zgetsqrhrt()

subroutine zgetsqrhrt	(	integer	m,
		integer	n,
		integer	mb1,
		integer	nb1,
		integer	nb2,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldt, * )	t,
		integer	ldt,
		complex*16, dimension( * )	work,
		integer	lwork,
		integer	info )

ZGETSQRHRT

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Purpose:

!>
!> ZGETSQRHRT computes a NB2-sized column blocked QR-factorization
!> of a complex M-by-N matrix A with M >= N,
!>
!>    A = Q * R.
!>
!> The routine uses internally a NB1-sized column blocked and MB1-sized
!> row blocked TSQR-factorization and perfors the reconstruction
!> of the Householder vectors from the TSQR output. The routine also
!> converts the R_tsqr factor from the TSQR-factorization output into
!> the R factor that corresponds to the Householder QR-factorization,
!>
!>    A = Q_tsqr * R_tsqr = Q * R.
!>
!> The output Q and R factors are stored in the same format as in ZGEQRT
!> (Q is in blocked compact WY-representation). See the documentation
!> of ZGEQRT for more details on the format.
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. M >= N >= 0. !>
[in]	MB1	!> MB1 is INTEGER !> The row block size to be used in the blocked TSQR. !> MB1 > N. !>
[in]	NB1	!> NB1 is INTEGER !> The column block size to be used in the blocked TSQR. !> N >= NB1 >= 1. !>
[in]	NB2	!> NB2 is INTEGER !> The block size to be used in the blocked QR that is !> output. NB2 >= 1. !>
[in,out]	A	!> A is COMPLEX*16 array, dimension (LDA,N) !> !> On entry: an M-by-N matrix A. !> !> On exit: !> a) the elements on and above the diagonal !> of the array contain the N-by-N upper-triangular !> matrix R corresponding to the Householder QR; !> b) the elements below the diagonal represent Q by !> the columns of blocked V (compact WY-representation). !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[out]	T	!> T is COMPLEX*16 array, dimension (LDT,N)) !> The upper triangular block reflectors stored in compact form !> as a sequence of upper triangular blocks. !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of the array T. LDT >= NB2. !>
[out]	WORK	!> (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. !> If MIN(M,N) = 0, LWORK >= 1, else !> LWORK >= MAX( 1, LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) ), !> where !> NUM_ALL_ROW_BLOCKS = CEIL((M-N)/(MB1-N)), !> NB1LOCAL = MIN(NB1,N). !> LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL, !> LW1 = NB1LOCAL * N, !> LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) ). !> !> If LWORK = -1, then a workspace query is assumed. !> The routine only calculates the optimal size of the WORK !> array, returns this value as the first entry of the WORK !> array, and no error message related to LWORK is issued !> by XERBLA. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

!>
!> November 2020, Igor Kozachenko,
!>                Computer Science Division,
!>                University of California, Berkeley
!>
!> 

Definition at line 178 of file zgetsqrhrt.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, LWORK, M, N, NB1, NB2, MB1
*     ..
*     .. Array Arguments ..
      COMPLEX*16        A( LDA, * ), T( LDT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         CONE
      parameter( cone = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IINFO, J, LW1, LW2, LWT, LDWT, LWORKOPT,
     $                   NB1LOCAL, NB2LOCAL, NUM_ALL_ROW_BLOCKS
*     ..
*     .. External Subroutines ..
      EXTERNAL           zcopy, zlatsqr, zungtsqr_row,
     $                   zunhr_col,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ceiling, dble, dcmplx, max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      lquery = ( lwork.EQ.-1 )
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
         info = -2
      ELSE IF( mb1.LE.n ) THEN
         info = -3
      ELSE IF( nb1.LT.1 ) THEN
         info = -4
      ELSE IF( nb2.LT.1 ) THEN
         info = -5
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -7
      ELSE IF( ldt.LT.max( 1, min( nb2, n ) ) ) THEN
         info = -9
      ELSE
*
*        Test the input LWORK for the dimension of the array WORK.
*        This workspace is used to store array:
*        a) Matrix T and WORK for ZLATSQR;
*        b) N-by-N upper-triangular factor R_tsqr;
*        c) Matrix T and array WORK for ZUNGTSQR_ROW;
*        d) Diagonal D for ZUNHR_COL.
*
         IF( lwork.LT.n*n+1 .AND. .NOT.lquery ) THEN
            info = -11
         ELSE
*
*           Set block size for column blocks
*
            nb1local = min( nb1, n )
*
            num_all_row_blocks = max( 1,
     $                   ceiling( dble( m - n ) / dble( mb1 - n ) ) )
*
*           Length and leading dimension of WORK array to place
*           T array in TSQR.
*
            lwt = num_all_row_blocks * n * nb1local
 
