LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsygvx.f
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00001 *> \brief \b DSYGST
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DSYGVX + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygvx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
00022 *                          VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
00023 *                          LWORK, IWORK, IFAIL, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE, UPLO
00027 *       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
00028 *       DOUBLE PRECISION   ABSTOL, VL, VU
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IFAIL( * ), IWORK( * )
00032 *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
00033 *      $                   Z( LDZ, * )
00034 *       ..
00035 *  
00036 *
00037 *> \par Purpose:
00038 *  =============
00039 *>
00040 *> \verbatim
00041 *>
00042 *> DSYGVX computes selected eigenvalues, and optionally, eigenvectors
00043 *> of a real generalized symmetric-definite eigenproblem, of the form
00044 *> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
00045 *> and B are assumed to be symmetric and B is also positive definite.
00046 *> Eigenvalues and eigenvectors can be selected by specifying either a
00047 *> range of values or a range of indices for the desired eigenvalues.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] ITYPE
00054 *> \verbatim
00055 *>          ITYPE is INTEGER
00056 *>          Specifies the problem type to be solved:
00057 *>          = 1:  A*x = (lambda)*B*x
00058 *>          = 2:  A*B*x = (lambda)*x
00059 *>          = 3:  B*A*x = (lambda)*x
00060 *> \endverbatim
00061 *>
00062 *> \param[in] JOBZ
00063 *> \verbatim
00064 *>          JOBZ is CHARACTER*1
00065 *>          = 'N':  Compute eigenvalues only;
00066 *>          = 'V':  Compute eigenvalues and eigenvectors.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] RANGE
00070 *> \verbatim
00071 *>          RANGE is CHARACTER*1
00072 *>          = 'A': all eigenvalues will be found.
00073 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00074 *>                 will be found.
00075 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] UPLO
00079 *> \verbatim
00080 *>          UPLO is CHARACTER*1
00081 *>          = 'U':  Upper triangle of A and B are stored;
00082 *>          = 'L':  Lower triangle of A and B are stored.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] N
00086 *> \verbatim
00087 *>          N is INTEGER
00088 *>          The order of the matrix pencil (A,B).  N >= 0.
00089 *> \endverbatim
00090 *>
00091 *> \param[in,out] A
00092 *> \verbatim
00093 *>          A is DOUBLE PRECISION array, dimension (LDA, N)
00094 *>          On entry, the symmetric matrix A.  If UPLO = 'U', the
00095 *>          leading N-by-N upper triangular part of A contains the
00096 *>          upper triangular part of the matrix A.  If UPLO = 'L',
00097 *>          the leading N-by-N lower triangular part of A contains
00098 *>          the lower triangular part of the matrix A.
00099 *>
00100 *>          On exit, the lower triangle (if UPLO='L') or the upper
00101 *>          triangle (if UPLO='U') of A, including the diagonal, is
00102 *>          destroyed.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] LDA
00106 *> \verbatim
00107 *>          LDA is INTEGER
00108 *>          The leading dimension of the array A.  LDA >= max(1,N).
00109 *> \endverbatim
00110 *>
00111 *> \param[in,out] B
00112 *> \verbatim
00113 *>          B is DOUBLE PRECISION array, dimension (LDB, N)
00114 *>          On entry, the symmetric matrix B.  If UPLO = 'U', the
00115 *>          leading N-by-N upper triangular part of B contains the
00116 *>          upper triangular part of the matrix B.  If UPLO = 'L',
00117 *>          the leading N-by-N lower triangular part of B contains
00118 *>          the lower triangular part of the matrix B.
00119 *>
00120 *>          On exit, if INFO <= N, the part of B containing the matrix is
00121 *>          overwritten by the triangular factor U or L from the Cholesky
00122 *>          factorization B = U**T*U or B = L*L**T.
