LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ clahr2()

subroutine clahr2 ( 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 
)

CLAHR2 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.

Download CLAHR2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CLAHR2 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 an 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 auxiliary routine called by CGEHRD.
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.
          K < N.
[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 >= 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   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 CLAHRD
  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 CLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's CLAHRD.)
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.

Definition at line 180 of file clahr2.f.

181*
182* -- LAPACK auxiliary routine --
183* -- LAPACK is a software package provided by Univ. of Tennessee, --
184* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
185*
186* .. Scalar Arguments ..
187 INTEGER K, LDA, LDT, LDY, N, NB
188* ..
189* .. Array Arguments ..
190 COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ),
191 $ Y( LDY, NB )
192* ..
193*
194* =====================================================================
195*
196* .. Parameters ..
197 COMPLEX ZERO, ONE
198 parameter( zero = ( 0.0e+0, 0.0e+0 ),
199 $ one = ( 1.0e+0, 0.0e+0 ) )
200* ..
201* .. Local Scalars ..
202 INTEGER I
203 COMPLEX EI
204* ..
205* .. External Subroutines ..
206 EXTERNAL caxpy, ccopy, cgemm, cgemv, clacpy,
208* ..
209* .. Intrinsic Functions ..
210 INTRINSIC min
211* ..
212* .. Executable Statements ..
213*
214* Quick return if possible
215*
216 IF( n.LE.1 )
217 $ RETURN
218*
219 DO 10 i = 1, nb
220 IF( i.GT.1 ) THEN
221*
222* Update A(K+1:N,I)
223*
224* Update I-th column of A - Y * V**H
225*
226 CALL clacgv( i-1, a( k+i-1, 1 ), lda )
227 CALL cgemv( 'NO TRANSPOSE', n-k, i-1, -one, y(k+1,1), ldy,
228 $ a( k+i-1, 1 ), lda, one, a( k+1, i ), 1 )
229 CALL clacgv( i-1, a( k+i-1, 1 ), lda )
230*
231* Apply I - V * T**H * V**H to this column (call it b) from the
232* left, using the last column of T as workspace
233*
234* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
235* ( V2 ) ( b2 )
236*
237* where V1 is unit lower triangular
238*
239* w := V1**H * b1
240*
241 CALL ccopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
242 CALL ctrmv( 'Lower', 'Conjugate transpose', 'UNIT',
243 $ i-1, a( k+1, 1 ),
244 $ lda, t( 1, nb ), 1 )
245*
246* w := w + V2**H * b2
247*
248 CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1,
249 $ one, a( k+i, 1 ),
250 $ lda, a( k+i, i ), 1, one, t( 1, nb ), 1 )
251*
252* w := T**H * w
253*
254 CALL ctrmv( 'Upper', 'Conjugate transpose', 'NON-UNIT',
255 $ i-1, t, ldt,
256 $ t( 1, nb ), 1 )
257*
258* b2 := b2 - V2*w
259*
260 CALL cgemv( 'NO TRANSPOSE', n-k-i+1, i-1, -one,
261 $ a( k+i, 1 ),
262 $ lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
263*
264* b1 := b1 - V1*w
265*
266 CALL ctrmv( 'Lower', 'NO TRANSPOSE',
267 $ 'UNIT', i-1,
268 $ a( k+1, 1 ), lda, t( 1, nb ), 1 )
269 CALL caxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
270*
271 a( k+i-1, i-1 ) = ei
272 END IF
273*
274* Generate the elementary reflector H(I) to annihilate
275* A(K+I+1:N,I)
276*
277 CALL clarfg( n-k-i+1, a( k+i, i ), a( min( k+i+1, n ), i ), 1,
278 $ tau( i ) )
279 ei = a( k+i, i )
280 a( k+i, i ) = one
281*
282* Compute Y(K+1:N,I)
283*
284 CALL cgemv( 'NO TRANSPOSE', n-k, n-k-i+1,
285 $ one, a( k+1, i+1 ),
286 $ lda, a( k+i, i ), 1, zero, y( k+1, i ), 1 )
287 CALL cgemv( 'Conjugate transpose', n-k-i+1, i-1,
288 $ one, a( k+i, 1 ), lda,
289 $ a( k+i, i ), 1, zero, t( 1, i ), 1 )
290 CALL cgemv( 'NO TRANSPOSE', n-k, i-1, -one,
291 $ y( k+1, 1 ), ldy,
292 $ t( 1, i ), 1, one, y( k+1, i ), 1 )
293 CALL cscal( n-k, tau( i ), y( k+1, i ), 1 )
294*
295* Compute T(1:I,I)
296*
297 CALL cscal( i-1, -tau( i ), t( 1, i ), 1 )
298 CALL ctrmv( 'Upper', 'No Transpose', 'NON-UNIT',
299 $ i-1, t, ldt,
300 $ t( 1, i ), 1 )
301 t( i, i ) = tau( i )
302*
303 10 CONTINUE
304 a( k+nb, nb ) = ei
305*
306* Compute Y(1:K,1:NB)
307*
308 CALL clacpy( 'ALL', k, nb, a( 1, 2 ), lda, y, ldy )
309 CALL ctrmm( 'RIGHT', 'Lower', 'NO TRANSPOSE',
310 $ 'UNIT', k, nb,
311 $ one, a( k+1, 1 ), lda, y, ldy )
312 IF( n.GT.k+nb )
313 $ CALL cgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', k,
314 $ nb, n-k-nb, one,
315 $ a( 1, 2+nb ), lda, a( k+1+nb, 1 ), lda, one, y,
316 $ ldy )
317 CALL ctrmm( 'RIGHT', 'Upper', 'NO TRANSPOSE',
318 $ 'NON-UNIT', k, nb,
319 $ one, t, ldt, y, ldy )
320*
321 RETURN
322*
323* End of CLAHR2
324*
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 cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine ctrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM
Definition: ctrmm.f:177
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
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
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