zunhr_col()

subroutine zunhr_col	(	integer	m,
		integer	n,
		integer	nb,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldt, * )	t,
		integer	ldt,
		complex*16, dimension( * )	d,
		integer	info )

ZUNHR_COL

Download ZUNHR_COL + dependencies [TGZ] [ZIP] [TXT]</a \par Purpose: @verbatim !> !> ZUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns !> as input, stored in A, and performs Householder Reconstruction (HR), !> i.e. reconstructs Householder vectors V(i) implicitly representing !> another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, !> where S is an N-by-N diagonal matrix with diagonal entries !> equal to +1 or -1. The Householder vectors (columns V(i) of V) are !> stored in A on output, and the diagonal entries of S are stored in D. !> Block reflectors are also returned in T !> (same output format as ZGEQRT). !> \endverbatim @param M @verbatim !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> \endverbatim @param N @verbatim !> N is INTEGER !> The number of columns of the matrix A. M >= N >= 0. !> \endverbatim @param NB @verbatim !> NB is INTEGER !> The column block size to be used in the reconstruction !> of Householder column vector blocks in the array A and !> corresponding block reflectors in the array T. NB >= 1. !> (Note that if NB > N, then N is used instead of NB !> as the column block size.) !> \endverbatim @param A @verbatim !> A is COMPLEX*16 array, dimension (LDA,N) !> !> On entry: !> !> The array A contains an M-by-N orthonormal matrix Q_in, !> i.e the columns of A are orthogonal unit vectors. !> !> On exit: !> !> The elements below the diagonal of A represent the unit !> lower-trapezoidal matrix V of Householder column vectors !> V(i). The unit diagonal entries of V are not stored !> (same format as the output below the diagonal in A from !> ZGEQRT). The matrix T and the matrix V stored on output !> in A implicitly define Q_out. !> !> The elements above the diagonal contain the factor U !> of the LU-decomposition: !> Q_in - ( S ) = V * U !> ( 0 ) !> where 0 is a (M-N)-by-(M-N) zero matrix. !> \endverbatim @param LDA @verbatim !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> \endverbatim @param T @verbatim !> T is COMPLEX*16 array, !> dimension (LDT, N) !> !> Let NOCB = Number_of_output_col_blocks !> = CEIL(N/NB) !> !> On exit, T(1:NB, 1:N) contains NOCB upper-triangular !> block reflectors used to define Q_out stored in compact !> form as a sequence of upper-triangular NB-by-NB column !> blocks (same format as the output T in ZGEQRT). !> The matrix T and the matrix V stored on output in A !> implicitly define Q_out. NOTE: The lower triangles !> below the upper-triangular blocks will be filled with !> zeros. See Further Details. !> \endverbatim @param LDT @verbatim !> LDT is INTEGER !> The leading dimension of the array T. !> LDT >= max(1,min(NB,N)). !> \endverbatim @param D @verbatim !> D is COMPLEX*16 array, dimension min(M,N). !> The elements can be only plus or minus one. !> !> D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where !> 1 <= i <= min(M,N), and Q_in_i is Q_in after performing !> i-1 steps of “modified” Gaussian elimination. !> See Further Details. !> \endverbatim @param INFO @verbatim !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> \endverbatim \par Further Details: @verbatim !> !> The computed M-by-M unitary factor Q_out is defined implicitly as !> a product of unitary matrices Q_out(i). Each Q_out(i) is stored in !> the compact WY-representation format in the corresponding blocks of !> matrices V (stored in A) and T. !> !> The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N !> matrix A contains the column vectors V(i) in NB-size column !> blocks VB(j). For example, VB(1) contains the columns !> V(1), V(2), ... V(NB). NOTE: The unit entries on !> the diagonal of Y are not stored in A. !> !> The number of column blocks is !> !> NOCB = Number_of_output_col_blocks = CEIL(N/NB) !> !> where each block is of order NB except for the last block, which !> is of order LAST_NB = N - (NOCB-1)*NB. !> !> For example, if M=6, N=5 and NB=2, the matrix V is !> !> !> V = ( VB(1), VB(2), VB(3) ) = !> !> = ( 1 ) !> ( v21 1 ) !> ( v31 v32 1 ) !> ( v41 v42 v43 1 ) !> ( v51 v52 v53 v54 1 ) !> ( v61 v62 v63 v54 v65 ) !> !> !> For each of the column blocks VB(i), an upper-triangular block !> reflector TB(i) is computed. These blocks are stored as !> a sequence of upper-triangular column blocks in the NB-by-N !> matrix T. The size of each TB(i) block is NB-by-NB, except !> for the last block, whose size is LAST_NB-by-LAST_NB. !> !> For example, if M=6, N=5 and NB=2, the matrix T is !> !> T = ( TB(1), TB(2), TB(3) ) = !> !> = ( t11 t12 t13 t14 t15 ) !> ( t22 t24 ) !> !> !> The M-by-M factor Q_out is given as a product of NOCB !> unitary M-by-M matrices Q_out(i). !> !> Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB), !> !> where each matrix Q_out(i) is given by the WY-representation !> using corresponding blocks from the matrices V and T: !> !> Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T, !> !> where I is the identity matrix. Here is the formula with matrix !> dimensions: !> !> Q(i){M-by-M} = I{M-by-M} - !> VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M}, !> !> where INB = NB, except for the last block NOCB !> for which INB=LAST_NB. !> !> ===== !> NOTE: !> ===== !> !> If Q_in is the result of doing a QR factorization !> B = Q_in * R_in, then: !> !> B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out. !> !> So if one wants to interpret Q_out as the result !> of the QR factorization of B, then the corresponding R_out !> should be equal to R_out = S * R_in, i.e. some rows of R_in !> should be multiplied by -1. !> !> For the details of the algorithm, see [1]. !> !> [1] , !> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, !> E. Solomonik, J. Parallel Distrib. Comput., !> vol. 85, pp. 3-31, 2015. !> \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \par Contributors: @verbatim !> !> November 2019, Igor Kozachenko, !> Computer Science Division, !> University of California, Berkeley !> !> \endverbatim

Definition at line 256 of file zunhr_col.f.

