Routine: PDGEBD2()  File: SRC\pdgebd2.f

 
 
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..
     .. Array Arguments ..
     ..
  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
  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) 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 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
          orthogonal 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 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
          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) 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,
                                                  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 >= 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
  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 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
  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 )
  =====================================================================
     .. Parameters ..
 
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001        SUBROUTINE PDGEBD2( M , N , A , IA , JA , DESCA , D , E , TAUQ , TAUP ,
002       $WORK , LWORK , INFO )
003  
004  *     -- ScaLAPACK auxiliary routine(version 1.7) --
005  *     University of Tennessee , Knoxville , Oak Ridge National Laboratory ,
006  *     and University of California , Berkeley.
007  *     May 1 , 1997
008  
009  *     .. Scalar Arguments ..
010        INTEGER IA , INFO , JA , LWORK , M , N
011        INTEGER BLOCK_CYCLIC_2D , CSRC_ , CTXT_ , DLEN_ , DTYPE_ ,
012       $LLD_ , MB_ , M_ , NB_ , N_ , RSRC_
013        PARAMETER( BLOCK_CYCLIC_2D = 1 , DLEN_ = 9 , DTYPE_ = 1 ,
014       $CTXT_ = 2 , M_ = 3 , N_ = 4 , MB_ = 5 , NB_ = 6 ,
015       $RSRC_ = 7 , CSRC_ = 8 , LLD_ = 9 )
016        DOUBLE PRECISION ONE , ZERO
017        PARAMETER( ONE = 1.0D + 0 , ZERO = 0.0D + 0 )
018  *     ..
019  *     .. Local Scalars ..
020        LOGICAL LQUERY
021        INTEGER I , IACOL , IAROW , ICOFFA , ICTXT , II , IROFFA , J ,
022       $JJ , K , LWMIN , MPA0 , MYCOL , MYROW , NPCOL , NPROW ,
023       $NQA0
024        DOUBLE PRECISION ALPHA
025  *     ..
026  *     .. Local Arrays ..
027        INTEGER DESCD( DLEN_ ) , DESCE( DLEN_ )
028  *     ..
029  *     .. External Subroutines ..
030        EXTERNAL BLACS_ABORT , BLACS_GRIDINFO , CHK1MAT , DESCSET ,
031       $DGEBR2D , DGEBS2D , DLARFG , INFOG2L ,
032       $PDLARF , PDLARFG , PDELSET , PXERBLA
033  *     ..
034  *     .. External Functions ..
035        INTEGER INDXG2P , NUMROC
036        EXTERNAL INDXG2P , NUMROC
037  *     ..
038  *     .. Intrinsic Functions ..
039        INTRINSIC DBLE , MAX , MIN , MOD
040  *     ..
041  *     .. Executable Statements ..
042  
043  *     Test the input parameters
044  
045        ICTXT = DESCA( CTXT_ )
046        CALL BLACS_GRIDINFO( ICTXT , NPROW , NPCOL , MYROW , MYCOL )
047  
048  *     Test the input parameters
049  
050        INFO = 0
051        IF( NPROW.EQ. - 1 ) THEN
052            INFO = - (600 + CTXT_)
053        ELSE
054            CALL CHK1MAT( M , 1 , N , 2 , IA , JA , DESCA , 6 , INFO )
055            IF( INFO.EQ.0 ) THEN
056                IROFFA = MOD( IA - 1 , DESCA( MB_ ) )
057                ICOFFA = MOD( JA - 1 , DESCA( NB_ ) )
058                IAROW = INDXG2P( IA , DESCA( MB_ ) , MYROW , DESCA( RSRC_ ) ,
059       $        NPROW )
060                IACOL = INDXG2P( JA , DESCA( NB_ ) , MYCOL , DESCA( CSRC_ ) ,
