LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
chetrd.f
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1 *> \brief \b CHETRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CHETRD( 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 * REAL D( * ), E( * )
29 * COMPLEX A( LDA, * ), TAU( * ), WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CHETRD 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 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 REAL 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 REAL 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 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 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 *> \ingroup complexHEcomputational
143 *
144 *> \par Further Details:
145 * =====================
146 *>
147 *> \verbatim
148 *>
149 *> If UPLO = 'U', the matrix Q is represented as a product of elementary
150 *> reflectors
151 *>
152 *> Q = H(n-1) . . . H(2) H(1).
153 *>
154 *> Each H(i) has the form
155 *>
156 *> H(i) = I - tau * v * v**H
157 *>
158 *> where tau is a complex scalar, and v is a complex vector with
159 *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
160 *> A(1:i-1,i+1), and tau in TAU(i).
161 *>
162 *> If UPLO = 'L', the matrix Q is represented as a product of elementary
163 *> reflectors
164 *>
165 *> Q = H(1) H(2) . . . H(n-1).
166 *>
167 *> Each H(i) has the form
168 *>
169 *> H(i) = I - tau * v * v**H
170 *>
171 *> where tau is a complex scalar, and v is a complex vector with
172 *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
173 *> and tau in TAU(i).
174 *>
175 *> The contents of A on exit are illustrated by the following examples
176 *> with n = 5:
177 *>
178 *> if UPLO = 'U': if UPLO = 'L':
179 *>
180 *> ( d e v2 v3 v4 ) ( d )
181 *> ( d e v3 v4 ) ( e d )
182 *> ( d e v4 ) ( v1 e d )
183 *> ( d e ) ( v1 v2 e d )
184 *> ( d ) ( v1 v2 v3 e d )
185 *>
186 *> where d and e denote diagonal and off-diagonal elements of T, and vi
187 *> denotes an element of the vector defining H(i).
188 *> \endverbatim
189 *>
190 * =====================================================================
191  SUBROUTINE chetrd( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
192 *
193 * -- LAPACK computational routine --
194 * -- LAPACK is a software package provided by Univ. of Tennessee, --
195 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196 *
197 * .. Scalar Arguments ..
198  CHARACTER UPLO
199  INTEGER INFO, LDA, LWORK, N
200 * ..
201 * .. Array Arguments ..
202  REAL D( * ), E( * )
203  COMPLEX A( LDA, * ), TAU( * ), WORK( * )
204 * ..
205 *
206 * =====================================================================
207 *
208 * .. Parameters ..
209  REAL ONE
210  parameter( one = 1.0e+0 )
211  COMPLEX CONE
212  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
213 * ..
214 * .. Local Scalars ..
215  LOGICAL LQUERY, UPPER
216  INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
217  \$ NBMIN, NX
218 * ..
219 * .. External Subroutines ..
220  EXTERNAL cher2k, chetd2, clatrd, xerbla
221 * ..
222 * .. Intrinsic Functions ..
223  INTRINSIC max
224 * ..
225 * .. External Functions ..
226  LOGICAL LSAME
227  INTEGER ILAENV
228  EXTERNAL lsame, ilaenv
229 * ..
230 * .. Executable Statements ..
231 *
232 * Test the input parameters
233 *
234  info = 0
235  upper = lsame( uplo, 'U' )
236  lquery = ( lwork.EQ.-1 )
237  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
238  info = -1
239  ELSE IF( n.LT.0 ) THEN
240  info = -2
241  ELSE IF( lda.LT.max( 1, n ) ) THEN
242  info = -4
243  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
244  info = -9
245  END IF
246 *
247  IF( info.EQ.0 ) THEN
248 *
249 * Determine the block size.
250 *
251  nb = ilaenv( 1, 'CHETRD', uplo, n, -1, -1, -1 )
252  lwkopt = n*nb
253  work( 1 ) = lwkopt
254  END IF
255 *
256  IF( info.NE.0 ) THEN
257  CALL xerbla( 'CHETRD', -info )
258  RETURN
259  ELSE IF( lquery ) THEN
260  RETURN
261  END IF
262 *
263 * Quick return if possible
264 *
265  IF( n.EQ.0 ) THEN
266  work( 1 ) = 1
267  RETURN
268  END IF
269 *
270  nx = n
271  iws = 1
272  IF( nb.GT.1 .AND. nb.LT.n ) THEN
273 *
274 * Determine when to cross over from blocked to unblocked code
275 * (last block is always handled by unblocked code).
