LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

◆ clahrd()

 subroutine clahrd ( integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex, dimension( ldy, nb ) Y, integer LDY )

CLAHRD 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.

Purpose:
``` This routine is deprecated and has been replaced by routine CLAHR2.

CLAHRD 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.```
Parameters
 [in] N ``` N is INTEGER The order of the matrix A.``` [in] K ``` K is INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero.``` [in] NB ``` NB is INTEGER The number of columns to be reduced.``` [in,out] A ``` A is COMPLEX 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] TAU ``` TAU is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.``` [out] T ``` T is COMPLEX array, dimension (LDT,NB) The upper triangular matrix T.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= NB.``` [out] Y ``` Y is COMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y.``` [in] LDY ``` LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).```
Date
December 2016
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**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).```

Definition at line 169 of file clahrd.f.

169 *
170 * -- LAPACK auxiliary routine (version 3.7.0) --
171 * -- LAPACK is a software package provided by Univ. of Tennessee, --
172 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
173 * December 2016
174 *
175 * .. Scalar Arguments ..
176  INTEGER K, LDA, LDT, LDY, N, NB
177 * ..
178 * .. Array Arguments ..
179  COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ),
180  \$ Y( LDY, NB )
181 * ..
182 *
183 * =====================================================================
184 *
185 * .. Parameters ..
186  COMPLEX ZERO, ONE
187  parameter( zero = ( 0.0e+0, 0.0e+0 ),
188  \$ one = ( 1.0e+0, 0.0e+0 ) )
189 * ..
190 * .. Local Scalars ..
191  INTEGER I
192  COMPLEX EI
193 * ..
194 * .. External Subroutines ..
195  EXTERNAL caxpy, ccopy, cgemv, clacgv, clarfg, cscal,
196  \$ ctrmv
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC min
200 * ..
201 * .. Executable Statements ..
202 *
203 * Quick return if possible
204 *
205  IF( n.LE.1 )
206  \$ RETURN
207 *
208  DO 10 i = 1, nb
209  IF( i.GT.1 ) THEN
210 *
211 * Update A(1:n,i)
212 *
213 * Compute i-th column of A - Y * V**H
214 *
215  CALL clacgv( i-1, a( k+i-1, 1 ), lda )
216  CALL cgemv( 'No transpose', n, i-1, -one, y, ldy,
217  \$ a( k+i-1, 1 ), lda, one, a( 1, i ), 1 )
218  CALL clacgv( i-1, a( k+i-1, 1 ), lda )
219 *
220 * Apply I - V * T**H * V**H to this column (call it b) from the
221 * left, using the last column of T as workspace
222 *
223 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
224 * ( V2 ) ( b2 )
225 *
226 * where V1 is unit lower triangular
227 *
228 * w := V1**H * b1
229 *
230  CALL ccopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
231  CALL ctrmv( 'Lower', 'Conjugate transpose', 'Unit', i-1,
232  \$ a( k+1, 1 ), lda, t( 1, nb ), 1 )
233 *
234 * w := w + V2**H *b2
235 *
236  CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
237  \$ a( k+i, 1 ), lda, a( k+i, i ), 1, one,
238  \$ t( 1, nb ), 1 )
239 *
240 * w := T**H *w
241 *
242  CALL ctrmv( 'Upper', 'Conjugate transpose', 'Non-unit', i-1,
243  \$ t, ldt, t( 1, nb ), 1 )
244 *
245 * b2 := b2 - V2*w
246 *
247  CALL cgemv( 'No transpose', n-k-i+1, i-1, -one, a( k+i, 1 ),
248  \$ lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
249 *
250 * b1 := b1 - V1*w
251 *
252  CALL ctrmv( 'Lower', 'No transpose', 'Unit', i-1,
253  \$ a( k+1, 1 ), lda, t( 1, nb ), 1 )
254  CALL caxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
255 *
256  a( k+i-1, i-1 ) = ei
257  END IF
258 *
259 * Generate the elementary reflector H(i) to annihilate
260 * A(k+i+1:n,i)
261 *
262  ei = a( k+i, i )
263  CALL clarfg( n-k-i+1, ei, a( min( k+i+1, n ), i ), 1,
264  \$ tau( i ) )
265  a( k+i, i ) = one
266 *
267 * Compute Y(1:n,i)
268 *
269  CALL cgemv( 'No transpose', n, n-k-i+1, one, a( 1, i+1 ), lda,
270  \$ a( k+i, i ), 1, zero, y( 1, i ), 1 )
271  CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
272  \$ a( k+i, 1 ), lda, a( k+i, i ), 1, zero, t( 1, i ),
273  \$ 1 )
274  CALL cgemv( 'No transpose', n, i-1, -one, y, ldy, t( 1, i ), 1,
275  \$ one, y( 1, i ), 1 )
276  CALL cscal( n, tau( i ), y( 1, i ), 1 )
277 *
278 * Compute T(1:i,i)
279 *
280  CALL cscal( i-1, -tau( i ), t( 1, i ), 1 )
281  CALL ctrmv( 'Upper', 'No transpose', 'Non-unit', i-1, t, ldt,
282  \$ t( 1, i ), 1 )
283  t( i, i ) = tau( i )
284 *
285  10 CONTINUE
286  a( k+nb, nb ) = ei
287 *
288  RETURN
289 *
290 * End of CLAHRD
291 *
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clarfg
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:108
cgemv
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
clacgv
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76
ctrmv
subroutine ctrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
CTRMV
Definition: ctrmv.f:149
cscal
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:80
ccopy
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
caxpy
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:90