Routine: PSLABRD()  File: SRC\pslabrd.f

 
 
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..
     .. Array Arguments ..
     ..
  Purpose
  =======
  PSLABRD reduces the first NB rows and columns of 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 by an orthogonal transformation Q' * A * P,
  and returns the matrices X and Y which are needed to apply the
  transformation to the unreduced part of sub( A ).
  If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to
  lower bidiagonal form.
  This is an auxiliary routine called by PSGEBRD.
  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.
  NB      (global input) INTEGER
          The number of leading rows and columns of sub( A ) to be
          reduced.
  A       (local input/local output) REAL 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 ) to be reduced. On exit,
          the first NB rows and columns of the matrix are overwritten;
          the rest of the distributed matrix sub( A ) is unchanged.
          If m >= n, elements on and below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the
            array TAUP, represent the orthogonal matrix P as a product
            of elementary reflectors.
          If m < n, elements below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors, and
            elements on and above the diagonal in the first NB rows,
            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) REAL array, dimension
          LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise.
          The distributed diagonal elements of the bidiagonal matrix
          B: D(i) = A(ia+i-1,ja+i-1). D is tied to the distributed
          matrix A.
  E       (local output) REAL 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(ia+i-1,ja+i) for i = 1,2,...,n-1;
          if m < n, E(i) = A(ia+i,ja+i-1) for i = 1,2,...,m-1.
          E is tied to the distributed matrix A.
  TAUQ    (local output) REAL 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) REAL 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.
  X       (local output) REAL pointer into the local memory
          to an array of dimension (LLD_X,NB). On exit, the local
          pieces of the distributed M-by-NB matrix
          X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced
          part of sub( A ).
  IX      (global input) INTEGER
          The row index in the global array X indicating the first
          row of sub( X ).
  JX      (global input) INTEGER
          The column index in the global array X indicating the
          first column of sub( X ).
  DESCX   (global and local input) INTEGER array of dimension DLEN_.
          The array descriptor for the distributed matrix X.
  Y       (local output) REAL pointer into the local memory
          to an array of dimension (LLD_Y,NB).  On exit, the local
          pieces of the distributed N-by-NB matrix
          Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced
          part of sub( A ).
  IY      (global input) INTEGER
          The row index in the global array Y indicating the first
          row of sub( Y ).
  JY      (global input) INTEGER
          The column index in the global array Y indicating the
          first column of sub( Y ).
  DESCY   (global and local input) INTEGER array of dimension DLEN_.
          The array descriptor for the distributed matrix Y.
  WORK    (local workspace) REAL array, dimension (LWORK)
          LWORK >= NB_A + NQ, with
          NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL )
          IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL )
          INDXG2P and NUMROC are ScaLAPACK tool functions;
          MYROW, MYCOL, NPROW and NPCOL can be determined by calling
          the subroutine BLACS_GRIDINFO.
  Further Details
  ===============
  The matrices Q and P are represented as products of elementary
  reflectors:
     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
  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.
  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 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).
  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1: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: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 elements of the vectors v and u together form the m-by-nb matrix
  V and the nb-by-n matrix U' which are needed, with X and Y, to apply
  the transformation to the unreduced part of the matrix, using a block
  update of the form:  sub( A ) := sub( A ) - V*Y' - X*U'.
  The contents of sub( A ) on exit are illustrated by the following
  examples with nb = 2:
  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )
  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element
  of the vector defining G(i).
  =====================================================================
     .. Parameters ..
 
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001        SUBROUTINE PSLABRD( M , N , NB , A , IA , JA , DESCA , D , E , TAUQ , TAUP ,
002       $X , IX , JX , DESCX , Y , IY , JY , DESCY , WORK )
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 , IX , IY , JA , JX , JY , M , N , NB
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        REAL ONE , ZERO
017        PARAMETER( ONE = 1.0E + 0 , ZERO = 0.0E + 0 )
018  *     ..
019  *     .. Local Scalars ..