276 *
277  nx = max( nb, ilaenv( 3, 'CHETRD', uplo, n, -1, -1, -1 ) )
278  IF( nx.LT.n ) THEN
279 *
280 * Determine if workspace is large enough for blocked code.
281 *
282  ldwork = n
283  iws = ldwork*nb
284  IF( lwork.LT.iws ) THEN
285 *
286 * Not enough workspace to use optimal NB: determine the
287 * minimum value of NB, and reduce NB or force use of
288 * unblocked code by setting NX = N.
289 *
290  nb = max( lwork / ldwork, 1 )
291  nbmin = ilaenv( 2, 'CHETRD', uplo, n, -1, -1, -1 )
292  IF( nb.LT.nbmin )
293  \$ nx = n
294  END IF
295  ELSE
296  nx = n
297  END IF
298  ELSE
299  nb = 1
300  END IF
301 *
302  IF( upper ) THEN
303 *
304 * Reduce the upper triangle of A.
305 * Columns 1:kk are handled by the unblocked method.
306 *
307  kk = n - ( ( n-nx+nb-1 ) / nb )*nb
308  DO 20 i = n - nb + 1, kk + 1, -nb
309 *
310 * Reduce columns i:i+nb-1 to tridiagonal form and form the
311 * matrix W which is needed to update the unreduced part of
312 * the matrix
313 *
314  CALL clatrd( uplo, i+nb-1, nb, a, lda, e, tau, work,
315  \$ ldwork )
316 *
317 * Update the unreduced submatrix A(1:i-1,1:i-1), using an
318 * update of the form: A := A - V*W**H - W*V**H
319 *
320  CALL cher2k( uplo, 'No transpose', i-1, nb, -cone,
321  \$ a( 1, i ), lda, work, ldwork, one, a, lda )
322 *
323 * Copy superdiagonal elements back into A, and diagonal
324 * elements into D
325 *
326  DO 10 j = i, i + nb - 1
327  a( j-1, j ) = e( j-1 )
328  d( j ) = real( a( j, j ) )
329  10 CONTINUE
330  20 CONTINUE
331 *
332 * Use unblocked code to reduce the last or only block
333 *
334  CALL chetd2( uplo, kk, a, lda, d, e, tau, iinfo )
335  ELSE
336 *
337 * Reduce the lower triangle of A
338 *
339  DO 40 i = 1, n - nx, nb
340 *
341 * Reduce columns i:i+nb-1 to tridiagonal form and form the
342 * matrix W which is needed to update the unreduced part of
343 * the matrix
344 *
345  CALL clatrd( uplo, n-i+1, nb, a( i, i ), lda, e( i ),
346  \$ tau( i ), work, ldwork )
347 *
348 * Update the unreduced submatrix A(i+nb:n,i+nb:n), using
349 * an update of the form: A := A - V*W**H - W*V**H
350 *
351  CALL cher2k( uplo, 'No transpose', n-i-nb+1, nb, -cone,
352  \$ a( i+nb, i ), lda, work( nb+1 ), ldwork, one,
353  \$ a( i+nb, i+nb ), lda )
354 *
355 * Copy subdiagonal elements back into A, and diagonal
356 * elements into D
357 *
358  DO 30 j = i, i + nb - 1
359  a( j+1, j ) = e( j )
360  d( j ) = real( a( j, j ) )
361  30 CONTINUE
362  40 CONTINUE
363 *
364 * Use unblocked code to reduce the last or only block
365 *
366  CALL chetd2( uplo, n-i+1, a( i, i ), lda, d( i ), e( i ),
367  \$ tau( i ), iinfo )
368  END IF
369 *
370  work( 1 ) = lwkopt
371  RETURN
372 *
373 * End of CHETRD
374 *
375  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cher2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CHER2K
Definition: cher2k.f:197
subroutine chetd2(UPLO, N, A, LDA, D, E, TAU, INFO)
CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transfo...
Definition: chetd2.f:175
subroutine chetrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHETRD
Definition: chetrd.f:192
subroutine clatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition: clatrd.f:199