00123 *> \endverbatim
00124 *>
00125 *> \param[in] LDB
00126 *> \verbatim
00127 *>          LDB is INTEGER
00128 *>          The leading dimension of the array B.  LDB >= max(1,N).
00129 *> \endverbatim
00130 *>
00131 *> \param[in] VL
00132 *> \verbatim
00133 *>          VL is DOUBLE PRECISION
00134 *> \endverbatim
00135 *>
00136 *> \param[in] VU
00137 *> \verbatim
00138 *>          VU is DOUBLE PRECISION
00139 *>          If RANGE='V', the lower and upper bounds of the interval to
00140 *>          be searched for eigenvalues. VL < VU.
00141 *>          Not referenced if RANGE = 'A' or 'I'.
00142 *> \endverbatim
00143 *>
00144 *> \param[in] IL
00145 *> \verbatim
00146 *>          IL is INTEGER
00147 *> \endverbatim
00148 *>
00149 *> \param[in] IU
00150 *> \verbatim
00151 *>          IU is INTEGER
00152 *>          If RANGE='I', the indices (in ascending order) of the
00153 *>          smallest and largest eigenvalues to be returned.
00154 *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00155 *>          Not referenced if RANGE = 'A' or 'V'.
00156 *> \endverbatim
00157 *>
00158 *> \param[in] ABSTOL
00159 *> \verbatim
00160 *>          ABSTOL is DOUBLE PRECISION
00161 *>          The absolute error tolerance for the eigenvalues.
00162 *>          An approximate eigenvalue is accepted as converged
00163 *>          when it is determined to lie in an interval [a,b]
00164 *>          of width less than or equal to
00165 *>
00166 *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
00167 *>
00168 *>          where EPS is the machine precision.  If ABSTOL is less than
00169 *>          or equal to zero, then  EPS*|T|  will be used in its place,
00170 *>          where |T| is the 1-norm of the tridiagonal matrix obtained
00171 *>          by reducing C to tridiagonal form, where C is the symmetric
00172 *>          matrix of the standard symmetric problem to which the
00173 *>          generalized problem is transformed.
00174 *>
00175 *>          Eigenvalues will be computed most accurately when ABSTOL is
00176 *>          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
00177 *>          If this routine returns with INFO>0, indicating that some
00178 *>          eigenvectors did not converge, try setting ABSTOL to
00179 *>          2*DLAMCH('S').
00180 *> \endverbatim
00181 *>
00182 *> \param[out] M
00183 *> \verbatim
00184 *>          M is INTEGER
00185 *>          The total number of eigenvalues found.  0 <= M <= N.
00186 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00187 *> \endverbatim
00188 *>
00189 *> \param[out] W
00190 *> \verbatim
00191 *>          W is DOUBLE PRECISION array, dimension (N)
00192 *>          On normal exit, the first M elements contain the selected
00193 *>          eigenvalues in ascending order.
00194 *> \endverbatim
00195 *>
00196 *> \param[out] Z
00197 *> \verbatim
00198 *>          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
00199 *>          If JOBZ = 'N', then Z is not referenced.
00200 *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00201 *>          contain the orthonormal eigenvectors of the matrix A
00202 *>          corresponding to the selected eigenvalues, with the i-th
00203 *>          column of Z holding the eigenvector associated with W(i).
00204 *>          The eigenvectors are normalized as follows:
00205 *>          if ITYPE = 1 or 2, Z**T*B*Z = I;
00206 *>          if ITYPE = 3, Z**T*inv(B)*Z = I.
00207 *>
00208 *>          If an eigenvector fails to converge, then that column of Z
00209 *>          contains the latest approximation to the eigenvector, and the
00210 *>          index of the eigenvector is returned in IFAIL.
00211 *>          Note: the user must ensure that at least max(1,M) columns are
00212 *>          supplied in the array Z; if RANGE = 'V', the exact value of M
00213 *>          is not known in advance and an upper bound must be used.
