LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
zhetrd.f
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1 *> \brief \b ZHETRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, LWORK, N
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION D( * ), E( * )
29 * COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> ZHETRD reduces a complex Hermitian matrix A to real symmetric
39 *> tridiagonal form T by a unitary similarity transformation:
40 *> Q**H * A * Q = T.
41 *> \endverbatim
42 *
43 * Arguments:
44 * ==========
45 *
46 *> \param[in] UPLO
47 *> \verbatim
48 *> UPLO is CHARACTER*1
49 *> = 'U': Upper triangle of A is stored;
50 *> = 'L': Lower triangle of A is stored.
51 *> \endverbatim
52 *>
53 *> \param[in] N
54 *> \verbatim
55 *> N is INTEGER
56 *> The order of the matrix A. N >= 0.
57 *> \endverbatim
58 *>
59 *> \param[in,out] A
60 *> \verbatim
61 *> A is COMPLEX*16 array, dimension (LDA,N)
62 *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
63 *> N-by-N upper triangular part of A contains the upper
64 *> triangular part of the matrix A, and the strictly lower
65 *> triangular part of A is not referenced. If UPLO = 'L', the
66 *> leading N-by-N lower triangular part of A contains the lower
67 *> triangular part of the matrix A, and the strictly upper
68 *> triangular part of A is not referenced.
69 *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70 *> of A are overwritten by the corresponding elements of the
71 *> tridiagonal matrix T, and the elements above the first
72 *> superdiagonal, with the array TAU, represent the unitary
73 *> matrix Q as a product of elementary reflectors; if UPLO
74 *> = 'L', the diagonal and first subdiagonal of A are over-
75 *> written by the corresponding elements of the tridiagonal
76 *> matrix T, and the elements below the first subdiagonal, with
77 *> the array TAU, represent the unitary matrix Q as a product
78 *> of elementary reflectors. See Further Details.
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the array A. LDA >= max(1,N).
85 *> \endverbatim
86 *>
87 *> \param[out] D
88 *> \verbatim
89 *> D is DOUBLE PRECISION array, dimension (N)
90 *> The diagonal elements of the tridiagonal matrix T:
91 *> D(i) = A(i,i).
92 *> \endverbatim
93 *>
94 *> \param[out] E
95 *> \verbatim
96 *> E is DOUBLE PRECISION array, dimension (N-1)
97 *> The off-diagonal elements of the tridiagonal matrix T:
98 *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99 *> \endverbatim
100 *>
101 *> \param[out] TAU
102 *> \verbatim
103 *> TAU is COMPLEX*16 array, dimension (N-1)
104 *> The scalar factors of the elementary reflectors (see Further
105 *> Details).
106 *> \endverbatim
107 *>
108 *> \param[out] WORK
109 *> \verbatim
110 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
111 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
112 *> \endverbatim
113 *>
114 *> \param[in] LWORK
115 *> \verbatim
116 *> LWORK is INTEGER
117 *> The dimension of the array WORK. LWORK >= 1.
118 *> For optimum performance LWORK >= N*NB, where NB is the
119 *> optimal blocksize.
120 *>
121 *> If LWORK = -1, then a workspace query is assumed; the routine
122 *> only calculates the optimal size of the WORK array, returns
123 *> this value as the first entry of the WORK array, and no error
124 *> message related to LWORK is issued by XERBLA.
125 *> \endverbatim
126 *>
127 *> \param[out] INFO
128 *> \verbatim
129 *> INFO is INTEGER
130 *> = 0: successful exit
131 *> < 0: if INFO = -i, the i-th argument had an illegal value
132 *> \endverbatim
133 *
134 * Authors:
135 * ========
136 *
137 *> \author Univ. of Tennessee
138 *> \author Univ. of California Berkeley
139 *> \author Univ. of Colorado Denver
140 *> \author NAG Ltd.
141 *
142 *> \date December 2016
143 *
144 *> \ingroup complex16HEcomputational
145 *
146 *> \par Further Details:
147 * =====================
148 *>
149 *> \verbatim
150 *>
151 *> If UPLO = 'U', the matrix Q is represented as a product of elementary
152 *> reflectors
153 *>
154 *> Q = H(n-1) . . . H(2) H(1).
