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SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DTZRZF 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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= M.
*
* A (input/output) DOUBLE PRECISION 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) DOUBLE PRECISION array, dimension (M)
* The scalar factors of the elementary reflectors.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) 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.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* Based on contributions by
* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
* The factorization is obtained by Householder's method. The kth
* transformation matrix, Z( k ), which is used to introduce zeros into
* the ( m - k + 1 )th row of A, is given in the form
*
* Z( k ) = ( I 0 ),
* ( 0 T( k ) )
*
* where
*
* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
* ( 0 )
* ( z( k ) )
*
* tau is a scalar and z( k ) is an ( n - m ) element vector.
* tau and z( k ) are chosen to annihilate the elements of the kth row
* of X.
*
* The scalar tau is returned in the kth element of TAU and the vector
* u( k ) in the kth row of A, such that the elements of z( k ) are
* in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
* the upper triangular part of A.
*
* Z is given by
*
* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, IWS, KI, KK, LDWORK, LWKOPT, M1, MU, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARZB, DLARZT, DLATRZ, XERBLA
* ..
* .. 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
ELSE
*
* Determine the block size.
*
NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
LWKOPT = M*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTZRZF', -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, 'DGERQF', ' ', 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, 'DGERQF', ' ', 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 DLATRZ( 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 DLARZT( '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 DLARZB( '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 DLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DTZRZF
*
END
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