ztplqt2.f

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*> \brief \b ZTPLQT2 computes a LQ factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZTPLQT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER   INFO, LDA, LDB, LDT, N, M, L
*       ..
*       .. Array Arguments ..
*       COMPLEX*16   A( LDA, * ), B( LDB, * ), T( LDT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZTPLQT2 computes a LQ a factorization of a complex "triangular-pentagonal"
*> matrix C, which is composed of a triangular block A and pentagonal block B,
*> using the compact WY representation for Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The total number of rows of the matrix B.
*>          M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix B, and the order of
*>          the triangular matrix A.
*>          N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is INTEGER
*>          The number of rows of the lower trapezoidal part of B.
*>          MIN(M,N) >= L >= 0.  See Further Details.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA,M)
*>          On entry, the lower triangular M-by-M matrix A.
*>          On exit, the elements on and below the diagonal of the array
*>          contain the lower triangular matrix L.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension (LDB,N)
*>          On entry, the pentagonal M-by-N matrix B.  The first N-L columns
*>          are rectangular, and the last L columns are lower trapezoidal.
*>          On exit, B contains the pentagonal matrix V.  See Further Details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is COMPLEX*16 array, dimension (LDT,M)
*>          The N-by-N upper triangular factor T of the block reflector.
*>          See Further Details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= max(1,M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tplqt2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The input matrix C is a M-by-(M+N) matrix
*>
*>               C = [ A ][ B ]
*>
*>
*>  where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
*>  matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
*>  upper trapezoidal matrix B2:
*>
*>               B = [ B1 ][ B2 ]
*>                   [ B1 ]  <-     M-by-(N-L) rectangular
*>                   [ B2 ]  <-     M-by-L lower trapezoidal.
*>
*>  The lower trapezoidal matrix B2 consists of the first L columns of a
*>  N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
*>  B is rectangular M-by-N; if M=L=N, B is lower triangular.
*>
*>  The matrix W stores the elementary reflectors H(i) in the i-th row
*>  above the diagonal (of A) in the M-by-(M+N) input matrix C
*>
*>               C = [ A ][ B ]
*>                   [ A ]  <- lower triangular M-by-M
*>                   [ B ]  <- M-by-N pentagonal
*>
*>  so that W can be represented as
*>
*>               W = [ I ][ V ]
*>                   [ I ]  <- identity, M-by-M
*>                   [ V ]  <- M-by-N, same form as B.
*>
*>  Thus, all of information needed for W is contained on exit in B, which
*>  we call V above.  Note that V has the same form as B; that is,
*>
*>               W = [ V1 ][ V2 ]
*>                   [ V1 ] <-     M-by-(N-L) rectangular
*>                   [ V2 ] <-     M-by-L lower trapezoidal.
*>
*>  The rows of V represent the vectors which define the H(i)'s.
*>  The (M+N)-by-(M+N) block reflector H is then given by
*>
*>               H = I - W**T * T * W
*>
*>  where W^H is the conjugate transpose of W and T is the upper triangular
*>  factor of the block reflector.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE ztplqt2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER        INFO, LDA, LDB, LDT, N, M, L
*     ..
*     .. Array Arguments ..
      COMPLEX*16     A( LDA, * ), B( LDB, * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16  ONE, ZERO
      parameter( zero = ( 0.0d+0, 0.0d+0 ),one  = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER   I, J, P, MP, NP
      COMPLEX*16   ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL  zlarfg, zgemv, zgerc, ztrmv, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      ELSE IF( ldb.LT.max( 1, m ) ) THEN
         info = -7
      ELSE IF( ldt.LT.max( 1, m ) ) THEN
         info = -9
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZTPLQT2', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
*
      DO i = 1, m
*
*        Generate elementary reflector H(I) to annihilate B(I,:)
*
         p = n-l+min( l, i )
         CALL zlarfg( p+1, a( i, i ), b( i, 1 ), ldb, t( 1, i ) )
         t(1,i)=conjg(t(1,i))
         IF( i.LT.m ) THEN
            DO j = 1, p
               b( i, j ) = conjg(b(i,j))
            END DO
*
*           W(M-I:1) := C(I+1:M,I:N) * C(I,I:N) [use W = T(M,:)]
*
            DO j = 1, m-i
               t( m, j ) = (a( i+j, i ))
            END DO
            CALL zgemv( 'N', m-i, p, one, b( i+1, 1 ), ldb,
     $                  b( i, 1 ), ldb, one, t( m, 1 ), ldt )
*
*           C(I+1:M,I:N) = C(I+1:M,I:N) + alpha * C(I,I:N)*W(M-1:1)^H
*
            alpha = -(t( 1, i ))
            DO j = 1, m-i
               a( i+j, i ) = a( i+j, i ) + alpha*(t( m, j ))
            END DO
            CALL zgerc( m-i, p, (alpha),  t( m, 1 ), ldt,
     $          b( i, 1 ), ldb, b( i+1, 1 ), ldb )
            DO j = 1, p
               b( i, j ) = conjg(b(i,j))
            END DO
         END IF
      END DO
*
      DO i = 2, m
*
*        T(I,1:I-1) := C(I:I-1,1:N)**H * (alpha * C(I,I:N))
*
         alpha = -(t( 1, i ))
         DO j = 1, i-1
            t( i, j ) = zero
         END DO
         p = min( i-1, l )
         np = min( n-l+1, n )
         mp = min( p+1, m )
         DO j = 1, n-l+p
           b(i,j)=conjg(b(i,j))
         END DO
*
*        Triangular part of B2
*
         DO j = 1, p
            t( i, j ) = (alpha*b( i, n-l+j ))
         END DO
         CALL ztrmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
     $               t( i, 1 ), ldt )
*
*        Rectangular part of B2
*
         CALL zgemv( 'N', i-1-p, l,  alpha, b( mp, np ), ldb,
     $               b( i, np ), ldb, zero, t( i,mp ), ldt )
*
*        B1
 
*
         CALL zgemv( 'N', i-1, n-l, alpha, b, ldb, b( i, 1 ), ldb,
     $               one, t( i, 1 ), ldt )
*
 
*
*        T(1:I-1,I) := T(1:I-1,1:I-1) * T(I,1:I-1)
*
         DO j = 1, i-1
            t(i,j)=conjg(t(i,j))
         END DO
         CALL ztrmv( 'L', 'C', 'N', i-1, t, ldt, t( i, 1 ), ldt )
         DO j = 1, i-1
            t(i,j)=conjg(t(i,j))
         END DO
         DO j = 1, n-l+p
            b(i,j)=conjg(b(i,j))
         END DO
*
*        T(I,I) = tau(I)
*
         t( i, i ) = t( 1, i )
         t( 1, i ) = zero
      END DO
      DO i=1,m
         DO j= i+1,m
            t(i,j)=(t(j,i))
            t(j,i)=zero
         END DO
      END DO
 
*
*     End of ZTPLQT2
*
      END