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    1:       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
    2:      $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
    3:      $                   LWORK, IWORK, IFAIL, INFO )
    4: *
    5: *  -- LAPACK driver routine (version 3.2) --
    6: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    7: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    8: *     November 2006
    9: *
   10: *     .. Scalar Arguments ..
   11:       CHARACTER          JOBZ, RANGE, UPLO
   12:       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
   13:       DOUBLE PRECISION   ABSTOL, VL, VU
   14: *     ..
   15: *     .. Array Arguments ..
   16:       INTEGER            IFAIL( * ), IWORK( * )
   17:       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
   18:      $                   Z( LDZ, * )
   19: *     ..
   20: *
   21: *  Purpose
   22: *  =======
   23: *
   24: *  DSYGVX computes selected eigenvalues, and optionally, eigenvectors
   25: *  of a real generalized symmetric-definite eigenproblem, of the form
   26: *  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
   27: *  and B are assumed to be symmetric and B is also positive definite.
   28: *  Eigenvalues and eigenvectors can be selected by specifying either a
   29: *  range of values or a range of indices for the desired eigenvalues.
   30: *
   31: *  Arguments
   32: *  =========
   33: *
   34: *  ITYPE   (input) INTEGER
   35: *          Specifies the problem type to be solved:
   36: *          = 1:  A*x = (lambda)*B*x
   37: *          = 2:  A*B*x = (lambda)*x
   38: *          = 3:  B*A*x = (lambda)*x
   39: *
   40: *  JOBZ    (input) CHARACTER*1
   41: *          = 'N':  Compute eigenvalues only;
   42: *          = 'V':  Compute eigenvalues and eigenvectors.
   43: *
   44: *  RANGE   (input) CHARACTER*1
   45: *          = 'A': all eigenvalues will be found.
   46: *          = 'V': all eigenvalues in the half-open interval (VL,VU]
   47: *                 will be found.
   48: *          = 'I': the IL-th through IU-th eigenvalues will be found.
   49: *
   50: *  UPLO    (input) CHARACTER*1
   51: *          = 'U':  Upper triangle of A and B are stored;
   52: *          = 'L':  Lower triangle of A and B are stored.
   53: *
   54: *  N       (input) INTEGER
   55: *          The order of the matrix pencil (A,B).  N >= 0.
   56: *
   57: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
   58: *          On entry, the symmetric matrix A.  If UPLO = 'U', the
   59: *          leading N-by-N upper triangular part of A contains the
   60: *          upper triangular part of the matrix A.  If UPLO = 'L',
   61: *          the leading N-by-N lower triangular part of A contains
   62: *          the lower triangular part of the matrix A.
   63: *
   64: *          On exit, the lower triangle (if UPLO='L') or the upper
   65: *          triangle (if UPLO='U') of A, including the diagonal, is
   66: *          destroyed.
   67: *
   68: *  LDA     (input) INTEGER
   69: *          The leading dimension of the array A.  LDA >= max(1,N).
   70: *
   71: *  B       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
   72: *          On entry, the symmetric matrix B.  If UPLO = 'U', the
   73: *          leading N-by-N upper triangular part of B contains the
   74: *          upper triangular part of the matrix B.  If UPLO = 'L',
   75: *          the leading N-by-N lower triangular part of B contains
   76: *          the lower triangular part of the matrix B.
   77: *
   78: *          On exit, if INFO <= N, the part of B containing the matrix is
   79: *          overwritten by the triangular factor U or L from the Cholesky
   80: *          factorization B = U**T*U or B = L*L**T.
   81: *
   82: *  LDB     (input) INTEGER
   83: *          The leading dimension of the array B.  LDB >= max(1,N).
   84: *
   85: *  VL      (input) DOUBLE PRECISION
   86: *  VU      (input) DOUBLE PRECISION
   87: *          If RANGE='V', the lower and upper bounds of the interval to
   88: *          be searched for eigenvalues. VL < VU.
   89: *          Not referenced if RANGE = 'A' or 'I'.
   90: *
   91: *  IL      (input) INTEGER
   92: *  IU      (input) INTEGER
   93: *          If RANGE='I', the indices (in ascending order) of the
   94: *          smallest and largest eigenvalues to be returned.
   95: *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
   96: *          Not referenced if RANGE = 'A' or 'V'.
