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    1: *> \brief \b DSPGST
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DSPGVX + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspgvx.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspgvx.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspgvx.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
   22: *                          IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
   23: *                          IFAIL, INFO )
   24: * 
   25: *       .. Scalar Arguments ..
   26: *       CHARACTER          JOBZ, RANGE, UPLO
   27: *       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
   28: *       DOUBLE PRECISION   ABSTOL, VL, VU
   29: *       ..
   30: *       .. Array Arguments ..
   31: *       INTEGER            IFAIL( * ), IWORK( * )
   32: *       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
   33: *      $                   Z( LDZ, * )
   34: *       ..
   35: *  
   36: *
   37: *> \par Purpose:
   38: *  =============
   39: *>
   40: *> \verbatim
   41: *>
   42: *> DSPGVX computes selected eigenvalues, and optionally, eigenvectors
   43: *> of a real generalized symmetric-definite eigenproblem, of the form
   44: *> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
   45: *> and B are assumed to be symmetric, stored in packed storage, and B
   46: *> is also positive definite.  Eigenvalues and eigenvectors can be
   47: *> selected by specifying either a range of values or a range of indices
   48: *> for the desired eigenvalues.
   49: *> \endverbatim
   50: *
   51: *  Arguments:
   52: *  ==========
   53: *
   54: *> \param[in] ITYPE
   55: *> \verbatim
   56: *>          ITYPE is INTEGER
   57: *>          Specifies the problem type to be solved:
   58: *>          = 1:  A*x = (lambda)*B*x
   59: *>          = 2:  A*B*x = (lambda)*x
   60: *>          = 3:  B*A*x = (lambda)*x
   61: *> \endverbatim
   62: *>
   63: *> \param[in] JOBZ
   64: *> \verbatim
   65: *>          JOBZ is CHARACTER*1
   66: *>          = 'N':  Compute eigenvalues only;
   67: *>          = 'V':  Compute eigenvalues and eigenvectors.
   68: *> \endverbatim
   69: *>
   70: *> \param[in] RANGE
   71: *> \verbatim
   72: *>          RANGE is CHARACTER*1
   73: *>          = 'A': all eigenvalues will be found.
   74: *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
   75: *>                 will be found.
   76: *>          = 'I': the IL-th through IU-th eigenvalues will be found.
   77: *> \endverbatim
   78: *>
   79: *> \param[in] UPLO
   80: *> \verbatim
   81: *>          UPLO is CHARACTER*1
   82: *>          = 'U':  Upper triangle of A and B are stored;
   83: *>          = 'L':  Lower triangle of A and B are stored.
   84: *> \endverbatim
   85: *>
   86: *> \param[in] N
   87: *> \verbatim
   88: *>          N is INTEGER
   89: *>          The order of the matrix pencil (A,B).  N >= 0.
   90: *> \endverbatim
   91: *>
   92: *> \param[in,out] AP
   93: *> \verbatim
   94: *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
   95: *>          On entry, the upper or lower triangle of the symmetric matrix
   96: *>          A, packed columnwise in a linear array.  The j-th column of A
   97: *>          is stored in the array AP as follows:
   98: *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   99: *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
  100: *>
  101: *>          On exit, the contents of AP are destroyed.
  102: *> \endverbatim
  103: *>
  104: *> \param[in,out] BP
  105: *> \verbatim
  106: *>          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
  107: *>          On entry, the upper or lower triangle of the symmetric matrix
  108: *>          B, packed columnwise in a linear array.  The j-th column of B
  109: *>          is stored in the array BP as follows:
  110: *>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
  111: *>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
  112: *>
  113: *>          On exit, the triangular factor U or L from the Cholesky
  114: *>          factorization B = U**T*U or B = L*L**T, in the same storage
  115: *>          format as B.
  116: *> \endverbatim
  117: *>
  118: *> \param[in] VL
  119: *> \verbatim
  120: *>          VL is DOUBLE PRECISION
  121: *> \endverbatim
  122: *>
  123: *> \param[in] VU
  124: *> \verbatim
  125: *>          VU is DOUBLE PRECISION
  126: *>
  127: *>          If RANGE='V', the lower and upper bounds of the interval to
  128: *>          be searched for eigenvalues. VL < VU.
  129: *>          Not referenced if RANGE = 'A' or 'I'.
  130: *> \endverbatim
  131: *>
  132: *> \param[in] IL
  133: *> \verbatim
  134: *>          IL is INTEGER
  135: *> \endverbatim
  136: *>
  137: *> \param[in] IU
  138: *> \verbatim
  139: *>          IU is INTEGER
  140: *>
  141: *>          If RANGE='I', the indices (in ascending order) of the
  142: *>          smallest and largest eigenvalues to be returned.
