LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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.

Download CLAHRD + dependencies [TGZ] [ZIP] [TXT]

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 This routine is deprecated and has been replaced by routine CLAHR2.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).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 166 of file clahrd.f.

167*
168* -- LAPACK auxiliary routine --
169* -- LAPACK is a software package provided by Univ. of Tennessee, --
170* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
171*
172* .. Scalar Arguments ..
173 INTEGER K, LDA, LDT, LDY, N, NB
174* ..
175* .. Array Arguments ..
176 COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ),
177 $ Y( LDY, NB )
178* ..
179*
180* =====================================================================
181*
182* .. Parameters ..
183 COMPLEX ZERO, ONE
184 parameter( zero = ( 0.0e+0, 0.0e+0 ),
185 $ one = ( 1.0e+0, 0.0e+0 ) )
186* ..
187* .. Local Scalars ..
188 INTEGER I
189 COMPLEX EI
190* ..
191* .. External Subroutines ..
192 EXTERNAL caxpy, ccopy, cgemv, clacgv, clarfg, cscal,
193 $ ctrmv
194* ..
195* .. Intrinsic Functions ..
196 INTRINSIC min
197* ..
198* .. Executable Statements ..
199*
200* Quick return if possible
201*
202 IF( n.LE.1 )
203 $ RETURN
204*
205 DO 10 i = 1, nb
206 IF( i.GT.1 ) THEN
207*
208* Update A(1:n,i)
209*
210* Compute i-th column of A - Y * V**H
211*
212 CALL clacgv( i-1, a( k+i-1, 1 ), lda )
213 CALL cgemv( 'No transpose', n, i-1, -one, y, ldy,
214 $ a( k+i-1, 1 ), lda, one, a( 1, i ), 1 )
215 CALL clacgv( i-1, a( k+i-1, 1 ), lda )
216*
217* Apply I - V * T**H * V**H to this column (call it b) from the
218* left, using the last column of T as workspace
219*
220* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
221* ( V2 ) ( b2 )
222*
223* where V1 is unit lower triangular
224*
225* w := V1**H * b1
226*
227 CALL ccopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
228 CALL ctrmv( 'Lower', 'Conjugate transpose', 'Unit', i-1,
229 $ a( k+1, 1 ), lda, t( 1, nb ), 1 )
230*
231* w := w + V2**H *b2
232*
233 CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
234 $ a( k+i, 1 ), lda, a( k+i, i ), 1, one,
235 $ t( 1, nb ), 1 )
236*
237* w := T**H *w
238*
239 CALL ctrmv( 'Upper', 'Conjugate transpose', 'Non-unit', i-1,
240 $ t, ldt, t( 1, nb ), 1 )
241*
242* b2 := b2 - V2*w
243*
244 CALL cgemv( 'No transpose', n-k-i+1, i-1, -one, a( k+i, 1 ),
245 $ lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
246*
247* b1 := b1 - V1*w
248*
249 CALL ctrmv( 'Lower', 'No transpose', 'Unit', i-1,
250 $ a( k+1, 1 ), lda, t( 1, nb ), 1 )
251 CALL caxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
252*
253 a( k+i-1, i-1 ) = ei
254 END IF
255*
256* Generate the elementary reflector H(i) to annihilate
257* A(k+i+1:n,i)
258*
259 ei = a( k+i, i )
260 CALL clarfg( n-k-i+1, ei, a( min( k+i+1, n ), i ), 1,
261 $ tau( i ) )
262 a( k+i, i ) = one
263*
264* Compute Y(1:n,i)
265*
266 CALL cgemv( 'No transpose', n, n-k-i+1, one, a( 1, i+1 ), lda,
267 $ a( k+i, i ), 1, zero, y( 1, i ), 1 )
268 CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1, one,
269 $ a( k+i, 1 ), lda, a( k+i, i ), 1, zero, t( 1, i ),
270 $ 1 )
271 CALL cgemv( 'No transpose', n, i-1, -one, y, ldy, t( 1, i ), 1,
272 $ one, y( 1, i ), 1 )
273 CALL cscal( n, tau( i ), y( 1, i ), 1 )
274*
275* Compute T(1:i,i)
276*
277 CALL cscal( i-1, -tau( i ), t( 1, i ), 1 )
278 CALL ctrmv( 'Upper', 'No transpose', 'Non-unit', i-1, t, ldt,
279 $ t( 1, i ), 1 )
280 t( i, i ) = tau( i )
281*
282 10 CONTINUE
283 a( k+nb, nb ) = ei
284*
285 RETURN
286*
287* End of CLAHRD
288*
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine ctrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
CTRMV
Definition: ctrmv.f:147
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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