1*> \brief <b> ZHPEVD 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*
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16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZHPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
22*                          RWORK, LRWORK, IWORK, LIWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          JOBZ, UPLO
26*       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
27*       ..
28*       .. Array Arguments ..
29*       INTEGER            IWORK( * )
30*       DOUBLE PRECISION   RWORK( * ), W( * )
31*       COMPLEX*16         AP( * ), WORK( * ), Z( LDZ, * )
32*       ..
33*
34*
35*> \par Purpose:
36*  =============
37*>
38*> \verbatim
39*>
40*> ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of
41*> a complex Hermitian matrix A in packed storage.  If eigenvectors are
42*> desired, it uses a divide and conquer algorithm.
43*>
44*> The divide and conquer algorithm makes very mild assumptions about
45*> floating point arithmetic. It will work on machines with a guard
46*> digit in add/subtract, or on those binary machines without guard
47*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
48*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
49*> without guard digits, but we know of none.
50*> \endverbatim
51*
52*  Arguments:
53*  ==========
54*
55*> \param[in] JOBZ
56*> \verbatim
57*>          JOBZ is CHARACTER*1
58*>          = 'N':  Compute eigenvalues only;
59*>          = 'V':  Compute eigenvalues and eigenvectors.
60*> \endverbatim
61*>
62*> \param[in] UPLO
63*> \verbatim
64*>          UPLO is CHARACTER*1
65*>          = 'U':  Upper triangle of A is stored;
66*>          = 'L':  Lower triangle of A is stored.
67*> \endverbatim
68*>
69*> \param[in] N
70*> \verbatim
71*>          N is INTEGER
72*>          The order of the matrix A.  N >= 0.
73*> \endverbatim
74*>
75*> \param[in,out] AP
76*> \verbatim
77*>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
78*>          On entry, the upper or lower triangle of the Hermitian matrix
79*>          A, packed columnwise in a linear array.  The j-th column of A
80*>          is stored in the array AP as follows:
81*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
82*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
83*>
84*>          On exit, AP is overwritten by values generated during the
85*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
86*>          and first superdiagonal of the tridiagonal matrix T overwrite
87*>          the corresponding elements of A, and if UPLO = 'L', the
88*>          diagonal and first subdiagonal of T overwrite the
89*>          corresponding elements of A.
90*> \endverbatim
91*>
92*> \param[out] W
93*> \verbatim
94*>          W is DOUBLE PRECISION array, dimension (N)
95*>          If INFO = 0, the eigenvalues in ascending order.
96*> \endverbatim
97*>
98*> \param[out] Z
99*> \verbatim
100*>          Z is COMPLEX*16 array, dimension (LDZ, N)
101*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
102*>          eigenvectors of the matrix A, with the i-th column of Z
103*>          holding the eigenvector associated with W(i).
104*>          If JOBZ = 'N', then Z is not referenced.
105*> \endverbatim
106*>
107*> \param[in] LDZ
108*> \verbatim
109*>          LDZ is INTEGER
110*>          The leading dimension of the array Z.  LDZ >= 1, and if
111*>          JOBZ = 'V', LDZ >= max(1,N).
112*> \endverbatim
113*>
114*> \param[out] WORK
115*> \verbatim
116*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
117*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
118*> \endverbatim
119*>
120*> \param[in] LWORK
121*> \verbatim
122*>          LWORK is INTEGER
123*>          The dimension of array WORK.
124*>          If N <= 1,               LWORK must be at least 1.
125*>          If JOBZ = 'N' and N > 1, LWORK must be at least N.
126*>          If JOBZ = 'V' and N > 1, LWORK must be at least 2*N.
127*>
128*>          If LWORK = -1, then a workspace query is assumed; the routine
129*>          only calculates the required sizes of the WORK, RWORK and
130*>          IWORK arrays, returns these values as the first entries of
131*>          the WORK, RWORK and IWORK arrays, and no error message
132*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
133*> \endverbatim
134*>
135*> \param[out] RWORK
136*> \verbatim
137*>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
138*>          On exit, if INFO = 0, RWORK(1) returns the required LRWORK.
