPLEX*16   array,   dimension
               LOCr(IA+MIN(M,N)-1).  The  scalar  factors  of  the  elementary  reflectors  which
               represent the unitary matrix P. TAUP is tied to  the  distributed  matrix  A.  See
               Further   Details.   WORK     (local  workspace/local  output)  COMPLEX*16  array,
               dimension (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
               >= NB*( MpA0 + NqA0 + 1 ) + NqA0

               where  NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-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+ICOFFA, 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    (global 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 reflectors:

       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 complex scalars, and v and u are complex 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 complex scalars, and v and u are complex 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 examples:

       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 )