1       SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
  2      $                   Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK,
  3      $                   LIWORK, INFO )
  4 *
  5 *  -- LAPACK driver routine (version 3.3.1) --
  6 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  7 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  8 *  -- April 2011                                                      --
  9 * @precisions normal z -> c
 10 *
 11 *     .. Scalar Arguments ..
 12       CHARACTER          JOBZ, UPLO
 13       INTEGER            INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK,
 14      $                   LWORK, N
 15 *     ..
 16 *     .. Array Arguments ..
 17       INTEGER            IWORK( * )
 18       DOUBLE PRECISION   RWORK( * ), W( * )
 19       COMPLEX*16         AB( LDAB, * ), BB( LDBB, * ), WORK( * ),
 20      $                   Z( LDZ, * )
 21 *     ..
 22 *
 23 *  Purpose
 24 *  =======
 25 *
 26 *  ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors
 27 *  of a complex generalized Hermitian-definite banded eigenproblem, of
 28 *  the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
 29 *  and banded, and B is also positive definite.  If eigenvectors are
 30 *  desired, it uses a divide and conquer algorithm.
 31 *
 32 *  The divide and conquer algorithm makes very mild assumptions about
 33 *  floating point arithmetic. It will work on machines with a guard
 34 *  digit in add/subtract, or on those binary machines without guard
 35 *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 36 *  Cray-2. It could conceivably fail on hexadecimal or decimal machines
 37 *  without guard digits, but we know of none.
 38 *
 39 *  Arguments
 40 *  =========
 41 *
 42 *  JOBZ    (input) CHARACTER*1
 43 *          = 'N':  Compute eigenvalues only;
 44 *          = 'V':  Compute eigenvalues and eigenvectors.
 45 *
 46 *  UPLO    (input) CHARACTER*1
 47 *          = 'U':  Upper triangles of A and B are stored;
 48 *          = 'L':  Lower triangles of A and B are stored.
 49 *
 50 *  N       (input) INTEGER
 51 *          The order of the matrices A and B.  N >= 0.
 52 *
 53 *  KA      (input) INTEGER
 54 *          The number of superdiagonals of the matrix A if UPLO = 'U',
 55 *          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
 56 *
 57 *  KB      (input) INTEGER
 58 *          The number of superdiagonals of the matrix B if UPLO = 'U',
 59 *          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
 60 *
 61 *  AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
 62 *          On entry, the upper or lower triangle of the Hermitian band
 63 *          matrix A, stored in the first ka+1 rows of the array.  The
 64 *          j-th column of A is stored in the j-th column of the array AB
 65 *          as follows:
 66 *          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
 67 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
 68 *
 69 *          On exit, the contents of AB are destroyed.
 70 *
 71 *  LDAB    (input) INTEGER
 72 *          The leading dimension of the array AB.  LDAB >= KA+1.
 73 *
 74 *  BB      (input/output) COMPLEX*16 array, dimension (LDBB, N)
 75 *          On entry, the upper or lower triangle of the Hermitian band
 76 *          matrix B, stored in the first kb+1 rows of the array.  The
 77 *          j-th column of B is stored in the j-th column of the array BB
 78 *          as follows:
 79 *          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
 80 *          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
 81 *
 82 *          On exit, the factor S from the split Cholesky factorization
 83 *          B = S**H*S, as returned by ZPBSTF.
 84 *
 85 *  LDBB    (input) INTEGER
 86 *          The leading dimension of the array BB.  LDBB >= KB+1.
 87 *
 88 *  W       (output) DOUBLE PRECISION array, dimension (N)
 89 *          If INFO = 0, the eigenvalues in ascending order.
 90 *
 91 *  Z       (output) COMPLEX*16 array, dimension (LDZ, N)
 92 *          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
 93 *          eigenvectors, with the i-th column of Z holding the
 94 *          eigenvector associated with W(i). The eigenvectors are
 95 *          normalized so that Z**H*B*Z = I.
 96 *          If JOBZ = 'N', then Z is not referenced.
 97 *
 98 *  LDZ     (input) INTEGER
 99 *          The leading dimension of the array Z.  LDZ >= 1, and if
100 *          JOBZ = 'V', LDZ >= N.
101 *
102 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
103 *          On exit, if INFO=0, WORK(1) returns the optimal LWORK.
104 *
105 *  LWORK   (input) INTEGER
106 *          The dimension of the array WORK.
107 *          If N <= 1,               LWORK >= 1.
108 *          If JOBZ = 'N' and N > 1, LWORK >= N.
109 *          If JOBZ = 'V' and N > 1, LWORK >= 2*N**2.
110 *
111 *          If LWORK = -1, then a workspace query is assumed; the routine
112 *          only calculates the optimal sizes of the WORK, RWORK and
113 *          IWORK arrays, returns these values as the first entries of
114 *          the WORK, RWORK and IWORK arrays, and no error message
115 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
116 *
117 *  RWORK   (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
118 *          On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.
119 *
120 *  LRWORK  (input) INTEGER
121 *          The dimension of array RWORK.
122 *          If N <= 1,               LRWORK >= 1.
123 *          If JOBZ = 'N' and N > 1, LRWORK >= N.
124 *          If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
125 *
126 *          If LRWORK = -1, then a workspace query is assumed; the
127 *          routine only calculates the optimal sizes of the WORK, RWORK
128 *          and IWORK arrays, returns these values as the first entries
129 *          of the WORK, RWORK and IWORK arrays, and no error message
130 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
131 *
132 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
133 *          On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.
134 *
135 *  LIWORK  (input) INTEGER
136 *          The dimension of array IWORK.
