1*> \brief <b> DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DSPEVD + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevd.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevd.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevd.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
22*                          IWORK, LIWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          JOBZ, UPLO
26*       INTEGER            INFO, LDZ, LIWORK, LWORK, N
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            IWORK( * )
30*       DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> DSPEVD computes all the eigenvalues and, optionally, eigenvectors
40*> of a real symmetric matrix A in packed storage. If eigenvectors are
41*> desired, it uses a 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*> \endverbatim
50*
51*  Arguments:
52*  ==========
53*
54*> \param[in] JOBZ
55*> \verbatim
56*>          JOBZ is CHARACTER*1
57*>          = 'N':  Compute eigenvalues only;
58*>          = 'V':  Compute eigenvalues and eigenvectors.
59*> \endverbatim
60*>
61*> \param[in] UPLO
62*> \verbatim
63*>          UPLO is CHARACTER*1
64*>          = 'U':  Upper triangle of A is stored;
65*>          = 'L':  Lower triangle of A is stored.
66*> \endverbatim
67*>
68*> \param[in] N
69*> \verbatim
70*>          N is INTEGER
71*>          The order of the matrix A.  N >= 0.
72*> \endverbatim
73*>
74*> \param[in,out] AP
75*> \verbatim
76*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
77*>          On entry, the upper or lower triangle of the symmetric matrix
78*>          A, packed columnwise in a linear array.  The j-th column of A
79*>          is stored in the array AP as follows:
80*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
81*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
82*>
83*>          On exit, AP is overwritten by values generated during the
84*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
85*>          and first superdiagonal of the tridiagonal matrix T overwrite
86*>          the corresponding elements of A, and if UPLO = 'L', the
87*>          diagonal and first subdiagonal of T overwrite the
88*>          corresponding elements of A.
89*> \endverbatim
90*>
91*> \param[out] W
92*> \verbatim
93*>          W is DOUBLE PRECISION array, dimension (N)
94*>          If INFO = 0, the eigenvalues in ascending order.
95*> \endverbatim
96*>
97*> \param[out] Z
98*> \verbatim
99*>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
100*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
101*>          eigenvectors of the matrix A, with the i-th column of Z
102*>          holding the eigenvector associated with W(i).
103*>          If JOBZ = 'N', then Z is not referenced.
104*> \endverbatim
105*>
106*> \param[in] LDZ
107*> \verbatim
108*>          LDZ is INTEGER
109*>          The leading dimension of the array Z.  LDZ >= 1, and if
110*>          JOBZ = 'V', LDZ >= max(1,N).
111*> \endverbatim
112*>
113*> \param[out] WORK
114*> \verbatim
115*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
116*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
117*> \endverbatim
118*>
119*> \param[in] LWORK
120*> \verbatim
121*>          LWORK is INTEGER
122*>          The dimension of the array WORK.
123*>          If N <= 1,               LWORK must be at least 1.
124*>          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
125*>          If JOBZ = 'V' and N > 1, LWORK must be at least
126*>                                                 1 + 6*N + N**2.
127*>
128*>          If LWORK = -1, then a workspace query is assumed; the routine
129*>          only calculates the required sizes of the WORK and IWORK
130*>          arrays, returns these values as the first entries of the WORK
131*>          and IWORK arrays, and no error message related to LWORK or
132*>          LIWORK is issued by XERBLA.
133*> \endverbatim
134*>
135*> \param[out] IWORK
136*> \verbatim
137*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
138*>          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
139*> \endverbatim
140*>
141*> \param[in] LIWORK
142*> \verbatim
143*>          LIWORK is INTEGER
144*>          The dimension of the array IWORK.
145*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
146*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
147*>
148*>          If LIWORK = -1, then a workspace query is assumed; the
149*>          routine only calculates the required sizes of the WORK and
150*>          IWORK arrays, returns these values as the first entries of
151*>          the WORK and IWORK arrays, and no error message related to
152*>          LWORK or LIWORK is issued by XERBLA.
153*> \endverbatim
154*>
155*> \param[out] INFO
156*> \verbatim
157*>          INFO is INTEGER
158*>          = 0:  successful exit
159*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
160*>          > 0:  if INFO = i, the algorithm failed to converge; i
161*>                off-diagonal elements of an intermediate tridiagonal
162*>                form did not converge to zero.
163*> \endverbatim
164*
165*  Authors:
166*  ========
167*
168*> \author Univ. of Tennessee
169*> \author Univ. of California Berkeley
170*> \author Univ. of Colorado Denver
171*> \author NAG Ltd.
172*
173*> \date June 2017
174*
175*> \ingroup doubleOTHEReigen
176*
177*  =====================================================================
178      SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
179     $                   IWORK, LIWORK, INFO )
180*
181*  -- LAPACK driver routine (version 3.7.1) --
182*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
183*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184*     June 2017
185*
186*     .. Scalar Arguments ..
