*> \brief \b STZRZF
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE STZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> STZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
*> to upper triangular form by means of orthogonal transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*> A = ( R 0 ) * Z,
*>
*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
*> triangular matrix.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> On entry, the leading M-by-N upper trapezoidal part of the
*> array A must contain the matrix to be factorized.
*> On exit, the leading M-by-M upper triangular part of A
*> contains the upper triangular matrix R, and elements M+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is REAL array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M).
*> For optimum performance LWORK >= M*NB, where NB is
*> the optimal blocksize.
*>
*> 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.
*> \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.
*
*> \date April 2012
*
*> \ingroup realOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The N-by-N matrix Z can be computed by
*>
*> Z = Z(1)*Z(2)* ... *Z(M)
*>
*> where each N-by-N Z(k) is given by
*>
*> Z(k) = I - tau(k)*v(k)*v(k)**T
*>
*> with v(k) is the kth row vector of the M-by-N matrix
*>
*> V = ( I A(:,M+1:N) )
*>
*> I is the M-by-M identity matrix, A(:,M+1:N)
*> is the output stored in A on exit from DTZRZF,
*> and tau(k) is the kth element of the array TAU.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE STZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IWS, KI, KK, LDWORK, LWKMIN, LWKOPT,
$ M1, MU, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, SLARZB, SLARZT, SLATRZ
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.EQ.0 .OR. M.EQ.N ) THEN
LWKOPT = 1
LWKMIN = 1
ELSE
*
* Determine the block size.
*
NB = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 )
LWKOPT = M*NB
LWKMIN = MAX( 1, M )
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'STZRZF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 ) THEN
RETURN
ELSE IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
RETURN
END IF
*
NBMIN = 2
NX = 1
IWS = M
IF( NB.GT.1 .AND. NB.LT.M ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'SGERQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.M ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'SGERQF', ' ', M, N, -1,
$ -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN
*
* Use blocked code initially.
* The last kk rows are handled by the block method.
*
M1 = MIN( M+1, N )
KI = ( ( M-NX-1 ) / NB )*NB
KK = MIN( M, KI+NB )
*
DO 20 I = M - KK + KI + 1, M - KK + 1, -NB
IB = MIN( M-I+1, NB )
*
* Compute the TZ factorization of the current block
* A(i:i+ib-1,i:n)
*
CALL SLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ),
$ WORK )
IF( I.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL SLARZT( 'Backward', 'Rowwise', N-M, IB, A( I, M1 ),
$ LDA, TAU( I ), WORK, LDWORK )
*
* Apply H to A(1:i-1,i:n) from the right
*
CALL SLARZB( 'Right', 'No transpose', 'Backward',
$ 'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ),
$ LDA, WORK, LDWORK, A( 1, I ), LDA,
$ WORK( IB+1 ), LDWORK )
END IF
20 CONTINUE
MU = I + NB - 1
ELSE
MU = M
END IF
*
* Use unblocked code to factor the last or only block
*
IF( MU.GT.0 )
$ CALL SLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of STZRZF
*
END