 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zlahr2()

 subroutine zlahr2 ( 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 )

ZLAHR2 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.

Purpose:
``` ZLAHR2 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 ZGEHRD.```
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*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 >= 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   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 ZLAHRD
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 ZLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)```
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 zlahr2.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*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
191 \$ Y( LDY, NB )
192* ..
193*
194* =====================================================================
195*
196* .. Parameters ..
197 COMPLEX*16 ZERO, ONE
198 parameter( zero = ( 0.0d+0, 0.0d+0 ),
199 \$ one = ( 1.0d+0, 0.0d+0 ) )
200* ..
201* .. Local Scalars ..
202 INTEGER I
203 COMPLEX*16 EI
204* ..
205* .. External Subroutines ..
206 EXTERNAL zaxpy, zcopy, zgemm, zgemv, zlacpy,
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 zlacgv( i-1, a( k+i-1, 1 ), lda )
227 CALL zgemv( '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 zlacgv( 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 zcopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
242 CALL ztrmv( '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 zgemv( '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 ztrmv( 'Upper', 'Conjugate transpose', 'NON-UNIT',
255 \$ i-1, t, ldt,
256 \$ t( 1, nb ), 1 )
257*
258* b2 := b2 - V2*w
259*
260 CALL zgemv( '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 ztrmv( 'Lower', 'NO TRANSPOSE',
267 \$ 'UNIT', i-1,
268 \$ a( k+1, 1 ), lda, t( 1, nb ), 1 )
269 CALL zaxpy( 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 zlarfg( 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 zgemv( '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 zgemv( '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 zgemv( '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 zscal( n-k, tau( i ), y( k+1, i ), 1 )
294*
295* Compute T(1:I,I)
296*
297 CALL zscal( i-1, -tau( i ), t( 1, i ), 1 )
298 CALL ztrmv( '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 zlacpy( 'ALL', k, nb, a( 1, 2 ), lda, y, ldy )
309 CALL ztrmm( '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 zgemm( '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 ztrmm( 'RIGHT', 'Upper', 'NO TRANSPOSE',
318 \$ 'NON-UNIT', k, nb,
319 \$ one, t, ldt, y, ldy )
320*
321 RETURN
322*
323* End of ZLAHR2
324*
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 zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine ztrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRMM
Definition: ztrmm.f:177
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
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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