020        INTEGER I , IACOL , IAROW , ICTXT , II , IPY , IW , J , JJ ,
021       $JWY , K , MYCOL , MYROW , NPCOL , NPROW
022        REAL ALPHA , TAU
023        INTEGER DESCD( DLEN_ ) , DESCE( DLEN_ ) ,
024       $DESCTP( DLEN_ ) , DESCTQ( DLEN_ ) ,
025       $DESCW( DLEN_ ) , DESCWY( DLEN_ )
026  *     ..
027  *     .. External Subroutines ..
028        EXTERNAL BLACS_GRIDINFO , DESCSET , INFOG2L , PSCOPY ,
029       $PSELGET , PSELSET , PSGEMV , PSLARFG ,
030       $PSSCAL
031  *     ..
032  *     .. Intrinsic Functions ..
033        INTRINSIC MIN , MOD
034  *     ..
035  *     .. Executable Statements ..
036  
037  *     Quick return if possible
038  
039        IF( M.LE.0 .OR. N.LE.0 )
040       $    RETURN
041  
042            ICTXT = DESCA( CTXT_ )
043            CALL BLACS_GRIDINFO( ICTXT , NPROW , NPCOL , MYROW , MYCOL )
044            CALL INFOG2L( IA , JA , DESCA , NPROW , NPCOL , MYROW , MYCOL , II , JJ ,
045       $    IAROW , IACOL )
046            IPY = DESCA( MB_ ) + 1
047            IW = MOD( IA - 1 , DESCA( NB_ ) ) + 1
048            ALPHA = ZERO
049  
050            CALL DESCSET( DESCWY , 1 , N + MOD( IA - 1 , DESCY( NB_ ) ) , 1 ,
051       $    DESCA( NB_ ) , IAROW , IACOL , ICTXT , 1 )
052            CALL DESCSET( DESCW , DESCA( MB_ ) , 1 , DESCA( MB_ ) , 1 , IAROW ,
053       $    IACOL , ICTXT , DESCA( MB_ ) )
054            CALL DESCSET( DESCTQ , 1 , JA + MIN(M , N) - 1 , 1 , DESCA( NB_ ) , IAROW ,
055       $    DESCA( CSRC_ ) , DESCA( CTXT_ ) , 1 )
056            CALL DESCSET( DESCTP , IA + MIN(M , N) - 1 , 1 , DESCA( MB_ ) , 1 ,
057       $    DESCA( RSRC_ ) , IACOL , DESCA( CTXT_ ) ,
058       $    DESCA( LLD_ ) )
059  
060            IF( M.GE.N ) THEN
061  
062  *             Reduce to upper bidiagonal form
063  
064                CALL DESCSET( DESCD , 1 , JA + MIN(M , N) - 1 , 1 , DESCA( NB_ ) , MYROW ,
065       $        DESCA( CSRC_ ) , DESCA( CTXT_ ) , 1 )
066                CALL DESCSET( DESCE , IA + MIN(M , N) - 1 , 1 , DESCA( MB_ ) , 1 ,
067       $        DESCA( RSRC_ ) , MYCOL , DESCA( CTXT_ ) ,
068       $        DESCA( LLD_ ) )
069                DO 10 K = 1 , NB
070                    I = IA + K - 1
071                    J = JA + K - 1
072                    JWY = IW + K
073  
074  *                 Update A(i : ia + m - 1 , j)
075  
076                    IF( K.GT.1 ) THEN
077                        CALL PSGEMV( 'No transpose' , M - K + 1 , K - 1 , - ONE , A , I , JA ,
078       $                DESCA , Y , IY , JY + K - 1 , DESCY , 1 , ONE , A , I ,
079       $                J , DESCA , 1 )
080                        CALL PSGEMV( 'No transpose' , M - K + 1 , K - 1 , - ONE , X , IX + K - 1 ,
081       $                JX , DESCX , A , IA , J , DESCA , 1 , ONE , A , I , J ,
082       $                DESCA , 1 )
083                        CALL PSELSET( A , I - 1 , J , DESCA , ALPHA )
084                    END IF
085  
086  *                 Generate reflection Q(i) to annihilate A(i + 1 : ia + m - 1 , j)
087  
088                    CALL PSLARFG ( M - K + 1 , ALPHA , I , J , A , I + 1 , J , DESCA , 1 ,
089       $            TAUQ )
090                    CALL PSELSET( D , 1 , J , DESCD , ALPHA )
