version 1.1.1.1, 2010/01/26 15:22:46
|
version 1.17, 2017/06/17 10:53:53
|
Line 1
|
Line 1
|
|
*> \brief \b DLAHRD 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 DLAHRD + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE DLAHRD( 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 |
|
*> |
|
*> This routine is deprecated and has been replaced by routine DLAHR2. |
|
*> |
|
*> DLAHRD 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. |
|
*> \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 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 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 DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) |
SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine (version 3.7.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER K, LDA, LDT, LDY, N, NB |
INTEGER K, LDA, LDT, LDY, N, NB |
Line 13
|
Line 180
|
$ Y( LDY, NB ) |
$ Y( LDY, NB ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DLAHRD 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' * A * Q. The routine returns the matrices V and T which determine |
|
* Q as a block reflector I - V*T*V', 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 DLAHR2 instead. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* N (input) INTEGER |
|
* The order of the matrix A. |
|
* |
|
* K (input) INTEGER |
|
* The offset for the reduction. Elements below the k-th |
|
* subdiagonal in the first NB columns are reduced to zero. |
|
* |
|
* NB (input) INTEGER |
|
* The number of columns to be reduced. |
|
* |
|
* A (input/output) 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. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* TAU (output) DOUBLE PRECISION array, dimension (NB) |
|
* The scalar factors of the elementary reflectors. See Further |
|
* Details. |
|
* |
|
* T (output) DOUBLE PRECISION array, dimension (LDT,NB) |
|
* The upper triangular matrix T. |
|
* |
|
* LDT (input) INTEGER |
|
* The leading dimension of the array T. LDT >= NB. |
|
* |
|
* Y (output) DOUBLE PRECISION array, dimension (LDY,NB) |
|
* The n-by-nb matrix Y. |
|
* |
|
* LDY (input) INTEGER |
|
* The leading dimension of the array Y. LDY >= N. |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* 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' |
|
* |
|
* 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') * (A - Y*V'). |
|
* |
|
* 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). |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
Line 130
|
Line 208
|
* |
* |
* Update A(1:n,i) |
* Update A(1:n,i) |
* |
* |
* Compute i-th column of A - Y * V' |
* Compute i-th column of A - Y * V**T |
* |
* |
CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, |
CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, |
$ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) |
$ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) |
* |
* |
* Apply I - V * T' * V' to this column (call it b) from the |
* Apply I - V * T**T * V**T to this column (call it b) from the |
* left, using the last column of T as workspace |
* left, using the last column of T as workspace |
* |
* |
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) |
* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) |
Line 143
|
Line 221
|
* |
* |
* where V1 is unit lower triangular |
* where V1 is unit lower triangular |
* |
* |
* w := V1' * b1 |
* w := V1**T * b1 |
* |
* |
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) |
CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) |
CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), |
CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), |
$ LDA, T( 1, NB ), 1 ) |
$ LDA, T( 1, NB ), 1 ) |
* |
* |
* w := w + V2'*b2 |
* w := w + V2**T *b2 |
* |
* |
CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), |
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 ) |
$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) |
* |
* |
* w := T'*w |
* w := T**T *w |
* |
* |
CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, |
CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, |
$ T( 1, NB ), 1 ) |
$ T( 1, NB ), 1 ) |