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*
8*> \htmlonly
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhpevd.f">
15*> [TXT]</a>
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*> \date June 2017
196*
197*> \ingroup complex16OTHEReigen
198*
199*  =====================================================================
200      SUBROUTINE ZHPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
201     $                   RWORK, LRWORK, IWORK, LIWORK, INFO )
202*
203*  -- LAPACK driver routine (version 3.7.1) --
204*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
205*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
206*     June 2017
207*
208*     .. Scalar Arguments ..
209      CHARACTER          JOBZ, UPLO
210      INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
211*     ..
212*     .. Array Arguments ..
213      INTEGER            IWORK( * )
214      DOUBLE PRECISION   RWORK( * ), W( * )
215      COMPLEX*16         AP( * ), WORK( * ), Z( LDZ, * )
216*     ..
217*
218*  =====================================================================
219*
220*     .. Parameters ..
221      DOUBLE PRECISION   ZERO, ONE
222      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
223      COMPLEX*16         CONE
224      PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
225*     ..
226*     .. Local Scalars ..
227      LOGICAL            LQUERY, WANTZ
228      INTEGER            IINFO, IMAX, INDE, INDRWK, INDTAU, INDWRK,
229     $                   ISCALE, LIWMIN, LLRWK, LLWRK, LRWMIN, LWMIN
230      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
231     $                   SMLNUM
232*     ..
233*     .. External Functions ..
234      LOGICAL            LSAME
235      DOUBLE PRECISION   DLAMCH, ZLANHP
236      EXTERNAL           LSAME, DLAMCH, ZLANHP
237*     ..
238*     .. External Subroutines ..
239      EXTERNAL           DSCAL, DSTERF, XERBLA, ZDSCAL, ZHPTRD, ZSTEDC,
240     $                   ZUPMTR
241*     ..
242*     .. Intrinsic Functions ..
243      INTRINSIC          SQRT
244*     ..
245*     .. Executable Statements ..
246*
247*     Test the input parameters.
248*
249      WANTZ = LSAME( JOBZ, 'V' )
250      LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
251*
252      INFO = 0
253      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
254         INFO = -1
255      ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
256     $          THEN
257         INFO = -2
258      ELSE IF( N.LT.0 ) THEN
259         INFO = -3
260      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
261         INFO = -7
262      END IF
263*
264      IF( INFO.EQ.0 ) THEN
265         IF( N.LE.1 ) THEN
266            LWMIN = 1
267            LIWMIN = 1
268            LRWMIN = 1
269         ELSE
270            IF( WANTZ ) THEN
271               LWMIN = 2*N
272               LRWMIN = 1 + 5*N + 2*N**2
273               LIWMIN = 3 + 5*N
274            ELSE
275               LWMIN = N
276               LRWMIN = N
277               LIWMIN = 1
278            END IF
279         END IF
280         WORK( 1 ) = LWMIN
281         RWORK( 1 ) = LRWMIN
282         IWORK( 1 ) = LIWMIN
283*
284         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
285            INFO = -9
286         ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
287            INFO = -11
288         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
289            INFO = -13
290         END IF
291      END IF
292*
293      IF( INFO.NE.0 ) THEN
294         CALL XERBLA( 'ZHPEVD', -INFO )
295         RETURN
296      ELSE IF( LQUERY ) THEN
297         RETURN
298      END IF
299*
300*     Quick return if possible
301*
302      IF( N.EQ.0 )
303     $   RETURN
304*
305      IF( N.EQ.1 ) THEN
306         W( 1 ) = AP( 1 )
307         IF( WANTZ )
308     $      Z( 1, 1 ) = CONE
309         RETURN
310      END IF
311*
312*     Get machine constants.
313*
314      SAFMIN = DLAMCH( 'Safe minimum' )
315      EPS = DLAMCH( 'Precision' )
316      SMLNUM = SAFMIN / EPS
317      BIGNUM = ONE / SMLNUM
318      RMIN = SQRT( SMLNUM )
319      RMAX = SQRT( BIGNUM )
320*
321*     Scale matrix to allowable range, if necessary.
322*
323      ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK )
324      ISCALE = 0
325      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
326         ISCALE = 1
327         SIGMA = RMIN / ANRM
328      ELSE IF( ANRM.GT.RMAX ) THEN
329         ISCALE = 1
330         SIGMA = RMAX / ANRM
331      END IF
332      IF( ISCALE.EQ.1 ) THEN
333         CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
334      END IF
335*
336*     Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
337*
338      INDE = 1
339      INDTAU = 1
340      INDRWK = INDE + N
341      INDWRK = INDTAU + N
342      LLWRK = LWORK - INDWRK + 1
343      LLRWK = LRWORK - INDRWK + 1
344      CALL ZHPTRD( UPLO, N, AP, W, RWORK( INDE ), WORK( INDTAU ),
345     $             IINFO )
346*
347*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
348*     ZUPGTR to generate the orthogonal matrix, then call ZSTEDC.
349*
350      IF( .NOT.WANTZ ) THEN
351         CALL DSTERF( N, W, RWORK( INDE ), INFO )
352      ELSE
353         CALL ZSTEDC( 'I', N, W, RWORK( INDE ), Z, LDZ, WORK( INDWRK ),
354     $                LLWRK, RWORK( INDRWK ), LLRWK, IWORK, LIWORK,
355     $                INFO )
356         CALL ZUPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ,
357     $                WORK( INDWRK ), IINFO )
358      END IF
359*
360*     If matrix was scaled, then rescale eigenvalues appropriately.
361*
362      IF( ISCALE.EQ.1 ) THEN
363         IF( INFO.EQ.0 ) THEN
364            IMAX = N
365         ELSE
366            IMAX = INFO - 1
367         END IF
368         CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
369      END IF
370*
371      WORK( 1 ) = LWMIN
372      RWORK( 1 ) = LRWMIN
373      IWORK( 1 ) = LIWMIN
374      RETURN
375*
376*     End of ZHPEVD
377*
378      END
379