zhegvx.f

          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 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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+0, 0.0D+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          MAX, MIN
    *     ..
    *     .. 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.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDB.LT.MAX( 1, 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.MAX( 1, 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.0) THEN
             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 = MAX( 1, ( NB + 1 )*N )
             WORK( 1 ) = LWKOPT
    *
             IF( LWORK.LT.MAX( 1, 2*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)'*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'*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