*> \brief \b ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th 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 ZLAHRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * .. Scalar Arguments .. * INTEGER K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), * \$ Y( LDY, NB ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) *> matrix A so that elements below the k-th subdiagonal are zero. The *> reduction is performed by a unitary similarity transformation *> Q**H * A * Q. The routine returns the matrices V and T which determine *> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T. *> *> This is an OBSOLETE auxiliary routine. *> This routine will be 'deprecated' in a future release. *> Please use the new routine ZLAHR2 instead. *> \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. *> \endverbatim *> *> \param[in] NB *> \verbatim *> NB is INTEGER *> The number of columns to be reduced. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 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 COMPLEX*16 array, dimension (NB) *> The scalar factors of the elementary reflectors. See Further *> Details. *> \endverbatim *> *> \param[out] T *> \verbatim *> T is COMPLEX*16 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 COMPLEX*16 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 >= max(1,N). *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date September 2012 * *> \ingroup complex16OTHERauxiliary * *> \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**H *> *> where tau is a complex scalar, and v is a complex 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**H) * (A - Y*V**H). *> *> The contents of A on exit are illustrated by the following example *> with n = 7, k = 3 and nb = 2: *> *> ( a h a a a ) *> ( a h a a a ) *> ( a h 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). *> \endverbatim *> * ===================================================================== SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * -- LAPACK auxiliary routine (version 3.4.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * September 2012 * * .. Scalar Arguments .. INTEGER K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), \$ Y( LDY, NB ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 ZERO, ONE PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), \$ ONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER I COMPLEX*16 EI * .. * .. External Subroutines .. EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL, \$ ZTRMV * .. * .. 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(1:n,i) * * Compute i-th column of A - Y * V**H * CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, \$ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) * * Apply I - V * T**H * V**H 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**H * b1 * CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) * * w := w + V2**H *b2 * CALL ZGEMV( 'Conjugate 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**H *w * CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1, \$ T, LDT, T( 1, NB ), 1 ) * * b2 := b2 - V2*w * CALL ZGEMV( '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 ZTRMV( 'Lower', 'No transpose', 'Unit', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL ZAXPY( 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) * EI = A( K+I, I ) CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1, \$ TAU( I ) ) A( K+I, I ) = ONE * * Compute Y(1:n,i) * CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, \$ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, \$ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ), \$ 1 ) CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, \$ ONE, Y( 1, I ), 1 ) CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 ) * * Compute T(1:i,i) * CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL ZTRMV( '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 * RETURN * * End of ZLAHRD * END