1*> \brief \b ZHPGVX
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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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
22*                          IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
23*                          IWORK, IFAIL, INFO )
24*
25*       .. Scalar Arguments ..
26*       CHARACTER          JOBZ, RANGE, UPLO
27*       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
28*       DOUBLE PRECISION   ABSTOL, VL, VU
29*       ..
30*       .. Array Arguments ..
31*       INTEGER            IFAIL( * ), IWORK( * )
32*       DOUBLE PRECISION   RWORK( * ), W( * )
33*       COMPLEX*16         AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
34*       ..
35*
36*
37*> \par Purpose:
38*  =============
39*>
40*> \verbatim
41*>
42*> ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
43*> of a complex generalized Hermitian-definite eigenproblem, of the form
44*> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
45*> B are assumed to be Hermitian, stored in packed format, and B is also
46*> positive definite.  Eigenvalues and eigenvectors can be selected by
47*> specifying either a range of values or a range of indices for the
48*> desired eigenvalues.
49*> \endverbatim
50*
51*  Arguments:
52*  ==========
53*
54*> \param[in] ITYPE
55*> \verbatim
56*>          ITYPE is INTEGER
57*>          Specifies the problem type to be solved:
58*>          = 1:  A*x = (lambda)*B*x
59*>          = 2:  A*B*x = (lambda)*x
60*>          = 3:  B*A*x = (lambda)*x
61*> \endverbatim
62*>
63*> \param[in] JOBZ
64*> \verbatim
65*>          JOBZ is CHARACTER*1
66*>          = 'N':  Compute eigenvalues only;
67*>          = 'V':  Compute eigenvalues and eigenvectors.
68*> \endverbatim
69*>
70*> \param[in] RANGE
71*> \verbatim
72*>          RANGE is CHARACTER*1
73*>          = 'A': all eigenvalues will be found;
74*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
75*>                 will be found;
76*>          = 'I': the IL-th through IU-th eigenvalues will be found.
77*> \endverbatim
78*>
79*> \param[in] UPLO
80*> \verbatim
81*>          UPLO is CHARACTER*1
82*>          = 'U':  Upper triangles of A and B are stored;
83*>          = 'L':  Lower triangles of A and B are stored.
84*> \endverbatim
85*>
86*> \param[in] N
87*> \verbatim
88*>          N is INTEGER
89*>          The order of the matrices A and B.  N >= 0.
90*> \endverbatim
91*>
92*> \param[in,out] AP
93*> \verbatim
94*>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
95*>          On entry, the upper or lower triangle of the Hermitian matrix
96*>          A, packed columnwise in a linear array.  The j-th column of A
97*>          is stored in the array AP as follows:
98*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
99*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
100*>
101*>          On exit, the contents of AP are destroyed.
102*> \endverbatim
103*>
104*> \param[in,out] BP
105*> \verbatim
106*>          BP is COMPLEX*16 array, dimension (N*(N+1)/2)
107*>          On entry, the upper or lower triangle of the Hermitian matrix
108*>          B, packed columnwise in a linear array.  The j-th column of B
109*>          is stored in the array BP as follows:
110*>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
111*>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
112*>
113*>          On exit, the triangular factor U or L from the Cholesky
114*>          factorization B = U**H*U or B = L*L**H, in the same storage
115*>          format as B.
116*> \endverbatim
117*>
118*> \param[in] VL
119*> \verbatim
120*>          VL is DOUBLE PRECISION
121*>
122*>          If RANGE='V', the lower bound of the interval to
123*>          be searched for eigenvalues. VL < VU.
124*>          Not referenced if RANGE = 'A' or 'I'.
125*> \endverbatim
126*>
127*> \param[in] VU
128*> \verbatim
129*>          VU is DOUBLE PRECISION
130*>
131*>          If RANGE='V', the upper bound of the interval to
132*>          be searched for eigenvalues. VL < VU.
133*>          Not referenced if RANGE = 'A' or 'I'.
134*> \endverbatim
135*>
136*> \param[in] IL
137*> \verbatim
138*>          IL is INTEGER
139*>
140*>          If RANGE='I', the index of the
141*>          smallest eigenvalue to be returned.
142*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
143*>          Not referenced if RANGE = 'A' or 'V'.
144*> \endverbatim
145*>
146*> \param[in] IU
147*> \verbatim
148*>          IU is INTEGER
149*>
150*>          If RANGE='I', the index of the
151*>          largest eigenvalue to be returned.
