LAPACK  3.9.0
LAPACK: Linear Algebra PACKage
dsytrd.f
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1 *> \brief \b DSYTRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYTRD( 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 A( LDA, * ), D( * ), E( * ), TAU( * ),
29 * $ WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> DSYTRD reduces a real symmetric matrix A to real symmetric
39 *> tridiagonal form T by an orthogonal similarity transformation:
40 *> Q**T * 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 DOUBLE PRECISION array, dimension (LDA,N)
62 *> On entry, the symmetric 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 orthogonal
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 orthogonal 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 doubleSYcomputational
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**T
159 *>
160 *> where tau is a real scalar, and v is a real 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**T
172 *>
173 *> where tau is a real scalar, and v is a real 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 dsytrd( 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 A( LDA, * ), D( * ), E( * ), TAU( * ),
206  $ WORK( * )
207 * ..
208 *
209 * =====================================================================
210 *
211 * .. Parameters ..
212  DOUBLE PRECISION ONE
213  parameter( one = 1.0d+0 )
214 * ..
215 * .. Local Scalars ..
216  LOGICAL LQUERY, UPPER
217  INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
218  $ NBMIN, NX
219 * ..
220 * .. External Subroutines ..
221  EXTERNAL dlatrd, dsyr2k, dsytd2, xerbla
222 * ..
223 * .. Intrinsic Functions ..
224  INTRINSIC max
225 * ..
226 * .. External Functions ..
227  LOGICAL LSAME
228  INTEGER ILAENV
229  EXTERNAL lsame, ilaenv
230 * ..
231 * .. Executable Statements ..
232 *
233 * Test the input parameters
234 *
235  info = 0
236  upper = lsame( uplo, 'U' )
237  lquery = ( lwork.EQ.-1 )
238  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
239  info = -1
240  ELSE IF( n.LT.0 ) THEN
241  info = -2
242  ELSE IF( lda.LT.max( 1, n ) ) THEN
243  info = -4
244  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
245  info = -9
246  END IF
247 *
248  IF( info.EQ.0 ) THEN
249 *
250 * Determine the block size.
251 *
252  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
253  lwkopt = n*nb
254  work( 1 ) = lwkopt
255  END IF
256 *
257  IF( info.NE.0 ) THEN
258  CALL xerbla( 'DSYTRD', -info )
259  RETURN
260  ELSE IF( lquery ) THEN
261  RETURN
262  END IF
263 *
264 * Quick return if possible
265 *
266  IF( n.EQ.0 ) THEN
267  work( 1 ) = 1
268  RETURN
269  END IF
270 *
271  nx = n
272  iws = 1
273  IF( nb.GT.1 .AND. nb.LT.n ) THEN
274 *
275 * Determine when to cross over from blocked to unblocked code
276 * (last block is always handled by unblocked code).
277 *
278  nx = max( nb, ilaenv( 3, 'DSYTRD', uplo, n, -1, -1, -1 ) )
279  IF( nx.LT.n ) THEN
280 *
281 * Determine if workspace is large enough for blocked code.
282 *
283  ldwork = n
284  iws = ldwork*nb
285  IF( lwork.LT.iws ) THEN
286 *
287 * Not enough workspace to use optimal NB: determine the
288 * minimum value of NB, and reduce NB or force use of
289 * unblocked code by setting NX = N.
290 *
291  nb = max( lwork / ldwork, 1 )
292  nbmin = ilaenv( 2, 'DSYTRD', uplo, n, -1, -1, -1 )
293  IF( nb.LT.nbmin )
294  $ nx = n
295  END IF
296  ELSE
297  nx = n
298  END IF
299  ELSE
300  nb = 1
301  END IF
302 *
303  IF( upper ) THEN
304 *
305 * Reduce the upper triangle of A.
306 * Columns 1:kk are handled by the unblocked method.
307 *
308  kk = n - ( ( n-nx+nb-1 ) / nb )*nb
309  DO 20 i = n - nb + 1, kk + 1, -nb
310 *
311 * Reduce columns i:i+nb-1 to tridiagonal form and form the
312 * matrix W which is needed to update the unreduced part of
313 * the matrix
314 *
315  CALL dlatrd( uplo, i+nb-1, nb, a, lda, e, tau, work,
316  $ ldwork )
317 *
318 * Update the unreduced submatrix A(1:i-1,1:i-1), using an
319 * update of the form: A := A - V*W**T - W*V**T
320 *
321  CALL dsyr2k( uplo, 'No transpose', i-1, nb, -one, a( 1, i ),
322  $ lda, work, ldwork, one, a, lda )
323 *
324 * Copy superdiagonal elements back into A, and diagonal
325 * elements into D
326 *
327  DO 10 j = i, i + nb - 1
328  a( j-1, j ) = e( j-1 )
329  d( j ) = a( j, j )
330  10 CONTINUE
331  20 CONTINUE
332 *
333 * Use unblocked code to reduce the last or only block
334 *
335  CALL dsytd2( uplo, kk, a, lda, d, e, tau, iinfo )
336  ELSE
337 *
338 * Reduce the lower triangle of A
339 *
340  DO 40 i = 1, n - nx, nb
341 *
342 * Reduce columns i:i+nb-1 to tridiagonal form and form the
343 * matrix W which is needed to update the unreduced part of
344 * the matrix
345 *
346  CALL dlatrd( uplo, n-i+1, nb, a( i, i ), lda, e( i ),
347  $ tau( i ), work, ldwork )
348 *
349 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using
350 * an update of the form: A := A - V*W**T - W*V**T
351 *
352  CALL dsyr2k( uplo, 'No transpose', n-i-nb+1, nb, -one,
353  $ a( i+nb, i ), lda, work( nb+1 ), ldwork, one,
354  $ a( i+nb, i+nb ), lda )
355 *
356 * Copy subdiagonal elements back into A, and diagonal
357 * elements into D
358 *
359  DO 30 j = i, i + nb - 1
360  a( j+1, j ) = e( j )
361  d( j ) = a( j, j )
362  30 CONTINUE
363  40 CONTINUE
364 *
365 * Use unblocked code to reduce the last or only block
366 *
367  CALL dsytd2( uplo, n-i+1, a( i, i ), lda, d( i ), e( i ),
368  $ tau( i ), iinfo )
369  END IF
370 *
371  work( 1 ) = lwkopt
372  RETURN
373 *
374 * End of DSYTRD
375 *
376  END
dsytd2
subroutine dsytd2(UPLO, N, A, LDA, D, E, TAU, INFO)
DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity tran...
Definition: dsytd2.f:175
dsytrd
subroutine dsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD
Definition: dsytrd.f:194
dsyr2k
subroutine dsyr2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYR2K
Definition: dsyr2k.f:194
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
dlatrd
subroutine dlatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition: dlatrd.f:200