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    1: *> \brief \b ZHEGV
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZHEGV + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhegv.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhegv.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhegv.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZHEGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
   22: *                         LWORK, RWORK, INFO )
   23: *
   24: *       .. Scalar Arguments ..
   25: *       CHARACTER          JOBZ, UPLO
   26: *       INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       DOUBLE PRECISION   RWORK( * ), W( * )
   30: *       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
   31: *       ..
   32: *
   33: *
   34: *> \par Purpose:
   35: *  =============
   36: *>
   37: *> \verbatim
   38: *>
   39: *> ZHEGV computes all the eigenvalues, and optionally, the eigenvectors
   40: *> of a complex generalized Hermitian-definite eigenproblem, of the form
   41: *> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
   42: *> Here A and B are assumed to be Hermitian and B is also
   43: *> positive definite.
   44: *> \endverbatim
   45: *
   46: *  Arguments:
   47: *  ==========
   48: *
   49: *> \param[in] ITYPE
   50: *> \verbatim
   51: *>          ITYPE is INTEGER
   52: *>          Specifies the problem type to be solved:
   53: *>          = 1:  A*x = (lambda)*B*x
   54: *>          = 2:  A*B*x = (lambda)*x
   55: *>          = 3:  B*A*x = (lambda)*x
   56: *> \endverbatim
   57: *>
   58: *> \param[in] JOBZ
   59: *> \verbatim
   60: *>          JOBZ is CHARACTER*1
   61: *>          = 'N':  Compute eigenvalues only;
   62: *>          = 'V':  Compute eigenvalues and eigenvectors.
   63: *> \endverbatim
   64: *>
   65: *> \param[in] UPLO
   66: *> \verbatim
   67: *>          UPLO is CHARACTER*1
   68: *>          = 'U':  Upper triangles of A and B are stored;
   69: *>          = 'L':  Lower triangles of A and B are stored.
   70: *> \endverbatim
   71: *>
   72: *> \param[in] N
   73: *> \verbatim
   74: *>          N is INTEGER
   75: *>          The order of the matrices A and B.  N >= 0.
   76: *> \endverbatim
   77: *>
   78: *> \param[in,out] A
   79: *> \verbatim
   80: *>          A is COMPLEX*16 array, dimension (LDA, N)
   81: *>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
   82: *>          leading N-by-N upper triangular part of A contains the
   83: *>          upper triangular part of the matrix A.  If UPLO = 'L',
   84: *>          the leading N-by-N lower triangular part of A contains
   85: *>          the lower triangular part of the matrix A.
   86: *>
   87: *>          On exit, if JOBZ = 'V', then if INFO = 0, A contains the
   88: *>          matrix Z of eigenvectors.  The eigenvectors are normalized
   89: *>          as follows:
   90: *>          if ITYPE = 1 or 2, Z**H*B*Z = I;
   91: *>          if ITYPE = 3, Z**H*inv(B)*Z = I.
   92: *>          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
   93: *>          or the lower triangle (if UPLO='L') of A, including the
   94: *>          diagonal, is destroyed.
   95: *> \endverbatim
   96: *>
   97: *> \param[in] LDA
   98: *> \verbatim
   99: *>          LDA is INTEGER
  100: *>          The leading dimension of the array A.  LDA >= max(1,N).
  101: *> \endverbatim
  102: *>
  103: *> \param[in,out] B
  104: *> \verbatim
  105: *>          B is COMPLEX*16 array, dimension (LDB, N)
  106: *>          On entry, the Hermitian positive definite matrix B.
  107: *>          If UPLO = 'U', the leading N-by-N upper triangular part of B
  108: *>          contains the upper triangular part of the matrix B.
  109: *>          If UPLO = 'L', the leading N-by-N lower triangular part of B
  110: *>          contains the lower triangular part of the matrix B.
  111: *>
  112: *>          On exit, if INFO <= N, the part of B containing the matrix is
  113: *>          overwritten by the triangular factor U or L from the Cholesky
  114: *>          factorization B = U**H*U or B = L*L**H.
  115: *> \endverbatim
  116: *>
  117: *> \param[in] LDB
  118: *> \verbatim
  119: *>          LDB is INTEGER
  120: *>          The leading dimension of the array B.  LDB >= max(1,N).
  121: *> \endverbatim
  122: *>
  123: *> \param[out] W
  124: *> \verbatim
  125: *>          W is DOUBLE PRECISION array, dimension (N)
  126: *>          If INFO = 0, the eigenvalues in ascending order.
  127: *> \endverbatim
  128: *>
  129: *> \param[out] WORK
  130: *> \verbatim
  131: *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
  132: *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  133: *> \endverbatim
  134: *>
  135: *> \param[in] LWORK
  136: *> \verbatim
  137: *>          LWORK is INTEGER
  138: *>          The length of the array WORK.  LWORK >= max(1,2*N-1).
  139: *>          For optimal efficiency, LWORK >= (NB+1)*N,
  140: *>          where NB is the blocksize for ZHETRD returned by ILAENV.
  141: *>
  142: *>          If LWORK = -1, then a workspace query is assumed; the routine
  143: *>          only calculates the optimal size of the WORK array, returns
  144: *>          this value as the first entry of the WORK array, and no error
  145: *>          message related to LWORK is issued by XERBLA.
