1
       2
       3
       4
       5
       6
       7
       8
       9
      10
      11
      12
      13
      14
      15
      16
      17
      18
      19
      20
      21
      22
      23
      24
      25
      26
      27
      28
      29
      30
      31
      32
      33
      34
      35
      36
      37
      38
      39
      40
      41
      42
      43
      44
      45
      46
      47
      48
      49
      50
      51
      52
      53
      54
      55
      56
      57
      58
      59
      60
      61
      62
      63
      64
      65
      66
      67
      68
      69
      70
      71
      72
      73
      74
      75
      76
      77
      78
      79
      80
      81
      82
      83
      84
      85
      86
      87
      88
      89
      90
      91
      92
      93
      94
      95
      96
      97
      98
      99
     100
     101
     102
     103
     104
     105
     106
     107
     108
     109
     110
     111
     112
     113
     114
     115
     116
     117
     118
     119
     120
     121
     122
     123
     124
     125
     126
     127
     128
     129
     130
     131
     132
     133
     134
     135
     136
     137
     138
     139
     140
     141
     142
     143
     144
     145
     146
     147
     148
     149
     150
     151
     152
     153
     154
     155
     156
     157
     158
     159
     160
     161
     162
     163
     164
     165
     166
     167
     168
     169
     170
     171
     172
     173
     174
     175
     176
     177
     178
     179
     180
     181
     182
     183
     184
     185
     186
     187
     188
     189
     190
     191
     192
     193
     194
     195
     196
     197
     198
     199
     200
     201
     202
     203
     204
     205
     206
     207
     208
     209
     210
     211
     212
     213
     214
     215
     216
     217
     218
     219
     220
     221
     222
     223
     224
     225
     226
     227
     228
     229
     230
     231
     232
     233
     234
     235
     236
     237
     238
     239
     240
     241
     242
     243
     244
     245
     246
     247
     248
     249
     250
     251
     252
     253
     254
     255
     256
     257
     258
     259
     260
     261
     262
     263
     264
     265
     266
     267
     268
     269
     270
     271
     272
     273
     274
     275
     276
     277
     278
     279
     280
     281
     282
     283
     284
     285
     286
     287
     288
     289
     290
     291
     292
     293
     294
     295
     296
     297
     298
     299
     300
     301
     302
     303
     304
     305
     306
     307
     308
     309
     310
     311
     312
     313
     314
     315
     316
     317
     318
     319
     320
     321
     322
     323
     324
     325
     326
     327
     328
     329
     330
     331
     332
     333
     334
     335
     336
     337
      SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
     $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
     $                   LWORK, RWORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine (version 3.3.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*  -- April 2011                                                      --
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
      DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
     $                   Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  ZHEGVX computes selected eigenvalues, and optionally, eigenvectors
*  of a complex generalized Hermitian-definite eigenproblem, of the form
*  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
*  B are assumed to be Hermitian and B is also positive definite.
*  Eigenvalues and eigenvectors can be selected by specifying either a
*  range of values or a range of indices for the desired eigenvalues.
*
*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          Specifies the problem type to be solved:
*          = 1:  A*x = (lambda)*B*x
*          = 2:  A*B*x = (lambda)*x
*          = 3:  B*A*x = (lambda)*x
*
*  JOBZ    (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only;
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  RANGE   (input) CHARACTER*1
*          = 'A': all eigenvalues will be found.
*          = 'V': all eigenvalues in the half-open interval (VL,VU]
*                 will be found.
*          = 'I': the IL-th through IU-th eigenvalues will be found.
**
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangles of A and B are stored;
*          = 'L':  Lower triangles of A and B are stored.
*
*  N       (input) INTEGER
*          The order of the matrices A and B.  N >= 0.
*
*  A       (input/output) COMPLEX*16 array, dimension (LDA, N)
*          On entry, the Hermitian matrix A.  If UPLO = 'U', the
*          leading N-by-N upper triangular part of A contains the
*          upper triangular part of the matrix A.  If UPLO = 'L',
*          the leading N-by-N lower triangular part of A contains
*          the lower triangular part of the matrix A.
*
*          On exit,  the lower triangle (if UPLO='L') or the upper
*          triangle (if UPLO='U') of A, including the diagonal, is
*          destroyed.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  B       (input/output) COMPLEX*16 array, dimension (LDB, N)
*          On entry, the Hermitian matrix B.  If UPLO = 'U', the
*          leading N-by-N upper triangular part of B contains the
*          upper triangular part of the matrix B.  If UPLO = 'L',
*          the leading N-by-N lower triangular part of B contains
*          the lower triangular part of the matrix B.
*
*          On exit, if INFO <= N, the part of B containing the matrix is
*          overwritten by the triangular factor U or L from the Cholesky
*          factorization B = U**H*U or B = L*L**H.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  VL      (input) DOUBLE PRECISION
*  VU      (input) DOUBLE PRECISION
*          If RANGE='V', the lower and upper bounds of the interval to
*          be searched for eigenvalues. VL < VU.
*          Not referenced if RANGE = 'A' or 'I'.
*
*  IL      (input) INTEGER
*  IU      (input) INTEGER
*          If RANGE='I', the indices (in ascending order) of the
*          smallest and largest eigenvalues to be returned.
*          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*          Not referenced if RANGE = 'A' or 'V'.