137 *          If JOBZ = 'N' or N <= 1, LIWORK >= 1.
138 *          If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
139 *
140 *          If LIWORK = -1, then a workspace query is assumed; the
141 *          routine only calculates the optimal sizes of the WORK, RWORK
142 *          and IWORK arrays, returns these values as the first entries
143 *          of the WORK, RWORK and IWORK arrays, and no error message
144 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
145 *
146 *  INFO    (output) INTEGER
147 *          = 0:  successful exit
148 *          < 0:  if INFO = -i, the i-th argument had an illegal value
149 *          > 0:  if INFO = i, and i is:
150 *             <= N:  the algorithm failed to converge:
151 *                    i off-diagonal elements of an intermediate
152 *                    tridiagonal form did not converge to zero;
153 *             > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
154 *                    returned INFO = i: B is not positive definite.
155 *                    The factorization of B could not be completed and
156 *                    no eigenvalues or eigenvectors were computed.
157 *
158 *  Further Details
159 *  ===============
160 *
161 *  Based on contributions by
162 *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
163 *
164 *  =====================================================================
165 *
166 *     .. Parameters ..
167       COMPLEX*16         CONE, CZERO
168       PARAMETER          ( CONE = ( 1.0D+00.0D+0 ),
169      $                   CZERO = ( 0.0D+00.0D+0 ) )
170 *     ..
171 *     .. Local Scalars ..
172       LOGICAL            LQUERY, UPPER, WANTZ
173       CHARACTER          VECT
174       INTEGER            IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLRWK,
175      $                   LLWK2, LRWMIN, LWMIN
176 *     ..
177 *     .. External Functions ..
178       LOGICAL            LSAME
179       EXTERNAL           LSAME
180 *     ..
181 *     .. External Subroutines ..
182       EXTERNAL           DSTERF, XERBLA, ZGEMM, ZHBGST, ZHBTRD, ZLACPY,
183      $                   ZPBSTF, ZSTEDC
184 *     ..
185 *     .. Executable Statements ..
186 *
187 *     Test the input parameters.
188 *
189       WANTZ = LSAME( JOBZ, 'V' )
190       UPPER = LSAME( UPLO, 'U' )
191       LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
192 *
193       INFO = 0
194       IF( N.LE.1 ) THEN
195          LWMIN = 1+N
196          LRWMIN = 1+N
197          LIWMIN = 1
198       ELSE IF( WANTZ ) THEN
199          LWMIN = 2*N**2
200          LRWMIN = 1 + 5*+ 2*N**2
201          LIWMIN = 3 + 5*N
202       ELSE
203          LWMIN = N
204          LRWMIN = N
205          LIWMIN = 1
206       END IF
207       IF.NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
208          INFO = -1
209       ELSE IF.NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
210          INFO = -2
211       ELSE IF( N.LT.0 ) THEN
212          INFO = -3
213       ELSE IF( KA.LT.0 ) THEN
214          INFO = -4
215       ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
216          INFO = -5
217       ELSE IF( LDAB.LT.KA+1 ) THEN
218          INFO = -7
219       ELSE IF( LDBB.LT.KB+1 ) THEN
220          INFO = -9
221       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
222          INFO = -12
223       END IF
224 *
225       IF( INFO.EQ.0 ) THEN
226          WORK( 1 ) = LWMIN
227          RWORK( 1 ) = LRWMIN
228          IWORK( 1 ) = LIWMIN
229 *
230          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
231             INFO = -14
232          ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
233             INFO = -16
234          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
235             INFO = -18
236          END IF
237       END IF
238 *
239       IF( INFO.NE.0 ) THEN
240          CALL XERBLA( 'ZHBGVD'-INFO )
241          RETURN
242       ELSE IF( LQUERY ) THEN
243          RETURN
244       END IF
245 *
246 *     Quick return if possible
247 *
248       IF( N.EQ.0 )
249      $   RETURN
250 *
251 *     Form a split Cholesky factorization of B.
252 *
253       CALL ZPBSTF( UPLO, N, KB, BB, LDBB, INFO )
254       IF( INFO.NE.0 ) THEN
255          INFO = N + INFO
256          RETURN
257       END IF
258 *
259 *     Transform problem to standard eigenvalue problem.
260 *
261       INDE = 1
262       INDWRK = INDE + N
263       INDWK2 = 1 + N*N
264       LLWK2 = LWORK - INDWK2 + 2
265       LLRWK = LRWORK - INDWRK + 2
266       CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
267      $             WORK, RWORK( INDWRK ), IINFO )
268 *
269 *     Reduce Hermitian band matrix to tridiagonal form.
270 *
271       IF( WANTZ ) THEN
272          VECT = 'U'
273       ELSE
274          VECT = 'N'
275       END IF
276       CALL ZHBTRD( VECT, UPLO, N, KA, AB, LDAB, W, RWORK( INDE ), Z,
277      $             LDZ, WORK, IINFO )
278 *
279 *     For eigenvalues only, call DSTERF.  For eigenvectors, call ZSTEDC.
280 *
281       IF.NOT.WANTZ ) THEN
282          CALL DSTERF( N, W, RWORK( INDE ), INFO )
283       ELSE
284          CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ),
285      $                LLWK2, RWORK( INDWRK ), LLRWK, IWORK, LIWORK,
286      $                INFO )
287          CALL ZGEMM( 'N''N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO,
288      $               WORK( INDWK2 ), N )
289          CALL ZLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
290       END IF
291 *
292       WORK( 1 ) = LWMIN
293       RWORK( 1 ) = LRWMIN
294       IWORK( 1 ) = LIWMIN
295       RETURN
296 *
297 *     End of ZHBGVD
298 *
299       END