139*> \endverbatim
140*>
141*> \param[in] LRWORK
142*> \verbatim
143*>          LRWORK is INTEGER
144*>          The dimension of array RWORK.
145*>          If N <= 1,               LRWORK must be at least 1.
146*>          If JOBZ = 'N' and N > 1, LRWORK must be at least N.
147*>          If JOBZ = 'V' and N > 1, LRWORK must be at least
148*>                    1 + 5*N + 2*N**2.
149*>
150*>          If LRWORK = -1, then a workspace query is assumed; the
151*>          routine only calculates the required sizes of the WORK, RWORK
152*>          and IWORK arrays, returns these values as the first entries
153*>          of the WORK, RWORK and IWORK arrays, and no error message
154*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
155*> \endverbatim
156*>
157*> \param[out] IWORK
158*> \verbatim
159*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
160*>          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
161*> \endverbatim
162*>
163*> \param[in] LIWORK
164*> \verbatim
165*>          LIWORK is INTEGER
166*>          The dimension of array IWORK.
167*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
168*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
169*>
170*>          If LIWORK = -1, then a workspace query is assumed; the
171*>          routine only calculates the required sizes of the WORK, RWORK
172*>          and IWORK arrays, returns these values as the first entries
173*>          of the WORK, RWORK and IWORK arrays, and no error message
174*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
175*> \endverbatim
176*>
177*> \param[out] INFO
178*> \verbatim
179*>          INFO is INTEGER
180*>          = 0:  successful exit
181*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
182*>          > 0:  if INFO = i, the algorithm failed to converge; i
183*>                off-diagonal elements of an intermediate tridiagonal
184*>                form did not converge to zero.
185*> \endverbatim
186*
187*  Authors:
188*  ========
189*
190*> \author Univ. of Tennessee
191*> \author Univ. of California Berkeley
192*> \author Univ. of Colorado Denver
193*> \author NAG Ltd.
194*
195*> \ingroup complex16OTHEReigen
196*
197*  =====================================================================
198      SUBROUTINE ZHPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
199     $                   RWORK, LRWORK, IWORK, LIWORK, INFO )
200*
201*  -- LAPACK driver routine --
202*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
203*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
204*
205*     .. Scalar Arguments ..
206      CHARACTER          JOBZ, UPLO
207      INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
208*     ..
209*     .. Array Arguments ..
210      INTEGER            IWORK( * )
211      DOUBLE PRECISION   RWORK( * ), W( * )
212      COMPLEX*16         AP( * ), WORK( * ), Z( LDZ, * )
213*     ..
214*
215*  =====================================================================
216*
217*     .. Parameters ..
218      DOUBLE PRECISION   ZERO, ONE
219      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
220      COMPLEX*16         CONE
221      PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
222*     ..
223*     .. Local Scalars ..
224      LOGICAL            LQUERY, WANTZ
225      INTEGER            IINFO, IMAX, INDE, INDRWK, INDTAU, INDWRK,
226     $                   ISCALE, LIWMIN, LLRWK, LLWRK, LRWMIN, LWMIN
227      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
228     $                   SMLNUM
229*     ..
230*     .. External Functions ..
231      LOGICAL            LSAME
232      DOUBLE PRECISION   DLAMCH, ZLANHP
233      EXTERNAL           LSAME, DLAMCH, ZLANHP
234*     ..
235*     .. External Subroutines ..
236      EXTERNAL           DSCAL, DSTERF, XERBLA, ZDSCAL, ZHPTRD, ZSTEDC,
237     $                   ZUPMTR
238*     ..
239*     .. Intrinsic Functions ..
240      INTRINSIC          SQRT
241*     ..
242*     .. Executable Statements ..
243*
244*     Test the input parameters.