187      CHARACTER          JOBZ, UPLO
188      INTEGER            INFO, LDZ, LIWORK, LWORK, N
189*     ..
190*     .. Array Arguments ..
191      INTEGER            IWORK( * )
192      DOUBLE PRECISION   AP( * ), W( * ), WORK( * ), Z( LDZ, * )
193*     ..
194*
195*  =====================================================================
196*
197*     .. Parameters ..
198      DOUBLE PRECISION   ZERO, ONE
199      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
200*     ..
201*     .. Local Scalars ..
202      LOGICAL            LQUERY, WANTZ
203      INTEGER            IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN,
204     $                   LLWORK, LWMIN
205      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
206     $                   SMLNUM
207*     ..
208*     .. External Functions ..
209      LOGICAL            LSAME
210      DOUBLE PRECISION   DLAMCH, DLANSP
211      EXTERNAL           LSAME, DLAMCH, DLANSP
212*     ..
213*     .. External Subroutines ..
214      EXTERNAL           DOPMTR, DSCAL, DSPTRD, DSTEDC, DSTERF, XERBLA
215*     ..
216*     .. Intrinsic Functions ..
217      INTRINSIC          SQRT
218*     ..
219*     .. Executable Statements ..
220*
221*     Test the input parameters.
222*
223      WANTZ = LSAME( JOBZ, 'V' )
224      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
225*
226      INFO = 0
227      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
228         INFO = -1
229      ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) )
230     $          THEN
231         INFO = -2
232      ELSE IF( N.LT.0 ) THEN
233         INFO = -3
234      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
235         INFO = -7
236      END IF
237*
238      IF( INFO.EQ.0 ) THEN
239         IF( N.LE.1 ) THEN
240            LIWMIN = 1
241            LWMIN = 1
242         ELSE
243            IF( WANTZ ) THEN
244               LIWMIN = 3 + 5*N
245               LWMIN = 1 + 6*N + N**2
246            ELSE
247               LIWMIN = 1
248               LWMIN = 2*N
249            END IF
250         END IF
251         IWORK( 1 ) = LIWMIN
252         WORK( 1 ) = LWMIN
253*
254         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
255            INFO = -9
256         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
257            INFO = -11
258         END IF
259      END IF
260*
261      IF( INFO.NE.0 ) THEN
262         CALL XERBLA( 'DSPEVD', -INFO )
263         RETURN
264      ELSE IF( LQUERY ) THEN
265         RETURN
266      END IF
267*
268*     Quick return if possible
269*
270      IF( N.EQ.0 )
271     $   RETURN
272*
273      IF( N.EQ.1 ) THEN
274         W( 1 ) = AP( 1 )
275         IF( WANTZ )
276     $      Z( 1, 1 ) = ONE
277         RETURN
278      END IF
279*
280*     Get machine constants.
281*
282      SAFMIN = DLAMCH( 'Safe minimum' )
283      EPS = DLAMCH( 'Precision' )
284      SMLNUM = SAFMIN / EPS
285      BIGNUM = ONE / SMLNUM
286      RMIN = SQRT( SMLNUM )
287      RMAX = SQRT( BIGNUM )
288*
289*     Scale matrix to allowable range, if necessary.
290*
291      ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
292      ISCALE = 0
293      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
294         ISCALE = 1
295         SIGMA = RMIN / ANRM
296      ELSE IF( ANRM.GT.RMAX ) THEN
297         ISCALE = 1
298         SIGMA = RMAX / ANRM
299      END IF
300      IF( ISCALE.EQ.1 ) THEN
301         CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
302      END IF
303*
304*     Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
305*
306      INDE = 1
307      INDTAU = INDE + N
308      CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ), IINFO )
309*
310*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
311*     DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
312*     tridiagonal matrix, then call DOPMTR to multiply it by the
313*     Householder transformations represented in AP.
314*
315      IF( .NOT.WANTZ ) THEN
316         CALL DSTERF( N, W, WORK( INDE ), INFO )
317      ELSE
318         INDWRK = INDTAU + N
319         LLWORK = LWORK - INDWRK + 1
320         CALL DSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ),
321     $                LLWORK, IWORK, LIWORK, INFO )
322         CALL DOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ,
323     $                WORK( INDWRK ), IINFO )
324      END IF
325*
326*     If matrix was scaled, then rescale eigenvalues appropriately.
327*
328      IF( ISCALE.EQ.1 )
329     $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
330*
331      WORK( 1 ) = LWMIN
332      IWORK( 1 ) = LIWMIN
333      RETURN
334*
335*     End of DSPEVD
336*
337      END
338