1*> \brief <b> DSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
22*                          LIWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          JOBZ, UPLO
26*       INTEGER            INFO, LDA, LIWORK, LWORK, N
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            IWORK( * )
30*       DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> DSYEVD computes all eigenvalues and, optionally, eigenvectors of a
40*> real symmetric matrix A. If eigenvectors are desired, it uses a
41*> divide and conquer algorithm.
42*>
43*> The divide and conquer algorithm makes very mild assumptions about
44*> floating point arithmetic. It will work on machines with a guard
45*> digit in add/subtract, or on those binary machines without guard
46*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
47*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
48*> without guard digits, but we know of none.
49*>
50*> Because of large use of BLAS of level 3, DSYEVD needs N**2 more
51*> workspace than DSYEVX.
52*> \endverbatim
53*
54*  Arguments:
55*  ==========
56*
57*> \param[in] JOBZ
58*> \verbatim
59*>          JOBZ is CHARACTER*1
60*>          = 'N':  Compute eigenvalues only;
61*>          = 'V':  Compute eigenvalues and eigenvectors.
62*> \endverbatim
63*>
64*> \param[in] UPLO
65*> \verbatim
66*>          UPLO is CHARACTER*1
67*>          = 'U':  Upper triangle of A is stored;
68*>          = 'L':  Lower triangle of A is stored.
69*> \endverbatim
70*>
71*> \param[in] N
72*> \verbatim
73*>          N is INTEGER
74*>          The order of the matrix A.  N >= 0.
75*> \endverbatim
76*>
77*> \param[in,out] A
78*> \verbatim
79*>          A is DOUBLE PRECISION array, dimension (LDA, N)
80*>          On entry, the symmetric matrix A.  If UPLO = 'U', the
81*>          leading N-by-N upper triangular part of A contains the
82*>          upper triangular part of the matrix A.  If UPLO = 'L',
83*>          the leading N-by-N lower triangular part of A contains
84*>          the lower triangular part of the matrix A.
85*>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
86*>          orthonormal eigenvectors of the matrix A.
87*>          If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
88*>          or the upper triangle (if UPLO='U') of A, including the
89*>          diagonal, is destroyed.
90*> \endverbatim
91*>
92*> \param[in] LDA
93*> \verbatim
94*>          LDA is INTEGER
95*>          The leading dimension of the array A.  LDA >= max(1,N).
96*> \endverbatim
97*>
98*> \param[out] W
99*> \verbatim
100*>          W is DOUBLE PRECISION array, dimension (N)
101*>          If INFO = 0, the eigenvalues in ascending order.
102*> \endverbatim
103*>
104*> \param[out] WORK
105*> \verbatim
106*>          WORK is DOUBLE PRECISION array,
107*>                                         dimension (LWORK)
108*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
109*> \endverbatim
110*>
111*> \param[in] LWORK
112*> \verbatim
113*>          LWORK is INTEGER
114*>          The dimension of the array WORK.
115*>          If N <= 1,               LWORK must be at least 1.
116*>          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
117*>          If JOBZ = 'V' and N > 1, LWORK must be at least
118*>                                                1 + 6*N + 2*N**2.
119*>
120*>          If LWORK = -1, then a workspace query is assumed; the routine
121*>          only calculates the optimal sizes of the WORK and IWORK
122*>          arrays, returns these values as the first entries of the WORK
123*>          and IWORK arrays, and no error message related to LWORK or
124*>          LIWORK is issued by XERBLA.
125*> \endverbatim
126*>
127*> \param[out] IWORK
128*> \verbatim
129*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
130*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
131*> \endverbatim
132*>
133*> \param[in] LIWORK
134*> \verbatim
135*>          LIWORK is INTEGER
136*>          The dimension of the array IWORK.
137*>          If N <= 1,                LIWORK must be at least 1.
138*>          If JOBZ  = 'N' and N > 1, LIWORK must be at least 1.
139*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
140*>
141*>          If LIWORK = -1, then a workspace query is assumed; the
142*>          routine only calculates the optimal sizes of the WORK and
143*>          IWORK arrays, returns these values as the first entries of
144*>          the WORK and IWORK arrays, and no error message related to
145*>          LWORK or LIWORK is issued by XERBLA.
146*> \endverbatim
147*>
148*> \param[out] INFO
149*> \verbatim
150*>          INFO is INTEGER
151*>          = 0:  successful exit
152*>          < 0:  if INFO = -i, the i-th argument had an illegal value
153*>          > 0:  if INFO = i and JOBZ = 'N', then the algorithm failed
154*>                to converge; i off-diagonal elements of an intermediate
155*>                tridiagonal form did not converge to zero;
156*>                if INFO = i and JOBZ = 'V', then the algorithm failed
157*>                to compute an eigenvalue while working on the submatrix
158*>                lying in rows and columns INFO/(N+1) through
159*>                mod(INFO,N+1).
160*> \endverbatim
161*
162*  Authors:
163*  ========
164*
165*> \author Univ. of Tennessee
166*> \author Univ. of California Berkeley
167*> \author Univ. of Colorado Denver
168*> \author NAG Ltd.