00214 *> \endverbatim
00215 *>
00216 *> \param[in] LDZ
00217 *> \verbatim
00218 *>          LDZ is INTEGER
00219 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00220 *>          JOBZ = 'V', LDZ >= max(1,N).
00221 *> \endverbatim
00222 *>
00223 *> \param[out] WORK
00224 *> \verbatim
00225 *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00226 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00227 *> \endverbatim
00228 *>
00229 *> \param[in] LWORK
00230 *> \verbatim
00231 *>          LWORK is INTEGER
00232 *>          The length of the array WORK.  LWORK >= max(1,8*N).
00233 *>          For optimal efficiency, LWORK >= (NB+3)*N,
00234 *>          where NB is the blocksize for DSYTRD returned by ILAENV.
00235 *>
00236 *>          If LWORK = -1, then a workspace query is assumed; the routine
00237 *>          only calculates the optimal size of the WORK array, returns
00238 *>          this value as the first entry of the WORK array, and no error
00239 *>          message related to LWORK is issued by XERBLA.
00240 *> \endverbatim
00241 *>
00242 *> \param[out] IWORK
00243 *> \verbatim
00244 *>          IWORK is INTEGER array, dimension (5*N)
00245 *> \endverbatim
00246 *>
00247 *> \param[out] IFAIL
00248 *> \verbatim
00249 *>          IFAIL is INTEGER array, dimension (N)
00250 *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
00251 *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00252 *>          indices of the eigenvectors that failed to converge.
00253 *>          If JOBZ = 'N', then IFAIL is not referenced.
00254 *> \endverbatim
00255 *>
00256 *> \param[out] INFO
00257 *> \verbatim
00258 *>          INFO is INTEGER
00259 *>          = 0:  successful exit
00260 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00261 *>          > 0:  DPOTRF or DSYEVX returned an error code:
00262 *>             <= N:  if INFO = i, DSYEVX failed to converge;
00263 *>                    i eigenvectors failed to converge.  Their indices
00264 *>                    are stored in array IFAIL.
00265 *>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
00266 *>                    minor of order i of B is not positive definite.
00267 *>                    The factorization of B could not be completed and
00268 *>                    no eigenvalues or eigenvectors were computed.
00269 *> \endverbatim
00270 *
00271 *  Authors:
00272 *  ========
00273 *
00274 *> \author Univ. of Tennessee 
00275 *> \author Univ. of California Berkeley 
00276 *> \author Univ. of Colorado Denver 
00277 *> \author NAG Ltd. 
00278 *
00279 *> \date November 2011
00280 *
00281 *> \ingroup doubleSYeigen
00282 *
00283 *> \par Contributors:
00284 *  ==================
00285 *>
00286 *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
00287 *
00288 *  =====================================================================
00289       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
00290      $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
00291      $                   LWORK, IWORK, IFAIL, INFO )
00292 *
00293 *  -- LAPACK driver routine (version 3.4.0) --
00294 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00295 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00296 *     November 2011
00297 *
00298 *     .. Scalar Arguments ..
00299       CHARACTER          JOBZ, RANGE, UPLO
00300       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
00301       DOUBLE PRECISION   ABSTOL, VL, VU
00302 *     ..
00303 *     .. Array Arguments ..
00304       INTEGER            IFAIL( * ), IWORK( * )
00305       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
00306      $                   Z( LDZ, * )
00307 *     ..
00308 *
00309 * =====================================================================
00310 *
00311 *     .. Parameters ..
00312       DOUBLE PRECISION   ONE
00313       PARAMETER          ( ONE = 1.0D+0 )
00314 *     ..
00315 *     .. Local Scalars ..
00316       LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
00317       CHARACTER          TRANS
00318       INTEGER            LWKMIN, LWKOPT, NB
00319 *     ..
00320 *     .. External Functions ..
00321       LOGICAL            LSAME
00322       INTEGER            ILAENV
00323       EXTERNAL           LSAME, ILAENV
00324 *     ..
