*> \brief \b DGEBRD
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBRD reduces a general real M-by-N matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
*>
*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N general matrix to be reduced.
*> On exit,
*> if m >= n, the diagonal and the first superdiagonal are
*> overwritten with the upper bidiagonal matrix B; the
*> elements below the diagonal, with the array TAUQ, represent
*> the orthogonal matrix Q as a product of elementary
*> reflectors, and the elements above the first superdiagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors;
*> if m < n, the diagonal and the first subdiagonal are
*> overwritten with the lower bidiagonal matrix B; the
*> elements below the first subdiagonal, with the array TAUQ,
*> represent the orthogonal matrix Q as a product of
*> elementary reflectors, and the elements above the diagonal,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
*> The off-diagonal elements of the bidiagonal matrix B:
*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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 length of the array WORK. LWORK >= max(1,M,N).
*> For optimum performance LWORK >= (M+N)*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.
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> If m >= n,
*>
*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n,
*>
*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors;
*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
*> tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The contents of A on exit are illustrated by the following examples:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
*> ( v1 v2 v3 v4 v5 )
*>
*> where d and e denote diagonal and off-diagonal elements of B, vi
*> denotes an element of the vector defining H(i), and ui an element of
*> the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
$ 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, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
$ NBMIN, NX, WS
* ..
* .. External Subroutines ..
EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
LWKOPT = ( M+N )*NB
WORK( 1 ) = DBLE( LWKOPT )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'DGEBRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
WS = MAX( M, N )
LDWRKX = M
LDWRKY = N
*
IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
*
* Set the crossover point NX.
*
NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
*
* Determine when to switch from blocked to unblocked code.
*
IF( NX.LT.MINMN ) THEN
WS = ( M+N )*NB
IF( LWORK.LT.WS ) THEN
*
* Not enough work space for the optimal NB, consider using
* a smaller block size.
*
NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
IF( LWORK.GE.( M+N )*NBMIN ) THEN
NB = LWORK / ( M+N )
ELSE
NB = 1
NX = MINMN
END IF
END IF
END IF
ELSE
NX = MINMN
END IF
*
DO 30 I = 1, MINMN - NX, NB
*
* Reduce rows and columns i:i+nb-1 to bidiagonal form and return
* the matrices X and Y which are needed to update the unreduced
* part of the matrix
*
CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
$ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
$ WORK( LDWRKX*NB+1 ), LDWRKY )
*
* Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
* of the form A := A - V*Y**T - X*U**T
*
CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
$ NB, -ONE, A( I+NB, I ), LDA,
$ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
$ A( I+NB, I+NB ), LDA )
CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
$ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
$ ONE, A( I+NB, I+NB ), LDA )
*
* Copy diagonal and off-diagonal elements of B back into A
*
IF( M.GE.N ) THEN
DO 10 J = I, I + NB - 1
A( J, J ) = D( J )
A( J, J+1 ) = E( J )
10 CONTINUE
ELSE
DO 20 J = I, I + NB - 1
A( J, J ) = D( J )
A( J+1, J ) = E( J )
20 CONTINUE
END IF
30 CONTINUE
*
* Use unblocked code to reduce the remainder of the matrix
*
CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAUQ( I ), TAUP( I ), WORK, IINFO )
WORK( 1 ) = WS
RETURN
*
* End of DGEBRD
*
END