061       $        NPCOL )
062                MPA0 = NUMROC( M + IROFFA , DESCA( MB_ ) , MYROW , IAROW , NPROW )
063                NQA0 = NUMROC( N + ICOFFA , DESCA( NB_ ) , MYCOL , IACOL , NPCOL )
064                LWMIN = MAX( MPA0 , NQA0 )
065  
066                WORK( 1 ) = DBLE( LWMIN )
067                LQUERY =( LWORK.EQ. - 1 )
068                IF( IROFFA.NE.ICOFFA ) THEN
069                    INFO = - 5
070                ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN
071                    INFO = - (600 + NB_)
072                ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
073                    INFO = - 12
074                END IF
075            END IF
076        END IF
077  
078        IF( INFO.LT.0 ) THEN
079            CALL PXERBLA( ICTXT , 'PDGEBD2' , - INFO )
080            CALL BLACS_ABORT( ICTXT , 1 )
081            RETURN
082        ELSE IF( LQUERY ) THEN
083            RETURN
084        END IF
085  
086        CALL INFOG2L( IA , JA , DESCA , NPROW , NPCOL , MYROW , MYCOL , II , JJ ,
087       $IAROW , IACOL )
088  
089        IF( M.EQ.1 .AND. N.EQ.1 ) THEN
090            IF( MYCOL.EQ.IACOL ) THEN
091                IF( MYROW.EQ.IAROW ) THEN
092                    I = II + (JJ - 1)*DESCA( LLD_ )
093                    CALL DLARFG( 1 , A( I ) , A( I ) , 1 , TAUQ( JJ ) )
094                    D( JJ ) = A( I )
095                    CALL DGEBS2D( ICTXT , 'Columnwise' , ' ' , 1 , 1 , D( JJ ) ,
096       $            1 )
097                    CALL DGEBS2D( ICTXT , 'Columnwise' , ' ' , 1 , 1 , TAUQ( JJ ) ,
098       $            1 )
099                ELSE
100                    CALL DGEBR2D( ICTXT , 'Columnwise' , ' ' , 1 , 1 , D( JJ ) ,
101       $            1 , IAROW , IACOL )
102                    CALL DGEBR2D( ICTXT , 'Columnwise' , ' ' , 1 , 1 , TAUQ( JJ ) ,
103       $            1 , IAROW , IACOL )
104                END IF
105            END IF
106            IF( MYROW.EQ.IAROW )
107       $        TAUP( II ) = ZERO
108                RETURN
109            END IF
110  
111            ALPHA = ZERO
112  
113            IF( M.GE.N ) THEN
114  
115  *             Reduce to upper bidiagonal form
116  
117                CALL DESCSET( DESCD , 1 , JA + MIN(M , N) - 1 , 1 , DESCA( NB_ ) , MYROW ,
118       $        DESCA( CSRC_ ) , DESCA( CTXT_ ) , 1 )
119                CALL DESCSET( DESCE , IA + MIN(M , N) - 1 , 1 , DESCA( MB_ ) , 1 ,
120       $        DESCA( RSRC_ ) , MYCOL , DESCA( CTXT_ ) ,
121       $        DESCA( LLD_ ) )
122                DO 10 K = 1 , N
123                    I = IA + K - 1
124                    J = JA + K - 1
125  
126  *                 Generate elementary reflector H(j) to annihilate
127  *                 A(ia + i : ia + m - 1 , j)
128  
129                    CALL PDLARFG ( M - K + 1 , ALPHA , I , J , A , MIN( I + 1 , M + IA - 1 ) ,
130       $            J , DESCA , 1 , TAUQ )
131                    CALL PDELSET( D , 1 , J , DESCD , ALPHA )
132                    CALL PDELSET( A , I , J , DESCA , ONE )
133  
134  *                 Apply H(i) to A(i : ia + m - 1 , i + 1 : ja + n - 1) from the left
135  
136                    CALL PDLARF ( 'Left' , M - K + 1 , N - K , A , I , J , DESCA , 1 , TAUQ , A ,
137       $            I , J + 1 , DESCA , WORK )
138                    CALL PDELSET( A , I , J , DESCA , ALPHA )
139  
140                    IF( K.LT.N ) THEN
141  