00325 *     .. External Subroutines ..
00326       EXTERNAL           DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM, XERBLA
00327 *     ..
00328 *     .. Intrinsic Functions ..
00329       INTRINSIC          MAX, MIN
00330 *     ..
00331 *     .. Executable Statements ..
00332 *
00333 *     Test the input parameters.
00334 *
00335       UPPER = LSAME( UPLO, 'U' )
00336       WANTZ = LSAME( JOBZ, 'V' )
00337       ALLEIG = LSAME( RANGE, 'A' )
00338       VALEIG = LSAME( RANGE, 'V' )
00339       INDEIG = LSAME( RANGE, 'I' )
00340       LQUERY = ( LWORK.EQ.-1 )
00341 *
00342       INFO = 0
00343       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
00344          INFO = -1
00345       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00346          INFO = -2
00347       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00348          INFO = -3
00349       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
00350          INFO = -4
00351       ELSE IF( N.LT.0 ) THEN
00352          INFO = -5
00353       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00354          INFO = -7
00355       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00356          INFO = -9
00357       ELSE
00358          IF( VALEIG ) THEN
00359             IF( N.GT.0 .AND. VU.LE.VL )
00360      $         INFO = -11
00361          ELSE IF( INDEIG ) THEN
00362             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00363                INFO = -12
00364             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00365                INFO = -13
00366             END IF
00367          END IF
00368       END IF
00369       IF (INFO.EQ.0) THEN
00370          IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
00371             INFO = -18
00372          END IF
00373       END IF
00374 *
00375       IF( INFO.EQ.0 ) THEN
00376          LWKMIN = MAX( 1, 8*N )
00377          NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
00378          LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
00379          WORK( 1 ) = LWKOPT
00380 *
00381          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
00382             INFO = -20
00383          END IF
00384       END IF
00385 *
00386       IF( INFO.NE.0 ) THEN
00387          CALL XERBLA( 'DSYGVX', -INFO )
00388          RETURN
00389       ELSE IF( LQUERY ) THEN
00390          RETURN
00391       END IF
00392 *
00393 *     Quick return if possible
00394 *
00395       M = 0
00396       IF( N.EQ.0 ) THEN
00397          RETURN
00398       END IF
00399 *
00400 *     Form a Cholesky factorization of B.
00401 *
00402       CALL DPOTRF( UPLO, N, B, LDB, INFO )
00403       IF( INFO.NE.0 ) THEN
00404          INFO = N + INFO
00405          RETURN
00406       END IF
00407 *
00408 *     Transform problem to standard eigenvalue problem and solve.
00409 *
00410       CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
00411       CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
00412      $             M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
00413 *
00414       IF( WANTZ ) THEN
00415 *
00416 *        Backtransform eigenvectors to the original problem.
00417 *
00418          IF( INFO.GT.0 )
00419      $      M = INFO - 1
00420          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
00421 *
00422 *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
00423 *           backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
00424 *
00425             IF( UPPER ) THEN
00426                TRANS = 'N'
00427             ELSE
00428                TRANS = 'T'
00429             END IF
00430 *
00431             CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
00432      $                  LDB, Z, LDZ )
00433 *
00434          ELSE IF( ITYPE.EQ.3 ) THEN
00435 *
00436 *           For B*A*x=(lambda)*x;
00437 *           backtransform eigenvectors: x = L*y or U**T*y
00438 *
00439             IF( UPPER ) THEN
00440                TRANS = 'T'
00441             ELSE
00442                TRANS = 'N'
00443             END IF
00444 *
00445             CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
00446      $                  LDB, Z, LDZ )
00447          END IF
00448       END IF
00449 *
00450 *     Set WORK(1) to optimal workspace size.
00451 *
00452       WORK( 1 ) = LWKOPT
00453 *
00454       RETURN
00455 *
00456 *     End of DSYGVX
00457 *
00458       END
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