169*
170*> \date September 2012
171*
172*> \ingroup doubleSYeigen
173*
174*> \par Contributors:
175*  ==================
176*>
177*> Jeff Rutter, Computer Science Division, University of California
178*> at Berkeley, USA \n
179*>  Modified by Francoise Tisseur, University of Tennessee \n
180*>  Modified description of INFO. Sven, 16 Feb 05. \n
181
182
183*>
184*  =====================================================================
185      SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK,
186     $                   LIWORK, INFO )
187*
188*  -- LAPACK driver routine (version 3.4.2) --
189*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
190*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191*     September 2012
192*
193*     .. Scalar Arguments ..
194      CHARACTER          JOBZ, UPLO
195      INTEGER            INFO, LDA, LIWORK, LWORK, N
196*     ..
197*     .. Array Arguments ..
198      INTEGER            IWORK( * )
199      DOUBLE PRECISION   A( LDA, * ), W( * ), WORK( * )
200*     ..
201*
202*  =====================================================================
203*
204*     .. Parameters ..
205      DOUBLE PRECISION   ZERO, ONE
206      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
207*     ..
208*     .. Local Scalars ..
209*
210      LOGICAL            LOWER, LQUERY, WANTZ
211      INTEGER            IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE,
212     $                   LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN
213      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
214     $                   SMLNUM
215*     ..
216*     .. External Functions ..
217      LOGICAL            LSAME
218      INTEGER            ILAENV
219      DOUBLE PRECISION   DLAMCH, DLANSY
220      EXTERNAL           LSAME, DLAMCH, DLANSY, ILAENV
221*     ..
222*     .. External Subroutines ..
223      EXTERNAL           DLACPY, DLASCL, DORMTR, DSCAL, DSTEDC, DSTERF,
224     $                   DSYTRD, XERBLA
225*     ..
226*     .. Intrinsic Functions ..
227      INTRINSIC          MAX, SQRT
228*     ..
229*     .. Executable Statements ..
230*
231*     Test the input parameters.
232*
233      WANTZ = LSAME( JOBZ, 'V' )
234      LOWER = LSAME( UPLO, 'L' )
235      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
236*
237      INFO = 0
238      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
239         INFO = -1
240      ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
241         INFO = -2
242      ELSE IF( N.LT.0 ) THEN
243         INFO = -3
244      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
245         INFO = -5
246      END IF
247*
248      IF( INFO.EQ.0 ) THEN
249         IF( N.LE.1 ) THEN
250            LIWMIN = 1
251            LWMIN = 1
252            LOPT = LWMIN
253            LIOPT = LIWMIN
254         ELSE
255            IF( WANTZ ) THEN
256               LIWMIN = 3 + 5*N
257               LWMIN = 1 + 6*N + 2*N**2
258            ELSE
259               LIWMIN = 1
260               LWMIN = 2*N + 1
261            END IF
262            LOPT = MAX( LWMIN, 2*N +
263     $                  ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
264            LIOPT = LIWMIN
265         END IF
266         WORK( 1 ) = LOPT
267         IWORK( 1 ) = LIOPT
268*
269         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
270            INFO = -8
271         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
272            INFO = -10
273         END IF
274      END IF
275*
276      IF( INFO.NE.0 ) THEN
277         CALL XERBLA( 'DSYEVD', -INFO )
278         RETURN
279      ELSE IF( LQUERY ) THEN
280         RETURN
281      END IF
282*
283*     Quick return if possible
284*
285      IF( N.EQ.0 )
286     $   RETURN
287*
288      IF( N.EQ.1 ) THEN
289         W( 1 ) = A( 1, 1 )
290         IF( WANTZ )
291     $      A( 1, 1 ) = ONE
292         RETURN
293      END IF
294*
295*     Get machine constants.
296*
297      SAFMIN = DLAMCH( 'Safe minimum' )
298      EPS = DLAMCH( 'Precision' )
299      SMLNUM = SAFMIN / EPS
300      BIGNUM = ONE / SMLNUM
301      RMIN = SQRT( SMLNUM )
302      RMAX = SQRT( BIGNUM )
303*
304*     Scale matrix to allowable range, if necessary.
305*
306      ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
307      ISCALE = 0
308      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
309         ISCALE = 1
310         SIGMA = RMIN / ANRM
311      ELSE IF( ANRM.GT.RMAX ) THEN
312         ISCALE = 1
313         SIGMA = RMAX / ANRM
314      END IF
315      IF( ISCALE.EQ.1 )
316     $   CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
317*
318*     Call DSYTRD to reduce symmetric matrix to tridiagonal form.
319*
320      INDE = 1
321      INDTAU = INDE + N
322      INDWRK = INDTAU + N
323      LLWORK = LWORK - INDWRK + 1
324      INDWK2 = INDWRK + N*N
325      LLWRK2 = LWORK - INDWK2 + 1
326*
327      CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ),
328     $             WORK( INDWRK ), LLWORK, IINFO )
329*
330*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
331*     DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
332*     tridiagonal matrix, then call DORMTR to multiply it by the
333*     Householder transformations stored in A.
334*
335      IF( .NOT.WANTZ ) THEN
336         CALL DSTERF( N, W, WORK( INDE ), INFO )
337      ELSE
338         CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
339     $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
340         CALL DORMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
341     $                WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
342         CALL DLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
343      END IF
344*
345*     If matrix was scaled, then rescale eigenvalues appropriately.
346*
347      IF( ISCALE.EQ.1 )
348     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
349*
350      WORK( 1 ) = LOPT
351      IWORK( 1 ) = LIOPT
352*
353      RETURN
354*
355*     End of DSYEVD
356*
357      END
358