*> \brief \b SLAHR2 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 SLAHR2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * .. Scalar Arguments .. * INTEGER K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. * REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), * \$ Y( LDY, NB ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAHR2 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 SGEHRD. *> \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 REAL 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 REAL array, dimension (NB) *> The scalar factors of the elementary reflectors. See Further *> Details. *> \endverbatim *> *> \param[out] T *> \verbatim *> T is REAL 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 REAL 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. * *> \ingroup realOTHERauxiliary * *> \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 SLAHRD *> 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 SLAHRD routine. (This *> subroutine is not backward compatible with LAPACK-3.0's SLAHRD.) *> \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 SLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * -- LAPACK auxiliary 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 K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), \$ Y( LDY, NB ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, \$ ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER I REAL EI * .. * .. External Subroutines .. EXTERNAL SAXPY, SCOPY, SGEMM, SGEMV, SLACPY, \$ SLARFG, SSCAL, STRMM, STRMV * .. * .. 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 SGEMV( '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 SCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL STRMV( 'Lower', 'Transpose', 'UNIT', \$ I-1, A( K+1, 1 ), \$ LDA, T( 1, NB ), 1 ) * * w := w + V2**T * b2 * CALL SGEMV( '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 STRMV( 'Upper', 'Transpose', 'NON-UNIT', \$ I-1, T, LDT, \$ T( 1, NB ), 1 ) * * b2 := b2 - V2*w * CALL SGEMV( '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 STRMV( 'Lower', 'NO TRANSPOSE', \$ 'UNIT', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL SAXPY( 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 SLARFG( 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 SGEMV( '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 SGEMV( 'Transpose', N-K-I+1, I-1, \$ ONE, A( K+I, 1 ), LDA, \$ A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) CALL SGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, \$ Y( K+1, 1 ), LDY, \$ T( 1, I ), 1, ONE, Y( K+1, I ), 1 ) CALL SSCAL( N-K, TAU( I ), Y( K+1, I ), 1 ) * * Compute T(1:I,I) * CALL SSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL STRMV( '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 SLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY ) CALL STRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE', \$ 'UNIT', K, NB, \$ ONE, A( K+1, 1 ), LDA, Y, LDY ) IF( N.GT.K+NB ) \$ CALL SGEMM( '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 STRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE', \$ 'NON-UNIT', K, NB, \$ ONE, T, LDT, Y, LDY ) * RETURN * * End of SLAHR2 * END