152*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
153*>          Not referenced if RANGE = 'A' or 'V'.
154*> \endverbatim
155*>
156*> \param[in] ABSTOL
157*> \verbatim
158*>          ABSTOL is DOUBLE PRECISION
159*>          The absolute error tolerance for the eigenvalues.
160*>          An approximate eigenvalue is accepted as converged
161*>          when it is determined to lie in an interval [a,b]
162*>          of width less than or equal to
163*>
164*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
165*>
166*>          where EPS is the machine precision.  If ABSTOL is less than
167*>          or equal to zero, then  EPS*|T|  will be used in its place,
168*>          where |T| is the 1-norm of the tridiagonal matrix obtained
169*>          by reducing AP to tridiagonal form.
170*>
171*>          Eigenvalues will be computed most accurately when ABSTOL is
172*>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
173*>          If this routine returns with INFO>0, indicating that some
174*>          eigenvectors did not converge, try setting ABSTOL to
175*>          2*DLAMCH('S').
176*> \endverbatim
177*>
178*> \param[out] M
179*> \verbatim
180*>          M is INTEGER
181*>          The total number of eigenvalues found.  0 <= M <= N.
182*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
183*> \endverbatim
184*>
185*> \param[out] W
186*> \verbatim
187*>          W is DOUBLE PRECISION array, dimension (N)
188*>          On normal exit, the first M elements contain the selected
189*>          eigenvalues in ascending order.
190*> \endverbatim
191*>
192*> \param[out] Z
193*> \verbatim
194*>          Z is COMPLEX*16 array, dimension (LDZ, N)
195*>          If JOBZ = 'N', then Z is not referenced.
196*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
197*>          contain the orthonormal eigenvectors of the matrix A
198*>          corresponding to the selected eigenvalues, with the i-th
199*>          column of Z holding the eigenvector associated with W(i).
200*>          The eigenvectors are normalized as follows:
201*>          if ITYPE = 1 or 2, Z**H*B*Z = I;
202*>          if ITYPE = 3, Z**H*inv(B)*Z = I.
203*>
204*>          If an eigenvector fails to converge, then that column of Z
205*>          contains the latest approximation to the eigenvector, and the
206*>          index of the eigenvector is returned in IFAIL.
207*>          Note: the user must ensure that at least max(1,M) columns are
208*>          supplied in the array Z; if RANGE = 'V', the exact value of M
209*>          is not known in advance and an upper bound must be used.
210*> \endverbatim
211*>
212*> \param[in] LDZ
213*> \verbatim
214*>          LDZ is INTEGER
215*>          The leading dimension of the array Z.  LDZ >= 1, and if
216*>          JOBZ = 'V', LDZ >= max(1,N).
217*> \endverbatim
218*>
219*> \param[out] WORK
220*> \verbatim
221*>          WORK is COMPLEX*16 array, dimension (2*N)
222*> \endverbatim
223*>
224*> \param[out] RWORK
225*> \verbatim
226*>          RWORK is DOUBLE PRECISION array, dimension (7*N)
227*> \endverbatim
228*>
229*> \param[out] IWORK
230*> \verbatim
231*>          IWORK is INTEGER array, dimension (5*N)
232*> \endverbatim
233*>
234*> \param[out] IFAIL
235*> \verbatim
236*>          IFAIL is INTEGER array, dimension (N)
237*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
238*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
239*>          indices of the eigenvectors that failed to converge.
240*>          If JOBZ = 'N', then IFAIL is not referenced.
241*> \endverbatim
242*>
243*> \param[out] INFO
244*> \verbatim
245*>          INFO is INTEGER
246*>          = 0:  successful exit
247*>          < 0:  if INFO = -i, the i-th argument had an illegal value
248*>          > 0:  ZPPTRF or ZHPEVX returned an error code:
249*>             <= N:  if INFO = i, ZHPEVX failed to converge;
250*>                    i eigenvectors failed to converge.  Their indices
251*>                    are stored in array IFAIL.
252*>             > N:   if INFO = N + i, for 1 <= i <= n, then the leading
253*>                    minor of order i of B is not positive definite.
254*>                    The factorization of B could not be completed and
255*>                    no eigenvalues or eigenvectors were computed.