  143: *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
  144: *>          Not referenced if RANGE = 'A' or 'V'.
  145: *> \endverbatim
  146: *>
  147: *> \param[in] ABSTOL
  148: *> \verbatim
  149: *>          ABSTOL is DOUBLE PRECISION
  150: *>          The absolute error tolerance for the eigenvalues.
  151: *>          An approximate eigenvalue is accepted as converged
  152: *>          when it is determined to lie in an interval [a,b]
  153: *>          of width less than or equal to
  154: *>
  155: *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
  156: *>
  157: *>          where EPS is the machine precision.  If ABSTOL is less than
  158: *>          or equal to zero, then  EPS*|T|  will be used in its place,
  159: *>          where |T| is the 1-norm of the tridiagonal matrix obtained
  160: *>          by reducing A to tridiagonal form.
  161: *>
  162: *>          Eigenvalues will be computed most accurately when ABSTOL is
  163: *>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
  164: *>          If this routine returns with INFO>0, indicating that some
  165: *>          eigenvectors did not converge, try setting ABSTOL to
  166: *>          2*DLAMCH('S').
  167: *> \endverbatim
  168: *>
  169: *> \param[out] M
  170: *> \verbatim
  171: *>          M is INTEGER
  172: *>          The total number of eigenvalues found.  0 <= M <= N.
  173: *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  174: *> \endverbatim
  175: *>
  176: *> \param[out] W
  177: *> \verbatim
  178: *>          W is DOUBLE PRECISION array, dimension (N)
  179: *>          On normal exit, the first M elements contain the selected
  180: *>          eigenvalues in ascending order.
  181: *> \endverbatim
  182: *>
  183: *> \param[out] Z
  184: *> \verbatim
  185: *>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
  186: *>          If JOBZ = 'N', then Z is not referenced.
  187: *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
  188: *>          contain the orthonormal eigenvectors of the matrix A
  189: *>          corresponding to the selected eigenvalues, with the i-th
  190: *>          column of Z holding the eigenvector associated with W(i).
  191: *>          The eigenvectors are normalized as follows:
  192: *>          if ITYPE = 1 or 2, Z**T*B*Z = I;
  193: *>          if ITYPE = 3, Z**T*inv(B)*Z = I.
  194: *>
  195: *>          If an eigenvector fails to converge, then that column of Z
  196: *>          contains the latest approximation to the eigenvector, and the
  197: *>          index of the eigenvector is returned in IFAIL.
  198: *>          Note: the user must ensure that at least max(1,M) columns are
  199: *>          supplied in the array Z; if RANGE = 'V', the exact value of M
  200: *>          is not known in advance and an upper bound must be used.
  201: *> \endverbatim
  202: *>
  203: *> \param[in] LDZ
  204: *> \verbatim
  205: *>          LDZ is INTEGER
  206: *>          The leading dimension of the array Z.  LDZ >= 1, and if
  207: *>          JOBZ = 'V', LDZ >= max(1,N).
  208: *> \endverbatim
  209: *>
  210: *> \param[out] WORK
  211: *> \verbatim
  212: *>          WORK is DOUBLE PRECISION array, dimension (8*N)
  213: *> \endverbatim
  214: *>
  215: *> \param[out] IWORK
  216: *> \verbatim
  217: *>          IWORK is INTEGER array, dimension (5*N)
  218: *> \endverbatim
  219: *>
  220: *> \param[out] IFAIL
  221: *> \verbatim
  222: *>          IFAIL is INTEGER array, dimension (N)
  223: *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
  224: *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
  225: *>          indices of the eigenvectors that failed to converge.
  226: *>          If JOBZ = 'N', then IFAIL is not referenced.
  227: *> \endverbatim
  228: *>
  229: *> \param[out] INFO
  230: *> \verbatim
  231: *>          INFO is INTEGER
  232: *>          = 0:  successful exit
  233: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  234: *>          > 0:  DPPTRF or DSPEVX returned an error code:
  235: *>             <= N:  if INFO = i, DSPEVX failed to converge;
  236: *>                    i eigenvectors failed to converge.  Their indices
  237: *>                    are stored in array IFAIL.
  238: *>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
  239: *>                    minor of order i of B is not positive definite.
  240: *>                    The factorization of B could not be completed and
  241: *>                    no eigenvalues or eigenvectors were computed.
  242: *> \endverbatim
  243: *
  244: *  Authors:
  245: *  ========
  246: *
  247: *> \author Univ. of Tennessee 
  248: *> \author Univ. of California Berkeley 
  249: *> \author Univ. of Colorado Denver 
  250: *> \author NAG Ltd. 