*
*  ABSTOL  (input) DOUBLE PRECISION
*          The absolute error tolerance for the eigenvalues.
*          An approximate eigenvalue is accepted as converged
*          when it is determined to lie in an interval [a,b]
*          of width less than or equal to
*
*                  ABSTOL + EPS *   max( |a|,|b| ) ,
*
*          where EPS is the machine precision.  If ABSTOL is less than
*          or equal to zero, then  EPS*|T|  will be used in its place,
*          where |T| is the 1-norm of the tridiagonal matrix obtained
*          by reducing A to tridiagonal form.
*
*          Eigenvalues will be computed most accurately when ABSTOL is
*          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*          If this routine returns with INFO>0, indicating that some
*          eigenvectors did not converge, try setting ABSTOL to
*          2*DLAMCH('S').
*
*  M       (output) INTEGER
*          The total number of eigenvalues found.  0 <= M <= N.
*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
*  W       (output) DOUBLE PRECISION array, dimension (N)
*          The first M elements contain the selected
*          eigenvalues in ascending order.
*
*  Z       (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
*          If JOBZ = 'N', then Z is not referenced.
*          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*          contain the orthonormal eigenvectors of the matrix A
*          corresponding to the selected eigenvalues, with the i-th
*          column of Z holding the eigenvector associated with W(i).
*          The eigenvectors are normalized as follows:
*          if ITYPE = 1 or 2, Z**T*B*Z = I;
*          if ITYPE = 3, Z**T*inv(B)*Z = I.
*
*          If an eigenvector fails to converge, then that column of Z
*          contains the latest approximation to the eigenvector, and the
*          index of the eigenvector is returned in IFAIL.
*          Note: the user must ensure that at least max(1,M) columns are
*          supplied in the array Z; if RANGE = 'V', the exact value of M
*          is not known in advance and an upper bound must be used.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The length of the array WORK.  LWORK >= max(1,2*N).
*          For optimal efficiency, LWORK >= (NB+1)*N,
*          where NB is the blocksize for ZHETRD returned by ILAENV.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (7*N)
*
*  IWORK   (workspace) INTEGER array, dimension (5*N)
*
*  IFAIL   (output) INTEGER array, dimension (N)
*          If JOBZ = 'V', then if INFO = 0, the first M elements of
*          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*          indices of the eigenvectors that failed to converge.
*          If JOBZ = 'N', then IFAIL is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  ZPOTRF or ZHEEVX returned an error code:
*             <= N:  if INFO = i, ZHEEVX failed to converge;
*                    i eigenvectors failed to converge.  Their indices
*                    are stored in array IFAIL.
*             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*                    minor of order i of B is not positive definite.
*                    The factorization of B could not be completed and
*                    no eigenvalues or eigenvectors were computed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         CONE
      PARAMETER          ( CONE = ( 1.0D+00.0D+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
      CHARACTER          TRANS
      INTEGER            LWKOPT, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           XERBLA, ZHEEVX, ZHEGST, ZPOTRF, ZTRMM, ZTRSM
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAXMIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      UPPER = LSAME( UPLO, 'U' )
      ALLEIG = LSAME( RANGE'A' )
      VALEIG = LSAME( RANGE'V' )
      INDEIG = LSAME( RANGE'I' )
      LQUERY = ( LWORK.EQ.-1 )
*
      INFO = 0
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         INFO = -1
      ELSE IF.NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF.NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -3
      ELSE IF.NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX1, N ) ) THEN
         INFO = -9
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -11
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX1, N ) ) THEN
               INFO = -12
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -13
            END IF
         END IF
      END IF
      IF (INFO.EQ.0THEN
         IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
            INFO = -18
         END IF
      END IF
*
      IF( INFO.EQ.0 ) THEN
         NB = ILAENV( 1'ZHETRD', UPLO, N, -1-1-1 )
         LWKOPT = MAX1, ( NB + 1 )*N )
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.MAX12*N ) .AND. .NOT.LQUERY ) THEN
            INFO = -20
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZHEGVX'-INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 ) THEN
         RETURN
      END IF
*
*     Form a Cholesky factorization of B.
*
      CALL ZPOTRF( UPLO, N, B, LDB, INFO )
      IF( INFO.NE.0 ) THEN
         INFO = N + INFO
         RETURN
      END IF
*
*     Transform problem to standard eigenvalue problem and solve.
*
      CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
      CALL ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
     $             M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,
     $             INFO )
*
      IF( WANTZ ) THEN
*
*        Backtransform eigenvectors to the original problem.
*
         IF( INFO.GT.0 )
     $      M = INFO - 1
         IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
*
*           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
*           backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
*
            IF( UPPER ) THEN
               TRANS = 'N'
            ELSE
               TRANS = 'C'
            END IF
*
            CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
     $                  LDB, Z, LDZ )
*
         ELSE IF( ITYPE.EQ.3 ) THEN
*
*           For B*A*x=(lambda)*x;
*           backtransform eigenvectors: x = L*y or U**H *y
*
            IF( UPPER ) THEN
               TRANS = 'C'
            ELSE
               TRANS = 'N'
            END IF
*
            CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
     $                  LDB, Z, LDZ )
         END IF
      END IF
*
*     Set WORK(1) to optimal complex workspace size.
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of ZHEGVX
*
      END