091                    CALL PSELSET( A , I , J , DESCA , ONE )
092  
093  *                 Compute Y(IA + I : IA + N - 1 , J)
094  
095                    CALL PSGEMV( 'Transpose' , M - K + 1 , N - K , ONE , A , I , J + 1 , DESCA ,
096       $            A , I , J , DESCA , 1 , ZERO , WORK( IPY ) , 1 , JWY ,
097       $            DESCWY , DESCWY( M_ ) )
098                    CALL PSGEMV( 'Transpose' , M - K + 1 , K - 1 , ONE , A , I , JA , DESCA ,
099       $            A , I , J , DESCA , 1 , ZERO , WORK , IW , 1 , DESCW ,
100       $            1 )
101                    CALL PSGEMV( 'Transpose' , K - 1 , N - K , - ONE , Y , IY , JY + K ,
102       $            DESCY , WORK , IW , 1 , DESCW , 1 , ONE , WORK( IPY ) ,
103       $            1 , JWY , DESCWY , DESCWY( M_ ) )
104                    CALL PSGEMV( 'Transpose' , M - K + 1 , K - 1 , ONE , X , IX + K - 1 , JX ,
105       $            DESCX , A , I , J , DESCA , 1 , ZERO , WORK , IW , 1 ,
106       $            DESCW , 1 )
107                    CALL PSGEMV( 'Transpose' , K - 1 , N - K , - ONE , A , IA , J + 1 , DESCA ,
108       $            WORK , IW , 1 , DESCW , 1 , ONE , WORK( IPY ) , 1 ,
109       $            JWY , DESCWY , DESCWY( M_ ) )
110  
111                    CALL PSELGET( 'Rowwise' , ' ' , TAU , TAUQ , 1 , J , DESCTQ )
112                    CALL PSSCAL( N - K , TAU , WORK( IPY ) , 1 , JWY , DESCWY ,
113       $            DESCWY( M_ ) )
114                    CALL PSCOPY( N - K , WORK( IPY ) , 1 , JWY , DESCWY , DESCWY( M_ ) ,
115       $            Y , IY + K - 1 , JY + K , DESCY , DESCY( M_ ) )
116  
117  *                 Update A(i , j + 1 : ja + n - 1)
118  
119                    CALL PSGEMV( 'Transpose' , K , N - K , - ONE , Y , IY , JY + K , DESCY ,
120       $            A , I , JA , DESCA , DESCA( M_ ) , ONE , A , I , J + 1 ,
121       $            DESCA , DESCA( M_ ) )
122                    CALL PSGEMV( 'Transpose' , K - 1 , N - K , - ONE , A , IA , J + 1 , DESCA ,
123       $            X , IX + K - 1 , JX , DESCX , DESCX( M_ ) , ONE , A , I ,
124       $            J + 1 , DESCA , DESCA( M_ ) )
125                    CALL PSELSET( A , I , J , DESCA , ALPHA )
126  
127  *                 Generate reflection P(i) to annihilate A(i , j + 2 : ja + n - 1)
128  
129                    CALL PSLARFG ( N - K , ALPHA , I , J + 1 , A , I ,
130       $            MIN( J + 2 , N + JA - 1 ) , DESCA , DESCA( M_ ) , TAUP )
131                    CALL PSELSET( E , I , 1 , DESCE , ALPHA )
132                    CALL PSELSET( A , I , J + 1 , DESCA , ONE )
133  
134  *                 Compute X(I + 1 : IA + M - 1 , J)
135  
136                    CALL PSGEMV( 'No transpose' , M - K , N - K , ONE , A , I + 1 , J + 1 ,
137       $            DESCA , A , I , J + 1 , DESCA , DESCA( M_ ) , ZERO , X ,
138       $            IX + K , JX + K - 1 , DESCX , 1 )
139                    CALL PSGEMV( 'No transpose' , K , N - K , ONE , Y , IY , JY + K ,
140       $            DESCY , A , I , J + 1 , DESCA , DESCA( M_ ) , ZERO ,
141       $            WORK , IW , 1 , DESCW , 1 )
142                    CALL PSGEMV( 'No transpose' , M - K , K , - ONE , A , I + 1 , JA ,