245*
246      WANTZ = LSAME( JOBZ, 'V' )
247      LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
248*
249      INFO = 0
250      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
251         INFO = -1
252      ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
253     $          THEN
254         INFO = -2
255      ELSE IF( N.LT.0 ) THEN
256         INFO = -3
257      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
258         INFO = -7
259      END IF
260*
261      IF( INFO.EQ.0 ) THEN
262         IF( N.LE.1 ) THEN
263            LWMIN = 1
264            LIWMIN = 1
265            LRWMIN = 1
266         ELSE
267            IF( WANTZ ) THEN
268               LWMIN = 2*N
269               LRWMIN = 1 + 5*N + 2*N**2
270               LIWMIN = 3 + 5*N
271            ELSE
272               LWMIN = N
273               LRWMIN = N
274               LIWMIN = 1
275            END IF
276         END IF
277         WORK( 1 ) = LWMIN
278         RWORK( 1 ) = LRWMIN
279         IWORK( 1 ) = LIWMIN
280*
281         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
282            INFO = -9
283         ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
284            INFO = -11
285         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
286            INFO = -13
287         END IF
288      END IF
289*
290      IF( INFO.NE.0 ) THEN
291         CALL XERBLA( 'ZHPEVD', -INFO )
292         RETURN
293      ELSE IF( LQUERY ) THEN
294         RETURN
295      END IF
296*
297*     Quick return if possible
298*
299      IF( N.EQ.0 )
300     $   RETURN
301*
302      IF( N.EQ.1 ) THEN
303         W( 1 ) = DBLE( AP( 1 ) )
304         IF( WANTZ )
305     $      Z( 1, 1 ) = CONE
306         RETURN
307      END IF
308*
309*     Get machine constants.
310*
311      SAFMIN = DLAMCH( 'Safe minimum' )
312      EPS = DLAMCH( 'Precision' )
313      SMLNUM = SAFMIN / EPS
314      BIGNUM = ONE / SMLNUM
315      RMIN = SQRT( SMLNUM )
316      RMAX = SQRT( BIGNUM )
317*
318*     Scale matrix to allowable range, if necessary.
319*
320      ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK )
321      ISCALE = 0
322      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
323         ISCALE = 1
324         SIGMA = RMIN / ANRM
325      ELSE IF( ANRM.GT.RMAX ) THEN
326         ISCALE = 1
327         SIGMA = RMAX / ANRM
328      END IF
329      IF( ISCALE.EQ.1 ) THEN
330         CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
331      END IF
332*
333*     Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
334*
335      INDE = 1
336      INDTAU = 1
337      INDRWK = INDE + N
338      INDWRK = INDTAU + N
339      LLWRK = LWORK - INDWRK + 1
340      LLRWK = LRWORK - INDRWK + 1
341      CALL ZHPTRD( UPLO, N, AP, W, RWORK( INDE ), WORK( INDTAU ),
342     $             IINFO )
343*
344*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
345*     ZUPGTR to generate the orthogonal matrix, then call ZSTEDC.
346*
347      IF( .NOT.WANTZ ) THEN
348         CALL DSTERF( N, W, RWORK( INDE ), INFO )
349      ELSE
350         CALL ZSTEDC( 'I', N, W, RWORK( INDE ), Z, LDZ, WORK( INDWRK ),
351     $                LLWRK, RWORK( INDRWK ), LLRWK, IWORK, LIWORK,
352     $                INFO )
353         CALL ZUPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ,
354     $                WORK( INDWRK ), IINFO )
355      END IF
356*
357*     If matrix was scaled, then rescale eigenvalues appropriately.
358*
359      IF( ISCALE.EQ.1 ) THEN
360         IF( INFO.EQ.0 ) THEN
361            IMAX = N
362         ELSE
363            IMAX = INFO - 1
364         END IF
365         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
366      END IF
367*
368      WORK( 1 ) = LWMIN
369      RWORK( 1 ) = LRWMIN
370      IWORK( 1 ) = LIWMIN
371      RETURN
372*
373*     End of ZHPEVD
374*
375      END
376