            ldwt = nb1local
*
*           Length of TSQR work array
*
            lw1 = nb1local * n
*
*           Length of ZUNGTSQR_ROW work array.
*
            lw2 = nb1local * max( nb1local, ( n - nb1local ) )
*
            lworkopt = max( lwt + lw1, max( lwt+n*n+lw2, lwt+n*n+n ) )
            lworkopt = max( 1, lworkopt )
*
            IF( lwork.LT.lworkopt .AND. .NOT.lquery ) THEN
               info = -11
            END IF
*
         END IF
      END IF
*
*     Handle error in the input parameters and return workspace query.
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZGETSQRHRT', -info )
         RETURN
      ELSE IF ( lquery ) THEN
         work( 1 ) = dcmplx( lworkopt )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( min( m, n ).EQ.0 ) THEN
         work( 1 ) = dcmplx( lworkopt )
         RETURN
      END IF
*
      nb2local = min( nb2, n )
*
*
*     (1) Perform TSQR-factorization of the M-by-N matrix A.
*
      CALL zlatsqr( m, n, mb1, nb1local, a, lda, work, ldwt,
     $              work(lwt+1), lw1, iinfo )
*
*     (2) Copy the factor R_tsqr stored in the upper-triangular part
*         of A into the square matrix in the work array
*         WORK(LWT+1:LWT+N*N) column-by-column.
*
      DO j = 1, n
         CALL zcopy( j, a( 1, j ), 1, work( lwt + n*(j-1)+1 ), 1 )
      END DO
*
*     (3) Generate a M-by-N matrix Q with orthonormal columns from
*     the result stored below the diagonal in the array A in place.
*
 
      CALL zungtsqr_row( m, n, mb1, nb1local, a, lda, work, ldwt,
     $                   work( lwt+n*n+1 ), lw2, iinfo )
*
*     (4) Perform the reconstruction of Householder vectors from
*     the matrix Q (stored in A) in place.
*
      CALL zunhr_col( m, n, nb2local, a, lda, t, ldt,
     $                work( lwt+n*n+1 ), iinfo )
*
*     (5) Copy the factor R_tsqr stored in the square matrix in the
*     work array WORK(LWT+1:LWT+N*N) into the upper-triangular
*     part of A.
*
*     (6) Compute from R_tsqr the factor R_hr corresponding to
*     the reconstructed Householder vectors, i.e. R_hr = S * R_tsqr.
*     This multiplication by the sign matrix S on the left means
*     changing the sign of I-th row of the matrix R_tsqr according
*     to sign of the I-th diagonal element DIAG(I) of the matrix S.
*     DIAG is stored in WORK( LWT+N*N+1 ) from the ZUNHR_COL output.
*
*     (5) and (6) can be combined in a single loop, so the rows in A
*     are accessed only once.
*
      DO i = 1, n
         IF( work( lwt+n*n+i ).EQ.-cone ) THEN
            DO j = i, n
               a( i, j ) = -cone * work( lwt+n*(j-1)+i )
            END DO
         ELSE
            CALL zcopy( n-i+1, work(lwt+n*(i-1)+i), n, a( i, i ),
     $                  lda )
         END IF
      END DO
*
      work( 1 ) = dcmplx( lworkopt )
      RETURN
*
*     End of ZGETSQRHRT
*

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