143       $            DESCA , WORK , IW , 1 , DESCW , 1 , ONE , X , IX + K ,
144       $            JX + K - 1 , DESCX , 1 )
145                    CALL PSGEMV( 'No transpose' , K - 1 , N - K , ONE , A , IA , J + 1 ,
146       $            DESCA , A , I , J + 1 , DESCA , DESCA( M_ ) , ZERO ,
147       $            WORK , IW , 1 , DESCW , 1 )
148                    CALL PSGEMV( 'No transpose' , M - K , K - 1 , - ONE , X , IX + K , JX ,
149       $            DESCX , WORK , IW , 1 , DESCW , 1 , ONE , X , IX + K ,
150       $            JX + K - 1 , DESCX , 1 )
151  
152                    CALL PSELGET( 'Columnwise' , ' ' , TAU , TAUP , I , 1 , DESCTP )
153                    CALL PSSCAL( M - K , TAU , X , IX + K , JX + K - 1 , DESCX , 1 )
154     10         CONTINUE
155  
156            ELSE
157  
158  *             Reduce to lower bidiagonal form
159  
160                CALL DESCSET( DESCD , IA + MIN(M , N) - 1 , 1 , DESCA( MB_ ) , 1 ,
161       $        DESCA( RSRC_ ) , MYCOL , DESCA( CTXT_ ) ,
162       $        DESCA( LLD_ ) )
163                CALL DESCSET( DESCE , 1 , JA + MIN(M , N) - 1 , 1 , DESCA( NB_ ) , MYROW ,
164       $        DESCA( CSRC_ ) , DESCA( CTXT_ ) , 1 )
165                DO 20 K = 1 , NB
166                    I = IA + K - 1
167                    J = JA + K - 1
168                    JWY = IW + K
169  
170  *                 Update A(i , j : ja + n - 1)
171  
172                    IF( K.GT.1 ) THEN
173                        CALL PSGEMV( 'Transpose' , K - 1 , N - K + 1 , - ONE , Y , IY ,
174       $                JY + K - 1 , DESCY , A , I , JA , DESCA , DESCA( M_ ) ,
175       $                ONE , A , I , J , DESCA , DESCA( M_ ) )
176                        CALL PSGEMV( 'Transpose' , K - 1 , N - K + 1 , - ONE , A , IA , J ,
177       $                DESCA , X , IX + K - 1 , JX , DESCX , DESCX( M_ ) ,
178       $                ONE , A , I , J , DESCA , DESCA( M_ ) )
179                        CALL PSELSET( A , I , J - 1 , DESCA , ALPHA )
180                    END IF
181  
182  *                 Generate reflection P(i) to annihilate A(i , j + 1 : ja + n - 1)
183  
184                    CALL PSLARFG ( N - K + 1 , ALPHA , I , J , A , I , J + 1 , DESCA ,
185       $            DESCA( M_ ) , TAUP )
186                    CALL PSELSET( D , I , 1 , DESCD , ALPHA )
187                    CALL PSELSET( A , I , J , DESCA , ONE )
188  
189  *                 Compute X(i + 1 : ia + m - 1 , j)
190  
191                    CALL PSGEMV( 'No transpose' , M - K , N - K + 1 , ONE , A , I + 1 , J ,
192       $            DESCA , A , I , J , DESCA , DESCA( M_ ) , ZERO , X ,
193       $            IX + K , JX + K - 1 , DESCX , 1 )
194                    CALL PSGEMV( 'No transpose' , K - 1 , N - K + 1 , ONE , Y , IY , JY + K - 1 ,
195       $            DESCY , A , I , J , DESCA , DESCA( M_ ) , ZERO ,
196       $            WORK , IW , 1 , DESCW , 1 )
197                    CALL PSGEMV( 'No transpose' , M - K , K - 1 , - ONE , A , I + 1 , JA ,
198       $            DESCA , WORK , IW , 1 , DESCW , 1 , ONE , X , IX + K ,
199       $            JX + K - 1 , DESCX , 1 )
