*> \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* .. Scalar Arguments ..
* INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
* $ Y( LDY, NB )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an orthogonal similarity transformation
*> Q**T * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
*>
*> This is an auxiliary routine called by DGEHRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The offset for the reduction. Elements below the k-th
*> subdiagonal in the first NB columns are reduced to zero.
*> K < N.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
*> On entry, the n-by-(n-k+1) general matrix A.
*> On exit, the elements on and above the k-th subdiagonal in
*> the first NB columns are overwritten with the corresponding
*> elements of the reduced matrix; the elements below the k-th
*> subdiagonal, with the array TAU, represent the matrix Q as a
*> product of elementary reflectors. The other columns of A are
*> unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (NB)
*> The scalar factors of the elementary reflectors. See Further
*> Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NB)
*> The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is DOUBLE PRECISION array, dimension (LDY,NB)
*> The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of nb elementary reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*> A(i+k+1:n,i), and tau in TAU(i).
*>
*> The elements of the vectors v together form the (n-k+1)-by-nb matrix
*> V which is needed, with T and Y, to apply the transformation to the
*> unreduced part of the matrix, using an update of the form:
*> A := (I - V*T*V**T) * (A - Y*V**T).
*>
*> The contents of A on exit are illustrated by the following example
*> with n = 7, k = 3 and nb = 2:
*>
*> ( a a a a a )
*> ( a a a a a )
*> ( a a a a a )
*> ( h h a a a )
*> ( v1 h a a a )
*> ( v1 v2 a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix A, h denotes a
*> modified element of the upper Hessenberg matrix H, and vi denotes an
*> element of the vector defining H(i).
*>
*> This subroutine is a slight modification of LAPACK-3.0's DLAHRD
*> incorporating improvements proposed by Quintana-Orti and Van de
*> Gejin. Note that the entries of A(1:K,2:NB) differ from those
*> returned by the original LAPACK-3.0's DLAHRD routine. (This
*> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
*> \endverbatim
*
*> \par References:
* ================
*>
*> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
*> performance of reduction to Hessenberg form," ACM Transactions on
*> Mathematical Software, 32(2):180-194, June 2006.
*>
* =====================================================================
SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
* -- LAPACK auxiliary 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..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER K, LDA, LDT, LDY, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
$ Y( LDY, NB )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0,
$ ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION EI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY,
$ DLARFG, DSCAL, DTRMM, DTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
DO 10 I = 1, NB
IF( I.GT.1 ) THEN
*
* Update A(K+1:N,I)
*
* Update I-th column of A - Y * V**T
*
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
$ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
*
* Apply I - V * T**T * V**T to this column (call it b) from the
* left, using the last column of T as workspace
*
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
* ( V2 ) ( b2 )
*
* where V1 is unit lower triangular
*
* w := V1**T * b1
*
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
CALL DTRMV( 'Lower', 'Transpose', 'UNIT',
$ I-1, A( K+1, 1 ),
$ LDA, T( 1, NB ), 1 )
*
* w := w + V2**T * b2
*
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ),
$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
* w := T**T * w
*
CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, NB ), 1 )
*
* b2 := b2 - V2*w
*
CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
$ A( K+I, 1 ),
$ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
* b1 := b1 - V1*w
*
CALL DTRMV( 'Lower', 'NO TRANSPOSE',
$ 'UNIT', I-1,
$ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
A( K+I-1, I-1 ) = EI
END IF
*
* Generate the elementary reflector H(I) to annihilate
* A(K+I+1:N,I)
*
CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
$ TAU( I ) )
EI = A( K+I, I )
A( K+I, I ) = ONE
*
* Compute Y(K+1:N,I)
*
CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
$ ONE, A( K+1, I+1 ),
$ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
CALL DGEMV( 'Transpose', N-K-I+1, I-1,
$ ONE, A( K+I, 1 ), LDA,
$ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
$ Y( K+1, 1 ), LDY,
$ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
*
* Compute T(1:I,I)
*
CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
$ I-1, T, LDT,
$ T( 1, I ), 1 )
T( I, I ) = TAU( I )
*
10 CONTINUE
A( K+NB, NB ) = EI
*
* Compute Y(1:K,1:NB)
*
CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
$ 'UNIT', K, NB,
$ ONE, A( K+1, 1 ), LDA, Y, LDY )
IF( N.GT.K+NB )
$ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
$ NB, N-K-NB, ONE,
$ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
$ LDY )
CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
$ 'NON-UNIT', K, NB,
$ ONE, T, LDT, Y, LDY )
*
RETURN
*
* End of DLAHR2
*
END