155 *>
156 *> Each H(i) has the form
157 *>
158 *> H(i) = I - tau * v * v**H
159 *>
160 *> where tau is a complex scalar, and v is a complex vector with
161 *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
162 *> A(1:i-1,i+1), and tau in TAU(i).
163 *>
164 *> If UPLO = 'L', the matrix Q is represented as a product of elementary
165 *> reflectors
166 *>
167 *> Q = H(1) H(2) . . . H(n-1).
168 *>
169 *> Each H(i) has the form
170 *>
171 *> H(i) = I - tau * v * v**H
172 *>
173 *> where tau is a complex scalar, and v is a complex vector with
174 *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
175 *> and tau in TAU(i).
176 *>
177 *> The contents of A on exit are illustrated by the following examples
178 *> with n = 5:
179 *>
180 *> if UPLO = 'U': if UPLO = 'L':
181 *>
182 *> ( d e v2 v3 v4 ) ( d )
183 *> ( d e v3 v4 ) ( e d )
184 *> ( d e v4 ) ( v1 e d )
185 *> ( d e ) ( v1 v2 e d )
186 *> ( d ) ( v1 v2 v3 e d )
187 *>
188 *> where d and e denote diagonal and off-diagonal elements of T, and vi
189 *> denotes an element of the vector defining H(i).
190 *> \endverbatim
191 *>
192 * =====================================================================
193  SUBROUTINE zhetrd( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
194 *
195 * -- LAPACK computational routine (version 3.7.0) --
196 * -- LAPACK is a software package provided by Univ. of Tennessee, --
197 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198 * December 2016
199 *
200 * .. Scalar Arguments ..
201  CHARACTER UPLO
202  INTEGER INFO, LDA, LWORK, N
203 * ..
204 * .. Array Arguments ..
205  DOUBLE PRECISION D( * ), E( * )
206  COMPLEX*16 A( lda, * ), TAU( * ), WORK( * )
207 * ..
208 *
209 * =====================================================================
210 *
211 * .. Parameters ..
212  DOUBLE PRECISION ONE
213  parameter( one = 1.0d+0 )
214  COMPLEX*16 CONE
215  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
216 * ..
217 * .. Local Scalars ..
218  LOGICAL LQUERY, UPPER
219  INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
220  $ nbmin, nx
221 * ..
222 * .. External Subroutines ..
223  EXTERNAL xerbla, zher2k, zhetd2, zlatrd
224 * ..
225 * .. Intrinsic Functions ..
226  INTRINSIC max
227 * ..
228 * .. External Functions ..
229  LOGICAL LSAME
230  INTEGER ILAENV
231  EXTERNAL lsame, ilaenv
232 * ..
233 * .. Executable Statements ..
234 *
235 * Test the input parameters
236 *
237  info = 0
238  upper = lsame( uplo, 'U' )
239  lquery = ( lwork.EQ.-1 )
240  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
241  info = -1
242  ELSE IF( n.LT.0 ) THEN
243  info = -2
244  ELSE IF( lda.LT.max( 1, n ) ) THEN
245  info = -4
246  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
247  info = -9
248  END IF
249 *
250  IF( info.EQ.0 ) THEN
251 *
252 * Determine the block size.
253 *
254  nb = ilaenv( 1, 'ZHETRD', uplo, n, -1, -1, -1 )
255  lwkopt = n*nb
256  work( 1 ) = lwkopt
257  END IF
258 *
259  IF( info.NE.0 ) THEN
260  CALL xerbla( 'ZHETRD', -info )
261  RETURN
262  ELSE IF( lquery ) THEN
263  RETURN
264  END IF
265 *
266 * Quick return if possible
267 *
268  IF( n.EQ.0 ) THEN
269  work( 1 ) = 1
270  RETURN
271  END IF
272 *
273  nx = n
274  iws = 1
275  IF( nb.GT.1 .AND. nb.LT.n ) THEN
276 *
277 * Determine when to cross over from blocked to unblocked code
278 * (last block is always handled by unblocked code).