200                    CALL PSGEMV( 'No transpose' , K - 1 , N - K + 1 , ONE , A , IA , J ,
201       $            DESCA , A , I , J , DESCA , DESCA( M_ ) , ZERO ,
202       $            WORK , IW , 1 , DESCW , 1 )
203                    CALL PSGEMV( 'No transpose' , M - K , K - 1 , - ONE , X , IX + K , JX ,
204       $            DESCX , WORK , IW , 1 , DESCW , 1 , ONE , X , IX + K ,
205       $            JX + K - 1 , DESCX , 1 )
206  
207                    CALL PSELGET( 'Columnwise' , ' ' , TAU , TAUP , I , 1 , DESCTP )
208                    CALL PSSCAL( M - K , TAU , X , IX + K , JX + K - 1 , DESCX , 1 )
209  
210  *                 Update A(i + 1 : ia + m - 1 , j)
211  
212                    CALL PSGEMV( 'No transpose' , M - K , K - 1 , - ONE , A , I + 1 , JA ,
213       $            DESCA , Y , IY , JY + K - 1 , DESCY , 1 , ONE , A , I + 1 , J ,
214       $            DESCA , 1 )
215                    CALL PSGEMV( 'No transpose' , M - K , K , - ONE , X , IX + K , JX ,
216       $            DESCX , A , IA , J , DESCA , 1 , ONE , A , I + 1 , J ,
217       $            DESCA , 1 )
218                    CALL PSELSET( A , I , J , DESCA , ALPHA )
219  
220  *                 Generate reflection Q(i) to annihilate A(i + 2 : ia + m - 1 , j)
221  
222                    CALL PSLARFG ( M - K , ALPHA , I + 1 , J , A , MIN( I + 2 , M + IA - 1 ) ,
223       $            J , DESCA , 1 , TAUQ )
224                    CALL PSELSET( E , 1 , J , DESCE , ALPHA )
225                    CALL PSELSET( A , I + 1 , J , DESCA , ONE )
226  
227  *                 Compute Y(ia + i : ia + n - 1 , j)
228  
229                    CALL PSGEMV( 'Transpose' , M - K , N - K , ONE , A , I + 1 , J + 1 , DESCA ,
230       $            A , I + 1 , J , DESCA , 1 , ZERO , WORK( IPY ) , 1 ,
231       $            JWY , DESCWY , DESCWY( M_ ) )
232                    CALL PSGEMV( 'Transpose' , M - K , K - 1 , ONE , A , I + 1 , JA , DESCA ,
233       $            A , I + 1 , J , DESCA , 1 , ZERO , WORK , IW , 1 , DESCW ,
234       $            1 )
235                    CALL PSGEMV( 'Transpose' , K - 1 , N - K , - ONE , Y , IY , JY + K ,
236       $            DESCY , WORK , IW , 1 , DESCW , 1 , ONE , WORK( IPY ) ,
237       $            1 , JWY , DESCWY , DESCWY( M_ ) )
238                    CALL PSGEMV( 'Transpose' , M - K , K , ONE , X , IX + K , JX , DESCX ,
239       $            A , I + 1 , J , DESCA , 1 , ZERO , WORK , IW , 1 , DESCW ,
240       $            1 )
241                    CALL PSGEMV( 'Transpose' , K , N - K , - ONE , A , IA , J + 1 , DESCA ,
242       $            WORK , IW , 1 , DESCW , 1 , ONE , WORK( IPY ) , 1 ,
243       $            JWY , DESCWY , DESCWY( M_ ) )
244  
245                    CALL PSELGET( 'Rowwise' , ' ' , TAU , TAUQ , 1 , J , DESCTQ )
246                    CALL PSSCAL( N - K , TAU , WORK( IPY ) , 1 , JWY , DESCWY ,
247       $            DESCWY( M_ ) )
248                    CALL PSCOPY( N - K , WORK( IPY ) , 1 , JWY , DESCWY , DESCWY( M_ ) ,
249       $            Y , IY + K - 1 , JY + K , DESCY , DESCY( M_ ) )
250     20         CONTINUE
251            END IF
252  
253            RETURN
254  
255  *         End of PSLABRD
256  
257        END