*> \brief \b DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEQRT2 computes a QR factorization of a real M-by-N matrix A,
*> using the compact WY representation of Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \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 geqrt2
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEQRT2( M, N, A, LDA, 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, LDT, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER( ONE = 1.0D+00, ZERO = 0.0D+00 )
* ..
* .. Local Scalars ..
INTEGER I, K
DOUBLE PRECISION AII, ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DGEMV, DGER, DTRMV, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( M.LT.N ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO I = 1, K
*
* Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1)
*
CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ T( I, 1 ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(I:M,I+1:N) from the left
*
AII = A( I, I )
A( I, I ) = ONE
*
* W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)]
*
CALL DGEMV( 'T',M-I+1, N-I, ONE, A( I, I+1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, N ), 1 )
*
* A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H
*
ALPHA = -(T( I, 1 ))
CALL DGER( M-I+1, N-I, ALPHA, A( I, I ), 1,
$ T( 1, N ), 1, A( I, I+1 ), LDA )
A( I, I ) = AII
END IF
END DO
*
DO I = 2, N
AII = A( I, I )
A( I, I ) = ONE
*
* T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I)
*
ALPHA = -T( I, 1 )
CALL DGEMV( 'T', M-I+1, I-1, ALPHA, A( I, 1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, I ), 1 )
A( I, I ) = AII
*
* T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
CALL DTRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
* T(I,I) = tau(I)
*
T( I, I ) = T( I, 1 )
T( I, 1) = ZERO
END DO
*
* End of DGEQRT2
*
END