LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ zhetd2()

subroutine zhetd2 ( character  UPLO,
integer  N,
complex*16, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
complex*16, dimension( * )  TAU,
integer  INFO 
)

ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).

Download ZHETD2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZHETD2 reduces a complex Hermitian matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T.
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.  N >= 0.
[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 diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, 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]D
          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).
[out]E
          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
[out]TAU
          TAU is COMPLEX*16 array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
  and tau in TAU(i).

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

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

    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i).

Definition at line 174 of file zhetd2.f.

175 *
176 * -- LAPACK computational routine --
177 * -- LAPACK is a software package provided by Univ. of Tennessee, --
178 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
179 *
180 * .. Scalar Arguments ..
181  CHARACTER UPLO
182  INTEGER INFO, LDA, N
183 * ..
184 * .. Array Arguments ..
185  DOUBLE PRECISION D( * ), E( * )
186  COMPLEX*16 A( LDA, * ), TAU( * )
187 * ..
188 *
189 * =====================================================================
190 *
191 * .. Parameters ..
192  COMPLEX*16 ONE, ZERO, HALF
193  parameter( one = ( 1.0d+0, 0.0d+0 ),
194  $ zero = ( 0.0d+0, 0.0d+0 ),
195  $ half = ( 0.5d+0, 0.0d+0 ) )
196 * ..
197 * .. Local Scalars ..
198  LOGICAL UPPER
199  INTEGER I
200  COMPLEX*16 ALPHA, TAUI
201 * ..
202 * .. External Subroutines ..
203  EXTERNAL xerbla, zaxpy, zhemv, zher2, zlarfg
204 * ..
205 * .. External Functions ..
206  LOGICAL LSAME
207  COMPLEX*16 ZDOTC
208  EXTERNAL lsame, zdotc
209 * ..
210 * .. Intrinsic Functions ..
211  INTRINSIC dble, max, min
212 * ..
213 * .. Executable Statements ..
214 *
215 * Test the input parameters
216 *
217  info = 0
218  upper = lsame( uplo, 'U')
219  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
220  info = -1
221  ELSE IF( n.LT.0 ) THEN
222  info = -2
223  ELSE IF( lda.LT.max( 1, n ) ) THEN
224  info = -4
225  END IF
226  IF( info.NE.0 ) THEN
227  CALL xerbla( 'ZHETD2', -info )
228  RETURN
229  END IF
230 *
231 * Quick return if possible
232 *
233  IF( n.LE.0 )
234  $ RETURN
235 *
236  IF( upper ) THEN
237 *
238 * Reduce the upper triangle of A
239 *
240  a( n, n ) = dble( a( n, n ) )
241  DO 10 i = n - 1, 1, -1
242 *
243 * Generate elementary reflector H(i) = I - tau * v * v**H
244 * to annihilate A(1:i-1,i+1)
245 *
246  alpha = a( i, i+1 )
247  CALL zlarfg( i, alpha, a( 1, i+1 ), 1, taui )
248  e( i ) = dble( alpha )
249 *
250  IF( taui.NE.zero ) THEN
251 *
252 * Apply H(i) from both sides to A(1:i,1:i)
253 *
254  a( i, i+1 ) = one
255 *
256 * Compute x := tau * A * v storing x in TAU(1:i)
257 *
258  CALL zhemv( uplo, i, taui, a, lda, a( 1, i+1 ), 1, zero,
259  $ tau, 1 )
260 *
261 * Compute w := x - 1/2 * tau * (x**H * v) * v
262 *
263  alpha = -half*taui*zdotc( i, tau, 1, a( 1, i+1 ), 1 )
264  CALL zaxpy( i, alpha, a( 1, i+1 ), 1, tau, 1 )
265 *
266 * Apply the transformation as a rank-2 update:
267 * A := A - v * w**H - w * v**H
268 *
269  CALL zher2( uplo, i, -one, a( 1, i+1 ), 1, tau, 1, a,
270  $ lda )
271 *
272  ELSE
273  a( i, i ) = dble( a( i, i ) )
274  END IF
275  a( i, i+1 ) = e( i )
276  d( i+1 ) = dble( a( i+1, i+1 ) )
277  tau( i ) = taui
278  10 CONTINUE
279  d( 1 ) = dble( a( 1, 1 ) )
280  ELSE
281 *
282 * Reduce the lower triangle of A
283 *
284  a( 1, 1 ) = dble( a( 1, 1 ) )
285  DO 20 i = 1, n - 1
286 *
287 * Generate elementary reflector H(i) = I - tau * v * v**H
288 * to annihilate A(i+2:n,i)
289 *
290  alpha = a( i+1, i )
291  CALL zlarfg( n-i, alpha, a( min( i+2, n ), i ), 1, taui )
292  e( i ) = dble( alpha )
293 *
294  IF( taui.NE.zero ) THEN
295 *
296 * Apply H(i) from both sides to A(i+1:n,i+1:n)
297 *
298  a( i+1, i ) = one
299 *
300 * Compute x := tau * A * v storing y in TAU(i:n-1)
301 *
302  CALL zhemv( uplo, n-i, taui, a( i+1, i+1 ), lda,
303  $ a( i+1, i ), 1, zero, tau( i ), 1 )
304 *
305 * Compute w := x - 1/2 * tau * (x**H * v) * v
306 *
307  alpha = -half*taui*zdotc( n-i, tau( i ), 1, a( i+1, i ),
308  $ 1 )
309  CALL zaxpy( n-i, alpha, a( i+1, i ), 1, tau( i ), 1 )
310 *
311 * Apply the transformation as a rank-2 update:
312 * A := A - v * w**H - w * v**H
313 *
314  CALL zher2( uplo, n-i, -one, a( i+1, i ), 1, tau( i ), 1,
315  $ a( i+1, i+1 ), lda )
316 *
317  ELSE
318  a( i+1, i+1 ) = dble( a( i+1, i+1 ) )
319  END IF
320  a( i+1, i ) = e( i )
321  d( i ) = dble( a( i, i ) )
322  tau( i ) = taui
323  20 CONTINUE
324  d( n ) = dble( a( n, n ) )
325  END IF
326 *
327  RETURN
328 *
329 * End of ZHETD2
330 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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 zhemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZHEMV
Definition: zhemv.f:154
subroutine zher2(UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZHER2
Definition: zher2.f:150
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
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