  146: *> \endverbatim
  147: *>
  148: *> \param[out] RWORK
  149: *> \verbatim
  150: *>          RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))
  151: *> \endverbatim
  152: *>
  153: *> \param[out] INFO
  154: *> \verbatim
  155: *>          INFO is INTEGER
  156: *>          = 0:  successful exit
  157: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  158: *>          > 0:  ZPOTRF or ZHEEV returned an error code:
  159: *>             <= N:  if INFO = i, ZHEEV failed to converge;
  160: *>                    i off-diagonal elements of an intermediate
  161: *>                    tridiagonal form did not converge to zero;
  162: *>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
  163: *>                    minor of order i of B is not positive definite.
  164: *>                    The factorization of B could not be completed and
  165: *>                    no eigenvalues or eigenvectors were computed.
  166: *> \endverbatim
  167: *
  168: *  Authors:
  169: *  ========
  170: *
  171: *> \author Univ. of Tennessee
  172: *> \author Univ. of California Berkeley
  173: *> \author Univ. of Colorado Denver
  174: *> \author NAG Ltd.
  175: *
  176: *> \date December 2016
  177: *
  178: *> \ingroup complex16HEeigen
  179: *
  180: *  =====================================================================
  181:       SUBROUTINE ZHEGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
  182:      $                  LWORK, RWORK, INFO )
  183: *
  184: *  -- LAPACK driver routine (version 3.7.0) --
  185: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  186: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  187: *     December 2016
  188: *
  189: *     .. Scalar Arguments ..
  190:       CHARACTER          JOBZ, UPLO
  191:       INTEGER            INFO, ITYPE, LDA, LDB, LWORK, N
  192: *     ..
  193: *     .. Array Arguments ..
  194:       DOUBLE PRECISION   RWORK( * ), W( * )
  195:       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * )
  196: *     ..
  197: *
  198: *  =====================================================================
  199: *
  200: *     .. Parameters ..
  201:       COMPLEX*16         ONE
  202:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
  203: *     ..
  204: *     .. Local Scalars ..
  205:       LOGICAL            LQUERY, UPPER, WANTZ
  206:       CHARACTER          TRANS
  207:       INTEGER            LWKOPT, NB, NEIG
  208: *     ..
  209: *     .. External Functions ..
  210:       LOGICAL            LSAME
  211:       INTEGER            ILAENV
  212:       EXTERNAL           LSAME, ILAENV
  213: *     ..
  214: *     .. External Subroutines ..
  215:       EXTERNAL           XERBLA, ZHEEV, ZHEGST, ZPOTRF, ZTRMM, ZTRSM
  216: *     ..
  217: *     .. Intrinsic Functions ..
  218:       INTRINSIC          MAX
  219: *     ..
  220: *     .. Executable Statements ..
  221: *
  222: *     Test the input parameters.
  223: *
  224:       WANTZ = LSAME( JOBZ, 'V' )
  225:       UPPER = LSAME( UPLO, 'U' )
  226:       LQUERY = ( LWORK.EQ.-1 )
  227: *
  228:       INFO = 0
  229:       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
  230:          INFO = -1
  231:       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
  232:          INFO = -2
  233:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
  234:          INFO = -3
  235:       ELSE IF( N.LT.0 ) THEN
  236:          INFO = -4
  237:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  238:          INFO = -6
  239:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  240:          INFO = -8
  241:       END IF
  242: *
  243:       IF( INFO.EQ.0 ) THEN
  244:          NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
  245:          LWKOPT = MAX( 1, ( NB + 1 )*N )
  246:          WORK( 1 ) = LWKOPT
  247: *
  248:          IF( LWORK.LT.MAX( 1, 2*N - 1 ) .AND. .NOT.LQUERY ) THEN
  249:             INFO = -11
  250:          END IF
  251:       END IF
  252: *
  253:       IF( INFO.NE.0 ) THEN
  254:          CALL XERBLA( 'ZHEGV ', -INFO )
  255:          RETURN
  256:       ELSE IF( LQUERY ) THEN
  257:          RETURN
  258:       END IF
  259: *
  260: *     Quick return if possible
  261: *
  262:       IF( N.EQ.0 )
  263:      $   RETURN
  264: *
  265: *     Form a Cholesky factorization of B.
  266: *
  267:       CALL ZPOTRF( UPLO, N, B, LDB, INFO )
  268:       IF( INFO.NE.0 ) THEN
  269:          INFO = N + INFO
  270:          RETURN
  271:       END IF
  272: *
  273: *     Transform problem to standard eigenvalue problem and solve.
  274: *
  275:       CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
  276:       CALL ZHEEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, INFO )
  277: *
  278:       IF( WANTZ ) THEN
  279: *
  280: *        Backtransform eigenvectors to the original problem.
  281: *
  282:          NEIG = N
  283:          IF( INFO.GT.0 )
  284:      $      NEIG = INFO - 1
  285:          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
  286: *
  287: *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
  288: *           backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
  289: *
  290:             IF( UPPER ) THEN
  291:                TRANS = 'N'
  292:             ELSE
  293:                TRANS = 'C'
  294:             END IF
  295: *
  296:             CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE,
  297:      $                  B, LDB, A, LDA )
  298: *
  299:          ELSE IF( ITYPE.EQ.3 ) THEN
  300: *
  301: *           For B*A*x=(lambda)*x;
  302: *           backtransform eigenvectors: x = L*y or U**H *y
  303: *
  304:             IF( UPPER ) THEN
  305:                TRANS = 'C'
  306:             ELSE
  307:                TRANS = 'N'
  308:             END IF
  309: *
  310:             CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE,
  311:      $                  B, LDB, A, LDA )
  312:          END IF
  313:       END IF
  314: *
  315:       WORK( 1 ) = LWKOPT
  316: *
  317:       RETURN
  318: *
  319: *     End of ZHEGV
  320: *
  321:       END