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

 subroutine zlatrd ( character UPLO, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, complex*16, dimension( * ) TAU, complex*16, dimension( ldw, * ) W, integer LDW )

ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:
``` ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
Hermitian tridiagonal form by a unitary similarity
transformation Q**H * A * Q, and returns the matrices V and W which are
needed to apply the transformation to the unreduced part of A.

If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A.``` [in] NB ``` NB is INTEGER The number of rows and columns to be reduced.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] E ``` E is DOUBLE PRECISION array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.``` [out] TAU ``` TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details.``` [out] W ``` W is COMPLEX*16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.``` [in] LDW ``` LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).```
Further Details:
```  If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).

If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.

The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:

if UPLO = 'U':                       if UPLO = 'L':

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).```

Definition at line 198 of file zlatrd.f.

199*
200* -- LAPACK auxiliary routine --
201* -- LAPACK is a software package provided by Univ. of Tennessee, --
202* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
203*
204* .. Scalar Arguments ..
205 CHARACTER UPLO
206 INTEGER LDA, LDW, N, NB
207* ..
208* .. Array Arguments ..
209 DOUBLE PRECISION E( * )
210 COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
211* ..
212*
213* =====================================================================
214*
215* .. Parameters ..
216 COMPLEX*16 ZERO, ONE, HALF
217 parameter( zero = ( 0.0d+0, 0.0d+0 ),
218 \$ one = ( 1.0d+0, 0.0d+0 ),
219 \$ half = ( 0.5d+0, 0.0d+0 ) )
220* ..
221* .. Local Scalars ..
222 INTEGER I, IW
223 COMPLEX*16 ALPHA
224* ..
225* .. External Subroutines ..
226 EXTERNAL zaxpy, zgemv, zhemv, zlacgv, zlarfg, zscal
227* ..
228* .. External Functions ..
229 LOGICAL LSAME
230 COMPLEX*16 ZDOTC
231 EXTERNAL lsame, zdotc
232* ..
233* .. Intrinsic Functions ..
234 INTRINSIC dble, min
235* ..
236* .. Executable Statements ..
237*
238* Quick return if possible
239*
240 IF( n.LE.0 )
241 \$ RETURN
242*
243 IF( lsame( uplo, 'U' ) ) THEN
244*
245* Reduce last NB columns of upper triangle
246*
247 DO 10 i = n, n - nb + 1, -1
248 iw = i - n + nb
249 IF( i.LT.n ) THEN
250*
251* Update A(1:i,i)
252*
253 a( i, i ) = dble( a( i, i ) )
254 CALL zlacgv( n-i, w( i, iw+1 ), ldw )
255 CALL zgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
256 \$ lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
257 CALL zlacgv( n-i, w( i, iw+1 ), ldw )
258 CALL zlacgv( n-i, a( i, i+1 ), lda )
259 CALL zgemv( 'No transpose', i, n-i, -one, w( 1, iw+1 ),
260 \$ ldw, a( i, i+1 ), lda, one, a( 1, i ), 1 )
261 CALL zlacgv( n-i, a( i, i+1 ), lda )
262 a( i, i ) = dble( a( i, i ) )
263 END IF
264 IF( i.GT.1 ) THEN
265*
266* Generate elementary reflector H(i) to annihilate
267* A(1:i-2,i)
268*
269 alpha = a( i-1, i )
270 CALL zlarfg( i-1, alpha, a( 1, i ), 1, tau( i-1 ) )
271 e( i-1 ) = dble( alpha )
272 a( i-1, i ) = one
273*
274* Compute W(1:i-1,i)
275*
276 CALL zhemv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
277 \$ zero, w( 1, iw ), 1 )
278 IF( i.LT.n ) THEN
279 CALL zgemv( 'Conjugate transpose', i-1, n-i, one,
280 \$ w( 1, iw+1 ), ldw, a( 1, i ), 1, zero,
281 \$ w( i+1, iw ), 1 )
282 CALL zgemv( 'No transpose', i-1, n-i, -one,
283 \$ a( 1, i+1 ), lda, w( i+1, iw ), 1, one,
284 \$ w( 1, iw ), 1 )
285 CALL zgemv( 'Conjugate transpose', i-1, n-i, one,
286 \$ a( 1, i+1 ), lda, a( 1, i ), 1, zero,
287 \$ w( i+1, iw ), 1 )
288 CALL zgemv( 'No transpose', i-1, n-i, -one,
289 \$ w( 1, iw+1 ), ldw, w( i+1, iw ), 1, one,
290 \$ w( 1, iw ), 1 )
291 END IF
292 CALL zscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
293 alpha = -half*tau( i-1 )*zdotc( i-1, w( 1, iw ), 1,
294 \$ a( 1, i ), 1 )
295 CALL zaxpy( i-1, alpha, a( 1, i ), 1, w( 1, iw ), 1 )
296 END IF
297*
298 10 CONTINUE
299 ELSE
300*
301* Reduce first NB columns of lower triangle
302*
303 DO 20 i = 1, nb
304*
305* Update A(i:n,i)
306*
307 a( i, i ) = dble( a( i, i ) )
308 CALL zlacgv( i-1, w( i, 1 ), ldw )
309 CALL zgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
310 \$ lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
311 CALL zlacgv( i-1, w( i, 1 ), ldw )
312 CALL zlacgv( i-1, a( i, 1 ), lda )
313 CALL zgemv( 'No transpose', n-i+1, i-1, -one, w( i, 1 ),
314 \$ ldw, a( i, 1 ), lda, one, a( i, i ), 1 )
315 CALL zlacgv( i-1, a( i, 1 ), lda )
316 a( i, i ) = dble( a( i, i ) )
317 IF( i.LT.n ) THEN
318*
319* Generate elementary reflector H(i) to annihilate
320* A(i+2:n,i)
321*
322 alpha = a( i+1, i )
323 CALL zlarfg( n-i, alpha, a( min( i+2, n ), i ), 1,
324 \$ tau( i ) )
325 e( i ) = dble( alpha )
326 a( i+1, i ) = one
327*
328* Compute W(i+1:n,i)
329*
330 CALL zhemv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
331 \$ a( i+1, i ), 1, zero, w( i+1, i ), 1 )
332 CALL zgemv( 'Conjugate transpose', n-i, i-1, one,
333 \$ w( i+1, 1 ), ldw, a( i+1, i ), 1, zero,
334 \$ w( 1, i ), 1 )
335 CALL zgemv( 'No transpose', n-i, i-1, -one, a( i+1, 1 ),
336 \$ lda, w( 1, i ), 1, one, w( i+1, i ), 1 )
337 CALL zgemv( 'Conjugate transpose', n-i, i-1, one,
338 \$ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
339 \$ w( 1, i ), 1 )
340 CALL zgemv( 'No transpose', n-i, i-1, -one, w( i+1, 1 ),
341 \$ ldw, w( 1, i ), 1, one, w( i+1, i ), 1 )
342 CALL zscal( n-i, tau( i ), w( i+1, i ), 1 )
343 alpha = -half*tau( i )*zdotc( n-i, w( i+1, i ), 1,
344 \$ a( i+1, i ), 1 )
345 CALL zaxpy( n-i, alpha, a( i+1, i ), 1, w( i+1, i ), 1 )
346 END IF
347*
348 20 CONTINUE
349 END IF
350*
351 RETURN
352*
353* End of ZLATRD
354*
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
complex *16 function zdotc(N, ZX, INCX, ZY, INCY)
ZDOTC
Definition: zdotc.f:83
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 zhemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZHEMV
Definition: zhemv.f:154
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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