 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zlahrd()

 subroutine zlahrd ( integer N, integer K, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( nb ) TAU, complex*16, dimension( ldt, nb ) T, integer LDT, complex*16, dimension( ldy, nb ) Y, integer LDY )

ZLAHRD 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``` This routine is deprecated and has been replaced by routine ZLAHR2.

ZLAHRD 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*16 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*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.``` [out] T ``` T is COMPLEX*16 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*16 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).```
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 zlahrd.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*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
177  \$ Y( LDY, NB )
178 * ..
179 *
180 * =====================================================================
181 *
182 * .. Parameters ..
183  COMPLEX*16 ZERO, ONE
184  parameter( zero = ( 0.0d+0, 0.0d+0 ),
185  \$ one = ( 1.0d+0, 0.0d+0 ) )
186 * ..
187 * .. Local Scalars ..
188  INTEGER I
189  COMPLEX*16 EI
190 * ..
191 * .. External Subroutines ..
192  EXTERNAL zaxpy, zcopy, zgemv, zlacgv, zlarfg, zscal,
193  \$ ztrmv
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 zlacgv( i-1, a( k+i-1, 1 ), lda )
213  CALL zgemv( 'No transpose', n, i-1, -one, y, ldy,
214  \$ a( k+i-1, 1 ), lda, one, a( 1, i ), 1 )
215  CALL zlacgv( 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 zcopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
228  CALL ztrmv( '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 zgemv( '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 ztrmv( 'Upper', 'Conjugate transpose', 'Non-unit', i-1,
240  \$ t, ldt, t( 1, nb ), 1 )
241 *
242 * b2 := b2 - V2*w
243 *
244  CALL zgemv( '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 ztrmv( 'Lower', 'No transpose', 'Unit', i-1,
250  \$ a( k+1, 1 ), lda, t( 1, nb ), 1 )
251  CALL zaxpy( 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 zlarfg( 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 zgemv( '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 zgemv( '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 zgemv( 'No transpose', n, i-1, -one, y, ldy, t( 1, i ), 1,
272  \$ one, y( 1, i ), 1 )
273  CALL zscal( n, tau( i ), y( 1, i ), 1 )
274 *
275 * Compute T(1:i,i)
276 *
277  CALL zscal( i-1, -tau( i ), t( 1, i ), 1 )
278  CALL ztrmv( '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 ZLAHRD
288 *
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:147
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
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