142  *                     Generate elementary reflector G(i) to annihilate
143  *                     A(i , ja + j + 1 : ja + n - 1)
144  
145                        CALL PDLARFG ( N - K , ALPHA , I , J + 1 , A , I ,
146       $                MIN( J + 2 , JA + N - 1 ) , DESCA , DESCA( M_ ) ,
147       $                TAUP )
148                        CALL PDELSET( E , I , 1 , DESCE , ALPHA )
149                        CALL PDELSET( A , I , J + 1 , DESCA , ONE )
150  
151  *                     Apply G(i) to A(i + 1 : ia + m - 1 , i + 1 : ja + n - 1) from the right
152  
153                        CALL PDLARF ( 'Right' , M - K , N - K , A , I , J + 1 , DESCA ,
154       $                DESCA( M_ ) , TAUP , A , I + 1 , J + 1 , DESCA ,
155       $                WORK )
156                        CALL PDELSET( A , I , J + 1 , DESCA , ALPHA )
157                    ELSE
158                        CALL PDELSET( TAUP , I , 1 , DESCE , ZERO )
159                    END IF
160     10         CONTINUE
161  
162            ELSE
163  
164  *             Reduce to lower bidiagonal form
165  
166                CALL DESCSET( DESCD , IA + MIN(M , N) - 1 , 1 , DESCA( MB_ ) , 1 ,
167       $        DESCA( RSRC_ ) , MYCOL , DESCA( CTXT_ ) ,
168       $        DESCA( LLD_ ) )
169                CALL DESCSET( DESCE , 1 , JA + MIN(M , N) - 1 , 1 , DESCA( NB_ ) , MYROW ,
170       $        DESCA( CSRC_ ) , DESCA( CTXT_ ) , 1 )
171                DO 20 K = 1 , M
172                    I = IA + K - 1
173                    J = JA + K - 1
174  
175  *                 Generate elementary reflector G(i) to annihilate
176  *                 A(i , ja + j : ja + n - 1)
177  
178                    CALL PDLARFG ( N - K + 1 , ALPHA , I , J , A , I ,
179       $            MIN( J + 1 , JA + N - 1 ) , DESCA , DESCA( M_ ) , TAUP )
180                    CALL PDELSET( D , I , 1 , DESCD , ALPHA )
181                    CALL PDELSET( A , I , J , DESCA , ONE )
182  
183  *                 Apply G(i) to A(i : ia + m - 1 , j : ja + n - 1) from the right
184  
185                    CALL PDLARF ( 'Right' , M - K , N - K + 1 , A , I , J , DESCA ,
186       $            DESCA( M_ ) , TAUP , A , MIN( I + 1 , IA + M - 1 ) , J ,
187       $            DESCA , WORK )
188                    CALL PDELSET( A , I , J , DESCA , ALPHA )
189  
190                    IF( K.LT.M ) THEN
191  
192  *                     Generate elementary reflector H(i) to annihilate
193  *                     A(i + 2 : ia + m - 1 , j)
194  
195                        CALL PDLARFG ( M - K , ALPHA , I + 1 , J , A ,
196       $                MIN( I + 2 , IA + M - 1 ) , J , DESCA , 1 , TAUQ )
197                        CALL PDELSET( E , 1 , J , DESCE , ALPHA )
198                        CALL PDELSET( A , I + 1 , J , DESCA , ONE )
199  
200  *                     Apply H(i) to A(i + 1 : ia + m - 1 , j + 1 : ja + n - 1) from the left
201  
202                        CALL PDLARF ( 'Left' , M - K , N - K , A , I + 1 , J , DESCA , 1 , TAUQ ,
203       $                A , I + 1 , J + 1 , DESCA , WORK )
204                        CALL PDELSET( A , I + 1 , J , DESCA , ALPHA )
205                    ELSE
206                        CALL PDELSET( TAUQ , 1 , J , DESCE , ZERO )
207                    END IF
208     20         CONTINUE
209            END IF
210  
211            WORK( 1 ) = DBLE( LWMIN )
212  
213            RETURN
214  
215  *         End of PDGEBD2
216  
217        END