  251: *
  252: *> \date November 2011
  253: *
  254: *> \ingroup doubleOTHEReigen
  255: *
  256: *> \par Contributors:
  257: *  ==================
  258: *>
  259: *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
  260: *
  261: *  =====================================================================
  262:       SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
  263:      $                   IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
  264:      $                   IFAIL, INFO )
  265: *
  266: *  -- LAPACK driver routine (version 3.4.0) --
  267: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  268: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  269: *     November 2011
  270: *
  271: *     .. Scalar Arguments ..
  272:       CHARACTER          JOBZ, RANGE, UPLO
  273:       INTEGER            IL, INFO, ITYPE, IU, LDZ, M, N
  274:       DOUBLE PRECISION   ABSTOL, VL, VU
  275: *     ..
  276: *     .. Array Arguments ..
  277:       INTEGER            IFAIL( * ), IWORK( * )
  278:       DOUBLE PRECISION   AP( * ), BP( * ), W( * ), WORK( * ),
  279:      $                   Z( LDZ, * )
  280: *     ..
  281: *
  282: * =====================================================================
  283: *
  284: *     .. Local Scalars ..
  285:       LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
  286:       CHARACTER          TRANS
  287:       INTEGER            J
  288: *     ..
  289: *     .. External Functions ..
  290:       LOGICAL            LSAME
  291:       EXTERNAL           LSAME
  292: *     ..
  293: *     .. External Subroutines ..
  294:       EXTERNAL           DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
  295: *     ..
  296: *     .. Intrinsic Functions ..
  297:       INTRINSIC          MIN
  298: *     ..
  299: *     .. Executable Statements ..
  300: *
  301: *     Test the input parameters.
  302: *
  303:       UPPER = LSAME( UPLO, 'U' )
  304:       WANTZ = LSAME( JOBZ, 'V' )
  305:       ALLEIG = LSAME( RANGE, 'A' )
  306:       VALEIG = LSAME( RANGE, 'V' )
  307:       INDEIG = LSAME( RANGE, 'I' )
  308: *
  309:       INFO = 0
  310:       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
  311:          INFO = -1
  312:       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
  313:          INFO = -2
  314:       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
  315:          INFO = -3
  316:       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
  317:          INFO = -4
  318:       ELSE IF( N.LT.0 ) THEN
  319:          INFO = -5
  320:       ELSE
  321:          IF( VALEIG ) THEN
  322:             IF( N.GT.0 .AND. VU.LE.VL ) THEN
  323:                INFO = -9
  324:             END IF
  325:          ELSE IF( INDEIG ) THEN
  326:             IF( IL.LT.1 ) THEN
  327:                INFO = -10
  328:             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
  329:                INFO = -11
  330:             END IF
  331:          END IF
  332:       END IF
  333:       IF( INFO.EQ.0 ) THEN
  334:          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
  335:             INFO = -16
  336:          END IF
  337:       END IF
  338: *
  339:       IF( INFO.NE.0 ) THEN
  340:          CALL XERBLA( 'DSPGVX', -INFO )
  341:          RETURN
  342:       END IF
  343: *
  344: *     Quick return if possible
  345: *
  346:       M = 0
  347:       IF( N.EQ.0 )
  348:      $   RETURN
  349: *
  350: *     Form a Cholesky factorization of B.
  351: *
  352:       CALL DPPTRF( UPLO, N, BP, INFO )
  353:       IF( INFO.NE.0 ) THEN
  354:          INFO = N + INFO
  355:          RETURN
  356:       END IF
  357: *
  358: *     Transform problem to standard eigenvalue problem and solve.
  359: *
  360:       CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
  361:       CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
  362:      $             W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
  363: *
  364:       IF( WANTZ ) THEN
  365: *
  366: *        Backtransform eigenvectors to the original problem.
  367: *
  368:          IF( INFO.GT.0 )
  369:      $      M = INFO - 1
  370:          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
  371: *
  372: *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
  373: *           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
  374: *
  375:             IF( UPPER ) THEN
  376:                TRANS = 'N'
  377:             ELSE
  378:                TRANS = 'T'
  379:             END IF
  380: *
  381:             DO 10 J = 1, M
  382:                CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
  383:      $                     1 )
  384:    10       CONTINUE
  385: *
  386:          ELSE IF( ITYPE.EQ.3 ) THEN
  387: *
  388: *           For B*A*x=(lambda)*x;
  389: *           backtransform eigenvectors: x = L*y or U**T*y
  390: *
  391:             IF( UPPER ) THEN
  392:                TRANS = 'T'
  393:             ELSE
  394:                TRANS = 'N'
  395:             END IF
  396: *
  397:             DO 20 J = 1, M
  398:                CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
  399:      $                     1 )
  400:    20       CONTINUE
  401:          END IF
  402:       END IF
  403: *
  404:       RETURN
  405: *
  406: *     End of DSPGVX
  407: *
  408:       END