   97: *
   98: *  ABSTOL  (input) DOUBLE PRECISION
   99: *          The absolute error tolerance for the eigenvalues.
  100: *          An approximate eigenvalue is accepted as converged
  101: *          when it is determined to lie in an interval [a,b]
  102: *          of width less than or equal to
  103: *
  104: *                  ABSTOL + EPS *   max( |a|,|b| ) ,
  105: *
  106: *          where EPS is the machine precision.  If ABSTOL is less than
  107: *          or equal to zero, then  EPS*|T|  will be used in its place,
  108: *          where |T| is the 1-norm of the tridiagonal matrix obtained
  109: *          by reducing A to tridiagonal form.
  110: *
  111: *          Eigenvalues will be computed most accurately when ABSTOL is
  112: *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
  113: *          If this routine returns with INFO>0, indicating that some
  114: *          eigenvectors did not converge, try setting ABSTOL to
  115: *          2*DLAMCH('S').
  116: *
  117: *  M       (output) INTEGER
  118: *          The total number of eigenvalues found.  0 <= M <= N.
  119: *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  120: *
  121: *  W       (output) DOUBLE PRECISION array, dimension (N)
  122: *          On normal exit, the first M elements contain the selected
  123: *          eigenvalues in ascending order.
  124: *
  125: *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
  126: *          If JOBZ = 'N', then Z is not referenced.
  127: *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
  128: *          contain the orthonormal eigenvectors of the matrix A
  129: *          corresponding to the selected eigenvalues, with the i-th
  130: *          column of Z holding the eigenvector associated with W(i).
  131: *          The eigenvectors are normalized as follows:
  132: *          if ITYPE = 1 or 2, Z**T*B*Z = I;
  133: *          if ITYPE = 3, Z**T*inv(B)*Z = I.
  134: *
  135: *          If an eigenvector fails to converge, then that column of Z
  136: *          contains the latest approximation to the eigenvector, and the
  137: *          index of the eigenvector is returned in IFAIL.
  138: *          Note: the user must ensure that at least max(1,M) columns are
  139: *          supplied in the array Z; if RANGE = 'V', the exact value of M
  140: *          is not known in advance and an upper bound must be used.
  141: *
  142: *  LDZ     (input) INTEGER
  143: *          The leading dimension of the array Z.  LDZ >= 1, and if
  144: *          JOBZ = 'V', LDZ >= max(1,N).
  145: *
  146: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
  147: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  148: *
  149: *  LWORK   (input) INTEGER
  150: *          The length of the array WORK.  LWORK >= max(1,8*N).
  151: *          For optimal efficiency, LWORK >= (NB+3)*N,
  152: *          where NB is the blocksize for DSYTRD returned by ILAENV.
  153: *
  154: *          If LWORK = -1, then a workspace query is assumed; the routine
  155: *          only calculates the optimal size of the WORK array, returns
  156: *          this value as the first entry of the WORK array, and no error
  157: *          message related to LWORK is issued by XERBLA.
  158: *
  159: *  IWORK   (workspace) INTEGER array, dimension (5*N)
  160: *
  161: *  IFAIL   (output) INTEGER array, dimension (N)
  162: *          If JOBZ = 'V', then if INFO = 0, the first M elements of
  163: *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
  164: *          indices of the eigenvectors that failed to converge.
  165: *          If JOBZ = 'N', then IFAIL is not referenced.
  166: *
  167: *  INFO    (output) INTEGER
  168: *          = 0:  successful exit
  169: *          < 0:  if INFO = -i, the i-th argument had an illegal value
  170: *          > 0:  DPOTRF or DSYEVX returned an error code:
  171: *             <= N:  if INFO = i, DSYEVX failed to converge;
  172: *                    i eigenvectors failed to converge.  Their indices
  173: *                    are stored in array IFAIL.
  174: *             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
  175: *                    minor of order i of B is not positive definite.
  176: *                    The factorization of B could not be completed and
  177: *                    no eigenvalues or eigenvectors were computed.
  178: *
  179: *  Further Details
  180: *  ===============
  181: *
  182: *  Based on contributions by
  183: *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
  184: *
  185: * =====================================================================
  186: *
  187: *     .. Parameters ..
  188:       DOUBLE PRECISION   ONE
  189:       PARAMETER          ( ONE = 1.0D+0 )
  190: *     ..
  191: *     .. Local Scalars ..