279 *
280  nx = max( nb, ilaenv( 3, 'ZHETRD', uplo, n, -1, -1, -1 ) )
281  IF( nx.LT.n ) THEN
282 *
283 * Determine if workspace is large enough for blocked code.
284 *
285  ldwork = n
286  iws = ldwork*nb
287  IF( lwork.LT.iws ) THEN
288 *
289 * Not enough workspace to use optimal NB: determine the
290 * minimum value of NB, and reduce NB or force use of
291 * unblocked code by setting NX = N.
292 *
293  nb = max( lwork / ldwork, 1 )
294  nbmin = ilaenv( 2, 'ZHETRD', uplo, n, -1, -1, -1 )
295  IF( nb.LT.nbmin )
296  $ nx = n
297  END IF
298  ELSE
299  nx = n
300  END IF
301  ELSE
302  nb = 1
303  END IF
304 *
305  IF( upper ) THEN
306 *
307 * Reduce the upper triangle of A.
308 * Columns 1:kk are handled by the unblocked method.
309 *
310  kk = n - ( ( n-nx+nb-1 ) / nb )*nb
311  DO 20 i = n - nb + 1, kk + 1, -nb
312 *
313 * Reduce columns i:i+nb-1 to tridiagonal form and form the
314 * matrix W which is needed to update the unreduced part of
315 * the matrix
316 *
317  CALL zlatrd( uplo, i+nb-1, nb, a, lda, e, tau, work,
318  $ ldwork )
319 *
320 * Update the unreduced submatrix A(1:i-1,1:i-1), using an
321 * update of the form: A := A - V*W**H - W*V**H
322 *
323  CALL zher2k( uplo, 'No transpose', i-1, nb, -cone,
324  $ a( 1, i ), lda, work, ldwork, one, a, lda )
325 *
326 * Copy superdiagonal elements back into A, and diagonal
327 * elements into D
328 *
329  DO 10 j = i, i + nb - 1
330  a( j-1, j ) = e( j-1 )
331  d( j ) = a( j, j )
332  10 CONTINUE
333  20 CONTINUE
334 *
335 * Use unblocked code to reduce the last or only block
336 *
337  CALL zhetd2( uplo, kk, a, lda, d, e, tau, iinfo )
338  ELSE
339 *
340 * Reduce the lower triangle of A
341 *
342  DO 40 i = 1, n - nx, nb
343 *
344 * Reduce columns i:i+nb-1 to tridiagonal form and form the
345 * matrix W which is needed to update the unreduced part of
346 * the matrix
347 *
348  CALL zlatrd( uplo, n-i+1, nb, a( i, i ), lda, e( i ),
349  $ tau( i ), work, ldwork )
350 *
351 * Update the unreduced submatrix A(i+nb:n,i+nb:n), using
352 * an update of the form: A := A - V*W**H - W*V**H
353 *
354  CALL zher2k( uplo, 'No transpose', n-i-nb+1, nb, -cone,
355  $ a( i+nb, i ), lda, work( nb+1 ), ldwork, one,
356  $ a( i+nb, i+nb ), lda )
357 *
358 * Copy subdiagonal elements back into A, and diagonal
359 * elements into D
360 *
361  DO 30 j = i, i + nb - 1
362  a( j+1, j ) = e( j )
363  d( j ) = a( j, j )
364  30 CONTINUE
365  40 CONTINUE
366 *
367 * Use unblocked code to reduce the last or only block
368 *
369  CALL zhetd2( uplo, n-i+1, a( i, i ), lda, d( i ), e( i ),
370  $ tau( i ), iinfo )
371  END IF
372 *
373  work( 1 ) = lwkopt
374  RETURN
375 *
376 * End of ZHETRD
377 *
378  END
subroutine zhetd2(UPLO, N, A, LDA, D, E, TAU, INFO)
ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transfo...
Definition: zhetd2.f:177
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zher2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZHER2K
Definition: zher2k.f:200
subroutine zlatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition: zlatrd.f:201
subroutine zhetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
ZHETRD
Definition: zhetrd.f:194