#include "ESMF_LapackBlas.inc" !> \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 !> Download DLAHR2 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahr2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahr2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f"> !> [TXT]</a> !> \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