LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zlarzt()

 subroutine zlarzt ( character DIRECT, character STOREV, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( ldt, * ) T, integer LDT )

ZLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:
``` ZLARZT forms the triangular factor T of a complex block reflector
H of order > n, which is defined as a product of k elementary
reflectors.

If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;

If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.

If STOREV = 'C', the vector which defines the elementary reflector
H(i) is stored in the i-th column of the array V, and

H  =  I - V * T * V**H

If STOREV = 'R', the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and

H  =  I - V**H * T * V

Currently, only STOREV = 'R' and DIRECT = 'B' are supported.```
Parameters
 [in] DIRECT ``` DIRECT is CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) = 'B': H = H(k) . . . H(2) H(1) (Backward)``` [in] STOREV ``` STOREV is CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): = 'C': columnwise (not supported yet) = 'R': rowwise``` [in] N ``` N is INTEGER The order of the block reflector H. N >= 0.``` [in] K ``` K is INTEGER The order of the triangular factor T (= the number of elementary reflectors). K >= 1.``` [in,out] V ``` V is COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details.``` [in] LDV ``` LDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.``` [in] TAU ``` TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i).``` [out] T ``` T is COMPLEX*16 array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= K.```
Date
December 2016
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
```  The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.

DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

______V_____
( v1 v2 v3 )                        /            \
( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
( v1 v2 v3 )
.  .  .
.  .  .
1  .  .
1  .
1

DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

______V_____
1                                          /            \
.  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
.  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
.  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
.  .  .
( v1 v2 v3 )
( v1 v2 v3 )
V = ( v1 v2 v3 )
( v1 v2 v3 )
( v1 v2 v3 )```

Definition at line 187 of file zlarzt.f.

187 *
188 * -- LAPACK computational routine (version 3.7.0) --
189 * -- LAPACK is a software package provided by Univ. of Tennessee, --
190 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191 * December 2016
192 *
193 * .. Scalar Arguments ..
194  CHARACTER direct, storev
195  INTEGER k, ldt, ldv, n
196 * ..
197 * .. Array Arguments ..
198  COMPLEX*16 t( ldt, * ), tau( * ), v( ldv, * )
199 * ..
200 *
201 * =====================================================================
202 *
203 * .. Parameters ..
204  COMPLEX*16 zero
205  parameter( zero = ( 0.0d+0, 0.0d+0 ) )
206 * ..
207 * .. Local Scalars ..
208  INTEGER i, info, j
209 * ..
210 * .. External Subroutines ..
211  EXTERNAL xerbla, zgemv, zlacgv, ztrmv
212 * ..
213 * .. External Functions ..
214  LOGICAL lsame
215  EXTERNAL lsame
216 * ..
217 * .. Executable Statements ..
218 *
219 * Check for currently supported options
220 *
221  info = 0
222  IF( .NOT.lsame( direct, 'B' ) ) THEN
223  info = -1
224  ELSE IF( .NOT.lsame( storev, 'R' ) ) THEN
225  info = -2
226  END IF
227  IF( info.NE.0 ) THEN
228  CALL xerbla( 'ZLARZT', -info )
229  RETURN
230  END IF
231 *
232  DO 20 i = k, 1, -1
233  IF( tau( i ).EQ.zero ) THEN
234 *
235 * H(i) = I
236 *
237  DO 10 j = i, k
238  t( j, i ) = zero
239  10 CONTINUE
240  ELSE
241 *
242 * general case
243 *
244  IF( i.LT.k ) THEN
245 *
246 * T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**H
247 *
248  CALL zlacgv( n, v( i, 1 ), ldv )
249  CALL zgemv( 'No transpose', k-i, n, -tau( i ),
250  \$ v( i+1, 1 ), ldv, v( i, 1 ), ldv, zero,
251  \$ t( i+1, i ), 1 )
252  CALL zlacgv( n, v( i, 1 ), ldv )
253 *
254 * T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
255 *
256  CALL ztrmv( 'Lower', 'No transpose', 'Non-unit', k-i,
257  \$ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
258  END IF
259  t( i, i ) = tau( i )
260  END IF
261  20 CONTINUE
262  RETURN
263 *
264 * End of ZLARZT
265 *
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:149
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
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