  192:       LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
  193:       CHARACTER          TRANS
  194:       INTEGER            LWKMIN, LWKOPT, NB
  195: *     ..
  196: *     .. External Functions ..
  197:       LOGICAL            LSAME
  198:       INTEGER            ILAENV
  199:       EXTERNAL           LSAME, ILAENV
  200: *     ..
  201: *     .. External Subroutines ..
  202:       EXTERNAL           DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM, XERBLA
  203: *     ..
  204: *     .. Intrinsic Functions ..
  205:       INTRINSIC          MAX, MIN
  206: *     ..
  207: *     .. Executable Statements ..
  208: *
  209: *     Test the input parameters.
  210: *
  211:       UPPER = LSAME( UPLO, 'U' )
  212:       WANTZ = LSAME( JOBZ, 'V' )
  213:       ALLEIG = LSAME( RANGE, 'A' )
  214:       VALEIG = LSAME( RANGE, 'V' )
  215:       INDEIG = LSAME( RANGE, 'I' )
  216:       LQUERY = ( LWORK.EQ.-1 )
  217: *
  218:       INFO = 0
  219:       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
  220:          INFO = -1
  221:       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
  222:          INFO = -2
  223:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
  224:          INFO = -3
  225:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
  226:          INFO = -4
  227:       ELSE IF( N.LT.0 ) THEN
  228:          INFO = -5
  229:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  230:          INFO = -7
  231:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  232:          INFO = -9
  233:       ELSE
  234:          IF( VALEIG ) THEN
  235:             IF( N.GT.0 .AND. VU.LE.VL )
  236:      $         INFO = -11
  237:          ELSE IF( INDEIG ) THEN
  238:             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
  239:                INFO = -12
  240:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
  241:                INFO = -13
  242:             END IF
  243:          END IF
  244:       END IF
  245:       IF (INFO.EQ.0) THEN
  246:          IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
  247:             INFO = -18
  248:          END IF
  249:       END IF
  250: *
  251:       IF( INFO.EQ.0 ) THEN
  252:          LWKMIN = MAX( 1, 8*N )
  253:          NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
  254:          LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
  255:          WORK( 1 ) = LWKOPT
  256: *
  257:          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
  258:             INFO = -20
  259:          END IF
  260:       END IF
  261: *
  262:       IF( INFO.NE.0 ) THEN
  263:          CALL XERBLA( 'DSYGVX', -INFO )
  264:          RETURN
  265:       ELSE IF( LQUERY ) THEN
  266:          RETURN
  267:       END IF
  268: *
  269: *     Quick return if possible
  270: *
  271:       M = 0
  272:       IF( N.EQ.0 ) THEN
  273:          RETURN
  274:       END IF
  275: *
  276: *     Form a Cholesky factorization of B.
  277: *
  278:       CALL DPOTRF( UPLO, N, B, LDB, INFO )
  279:       IF( INFO.NE.0 ) THEN
  280:          INFO = N + INFO
  281:          RETURN
  282:       END IF
  283: *
  284: *     Transform problem to standard eigenvalue problem and solve.
  285: *
  286:       CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
  287:       CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
  288:      $             M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
  289: *
  290:       IF( WANTZ ) THEN
  291: *
  292: *        Backtransform eigenvectors to the original problem.
  293: *
  294:          IF( INFO.GT.0 )
  295:      $      M = INFO - 1
  296:          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
  297: *
  298: *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
  299: *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
  300: *
  301:             IF( UPPER ) THEN
  302:                TRANS = 'N'
  303:             ELSE
  304:                TRANS = 'T'
  305:             END IF
  306: *
  307:             CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
  308:      $                  LDB, Z, LDZ )
  309: *
  310:          ELSE IF( ITYPE.EQ.3 ) THEN
  311: *
  312: *           For B*A*x=(lambda)*x;
  313: *           backtransform eigenvectors: x = L*y or U'*y
  314: *
  315:             IF( UPPER ) THEN
  316:                TRANS = 'T'
  317:             ELSE
  318:                TRANS = 'N'
  319:             END IF
  320: *
  321:             CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
  322:      $                  LDB, Z, LDZ )
  323:          END IF
  324:       END IF
  325: *
  326: *     Set WORK(1) to optimal workspace size.
  327: *
  328:       WORK( 1 ) = LWKOPT
  329: *
  330:       RETURN
  331: *
  332: *     End of DSYGVX
  333: *
  334:       END