256*> \endverbatim
257*
258*  Authors:
259*  ========
260*
261*> \author Univ. of Tennessee
262*> \author Univ. of California Berkeley
263*> \author Univ. of Colorado Denver
264*> \author NAG Ltd.
265*
266*> \date June 2016
267*
268*> \ingroup complex16OTHEReigen
269*
270*> \par Contributors:
271*  ==================
272*>
273*>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
274*
275*  =====================================================================
276      SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
277     $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
278     $                   IWORK, IFAIL, INFO )
279*
280*  -- LAPACK driver routine (version 3.7.0) --
281*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
282*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
283*     June 2016
284*
285*     .. Scalar Arguments ..
286      CHARACTER          JOBZ, RANGE, UPLO
287      INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
288      DOUBLE PRECISION   ABSTOL, VL, VU
289*     ..
290*     .. Array Arguments ..
291      INTEGER            IFAIL( * ), IWORK( * )
292      DOUBLE PRECISION   RWORK( * ), W( * )
293      COMPLEX*16         AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
294*     ..
295*
296*  =====================================================================
297*
298*     .. Local Scalars ..
299      LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
300      CHARACTER          TRANS
301      INTEGER            J
302*     ..
303*     .. External Functions ..
304      LOGICAL            LSAME
305      EXTERNAL           LSAME
306*     ..
307*     .. External Subroutines ..
308      EXTERNAL           XERBLA, ZHPEVX, ZHPGST, ZPPTRF, ZTPMV, ZTPSV
309*     ..
310*     .. Intrinsic Functions ..
311      INTRINSIC          MIN
312*     ..
313*     .. Executable Statements ..
314*
315*     Test the input parameters.
316*
317      WANTZ = LSAME( JOBZ, 'V' )
318      UPPER = LSAME( UPLO, 'U' )
319      ALLEIG = LSAME( RANGE, 'A' )
320      VALEIG = LSAME( RANGE, 'V' )
321      INDEIG = LSAME( RANGE, 'I' )
322*
323      INFO = 0
324      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
325         INFO = -1
326      ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
327         INFO = -2
328      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
329         INFO = -3
330      ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
331         INFO = -4
332      ELSE IF( N.LT.0 ) THEN
333         INFO = -5
334      ELSE
335         IF( VALEIG ) THEN
336            IF( N.GT.0 .AND. VU.LE.VL ) THEN
337               INFO = -9
338            END IF
339         ELSE IF( INDEIG ) THEN
340            IF( IL.LT.1 ) THEN
341               INFO = -10
342            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
343               INFO = -11
344            END IF
345         END IF
346      END IF
347      IF( INFO.EQ.0 ) THEN
348         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
349            INFO = -16
350         END IF
351      END IF
352*
353      IF( INFO.NE.0 ) THEN
354         CALL XERBLA( 'ZHPGVX', -INFO )
355         RETURN
356      END IF
357*
358*     Quick return if possible
359*
360      IF( N.EQ.0 )
361     $   RETURN
362*
363*     Form a Cholesky factorization of B.
364*
365      CALL ZPPTRF( UPLO, N, BP, INFO )
366      IF( INFO.NE.0 ) THEN
367         INFO = N + INFO
368         RETURN
369      END IF
370*
371*     Transform problem to standard eigenvalue problem and solve.
372*
373      CALL ZHPGST( ITYPE, UPLO, N, AP, BP, INFO )
374      CALL ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
375     $             W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
376*
377      IF( WANTZ ) THEN
378*
379*        Backtransform eigenvectors to the original problem.
380*
381         IF( INFO.GT.0 )
382     $      M = INFO - 1
383         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
384*
385*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
386*           backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
387*
388            IF( UPPER ) THEN
389               TRANS = 'N'
390            ELSE
391               TRANS = 'C'
392            END IF
393*
394            DO 10 J = 1, M
395               CALL ZTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
396     $                     1 )
397   10       CONTINUE
398*
399         ELSE IF( ITYPE.EQ.3 ) THEN
400*
401*           For B*A*x=(lambda)*x;
402*           backtransform eigenvectors: x = L*y or U**H *y
403*
404            IF( UPPER ) THEN
405               TRANS = 'C'
406            ELSE
407               TRANS = 'N'
408            END IF
409*
410            DO 20 J = 1, M
411               CALL ZTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
412     $                     1 )
413   20       CONTINUE
414         END IF
415      END IF
416*
417      RETURN
418*
419*     End of ZHPGVX
420*
421      END
422