      IMPLICIT NONE
*
*  -- 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, LDT, M, N, NB
*     ..
*     .. Array Arguments ..
      COMPLEX*16        A( LDA, * ), D( * ), T( LDT, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         CONE, CZERO
      parameter( cone = ( 1.0d+0, 0.0d+0 ),
     $                     czero = ( 0.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
     $                   NPLUSONE
*     ..
*     .. External Subroutines ..
      EXTERNAL           zcopy, zlaunhr_col_getrfnp, zscal,
     $                   ztrsm,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      info = 0
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 .OR. n.GT.m ) THEN
         info = -2
      ELSE IF( nb.LT.1 ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      ELSE IF( ldt.LT.max( 1, min( nb, n ) ) ) THEN
         info = -7
      END IF
*
*     Handle error in the input parameters.
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZUNHR_COL', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( min( m, n ).EQ.0 ) THEN
         RETURN
      END IF
*
*     On input, the M-by-N matrix A contains the unitary
*     M-by-N matrix Q_in.
*
*     (1) Compute the unit lower-trapezoidal V (ones on the diagonal
*     are not stored) by performing the "modified" LU-decomposition.
*
*     Q_in - ( S ) = V * U = ( V1 ) * U,
*            ( 0 )           ( V2 )
*
*     where 0 is an (M-N)-by-N zero matrix.
*
*     (1-1) Factor V1 and U.
 
      CALL zlaunhr_col_getrfnp( n, n, a, lda, d, iinfo )
*
*     (1-2) Solve for V2.
*
      IF( m.GT.n ) THEN
         CALL ztrsm( 'R', 'U', 'N', 'N', m-n, n, cone, a, lda,
     $               a( n+1, 1 ), lda )
      END IF
*
*     (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
*     as a sequence of upper-triangular blocks with NB-size column
*     blocking.
*
*     Loop over the column blocks of size NB of the array A(1:M,1:N)
*     and the array T(1:NB,1:N), JB is the column index of a column
*     block, JNB is the column block size at each step JB.
*
      nplusone = n + 1
      DO jb = 1, n, nb
*
*        (2-0) Determine the column block size JNB.
*
         jnb = min( nplusone-jb, nb )
*
*        (2-1) Copy the upper-triangular part of the current JNB-by-JNB
*        diagonal block U(JB) (of the N-by-N matrix U) stored
*        in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
*        of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
*        column-by-column, total JNB*(JNB+1)/2 elements.
*
         jbtemp1 = jb - 1
         DO j = jb, jb+jnb-1
            CALL zcopy( j-jbtemp1, a( jb, j ), 1, t( 1, j ), 1 )
         END DO
*
*        (2-2) Perform on the upper-triangular part of the current
*        JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
*        in T(1:JNB,JB:JB+JNB-1) the following operation in place:
*        (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
*        triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
*        of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
*        diagonal block S(JB) of the N-by-N sign matrix S from the
*        right means changing the sign of each J-th column of the block
*        U(JB) according to the sign of the diagonal element of the block
*        S(JB), i.e. S(J,J) that is stored in the array element D(J).
*
         DO j = jb, jb+jnb-1
            IF( d( j ).EQ.cone ) THEN
               CALL zscal( j-jbtemp1, -cone, t( 1, j ), 1 )
            END IF
         END DO
*
*        (2-3) Perform the triangular solve for the current block
*        matrix X(JB):
*
*               X(JB) * (A(JB)**T) = B(JB), where:
*
*               A(JB)**T  is a JNB-by-JNB unit upper-triangular
*                         coefficient block, and A(JB)=V1(JB), which
*                         is a JNB-by-JNB unit lower-triangular block
*                         stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
*                         The N-by-N matrix V1 is the upper part
*                         of the M-by-N lower-trapezoidal matrix V
*                         stored in A(1:M,1:N);
*
*               B(JB)     is a JNB-by-JNB  upper-triangular right-hand
*                         side block, B(JB) = (-1)*U(JB)*S(JB), and
*                         B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
*
*               X(JB)     is a JNB-by-JNB upper-triangular solution
*                         block, X(JB) is the upper-triangular block
*                         reflector T(JB), and X(JB) is stored
*                         in T(1:JNB,JB:JB+JNB-1).
*
*             In other words, we perform the triangular solve for the
*             upper-triangular block T(JB):
*
*               T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
*
*             Even though the blocks X(JB) and B(JB) are upper-
*             triangular, the routine ZTRSM will access all JNB**2
*             elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
*             we need to set to zero the elements of the block
*             T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
*             to ZTRSM.
*
*        (2-3a) Set the elements to zero.
*
         jbtemp2 = jb - 2
         DO j = jb, jb+jnb-2
            DO i = j-jbtemp2, min( nb, n )
               t( i, j ) = czero
            END DO
         END DO
*
*        (2-3b) Perform the triangular solve.
*
         CALL ztrsm( 'R', 'L', 'C', 'U', jnb, jnb, cone,
     $               a( jb, jb ), lda, t( 1, jb ), ldt )
*
      END DO
*
      RETURN
*
*     End of ZUNHR_COL
*

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