LAPACK 3.3.0

dsygvx.f

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00001       SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
00002      $                   VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
00003      $                   LWORK, IWORK, IFAIL, INFO )
00004 *
00005 *  -- LAPACK driver routine (version 3.2.2) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *     June 2010
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          JOBZ, RANGE, UPLO
00012       INTEGER            IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
00013       DOUBLE PRECISION   ABSTOL, VL, VU
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IFAIL( * ), IWORK( * )
00017       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
00018      $                   Z( LDZ, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  DSYGVX computes selected eigenvalues, and optionally, eigenvectors
00025 *  of a real generalized symmetric-definite eigenproblem, of the form
00026 *  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
00027 *  and B are assumed to be symmetric and B is also positive definite.
00028 *  Eigenvalues and eigenvectors can be selected by specifying either a
00029 *  range of values or a range of indices for the desired eigenvalues.
00030 *
00031 *  Arguments
00032 *  =========
00033 *
00034 *  ITYPE   (input) INTEGER
00035 *          Specifies the problem type to be solved:
00036 *          = 1:  A*x = (lambda)*B*x
00037 *          = 2:  A*B*x = (lambda)*x
00038 *          = 3:  B*A*x = (lambda)*x
00039 *
00040 *  JOBZ    (input) CHARACTER*1
00041 *          = 'N':  Compute eigenvalues only;
00042 *          = 'V':  Compute eigenvalues and eigenvectors.
00043 *
00044 *  RANGE   (input) CHARACTER*1
00045 *          = 'A': all eigenvalues will be found.
00046 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00047 *                 will be found.
00048 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00049 *
00050 *  UPLO    (input) CHARACTER*1
00051 *          = 'U':  Upper triangle of A and B are stored;
00052 *          = 'L':  Lower triangle of A and B are stored.
00053 *
00054 *  N       (input) INTEGER
00055 *          The order of the matrix pencil (A,B).  N >= 0.
00056 *
00057 *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
00058 *          On entry, the symmetric matrix A.  If UPLO = 'U', the
00059 *          leading N-by-N upper triangular part of A contains the
00060 *          upper triangular part of the matrix A.  If UPLO = 'L',
00061 *          the leading N-by-N lower triangular part of A contains
00062 *          the lower triangular part of the matrix A.
00063 *
00064 *          On exit, the lower triangle (if UPLO='L') or the upper
00065 *          triangle (if UPLO='U') of A, including the diagonal, is
00066 *          destroyed.
00067 *
00068 *  LDA     (input) INTEGER
00069 *          The leading dimension of the array A.  LDA >= max(1,N).
00070 *
00071 *  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
00072 *          On entry, the symmetric matrix B.  If UPLO = 'U', the
00073 *          leading N-by-N upper triangular part of B contains the
00074 *          upper triangular part of the matrix B.  If UPLO = 'L',
00075 *          the leading N-by-N lower triangular part of B contains
00076 *          the lower triangular part of the matrix B.
00077 *
00078 *          On exit, if INFO <= N, the part of B containing the matrix is
00079 *          overwritten by the triangular factor U or L from the Cholesky
00080 *          factorization B = U**T*U or B = L*L**T.
00081 *
00082 *  LDB     (input) INTEGER
00083 *          The leading dimension of the array B.  LDB >= max(1,N).
00084 *
00085 *  VL      (input) DOUBLE PRECISION
00086 *  VU      (input) DOUBLE PRECISION
00087 *          If RANGE='V', the lower and upper bounds of the interval to
00088 *          be searched for eigenvalues. VL < VU.
00089 *          Not referenced if RANGE = 'A' or 'I'.
00090 *
00091 *  IL      (input) INTEGER
00092 *  IU      (input) INTEGER
00093 *          If RANGE='I', the indices (in ascending order) of the
00094 *          smallest and largest eigenvalues to be returned.
00095 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00096 *          Not referenced if RANGE = 'A' or 'V'.
00097 *
00098 *  ABSTOL  (input) DOUBLE PRECISION
00099 *          The absolute error tolerance for the eigenvalues.
00100 *          An approximate eigenvalue is accepted as converged
00101 *          when it is determined to lie in an interval [a,b]
00102 *          of width less than or equal to
00103 *
00104 *                  ABSTOL + EPS *   max( |a|,|b| ) ,
00105 *
00106 *          where EPS is the machine precision.  If ABSTOL is less than
00107 *          or equal to zero, then  EPS*|T|  will be used in its place,
00108 *          where |T| is the 1-norm of the tridiagonal matrix obtained
00109 *          by reducing A to tridiagonal form.
00110 *
00111 *          Eigenvalues will be computed most accurately when ABSTOL is
00112 *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
00113 *          If this routine returns with INFO>0, indicating that some
00114 *          eigenvectors did not converge, try setting ABSTOL to
00115 *          2*DLAMCH('S').
00116 *
00117 *  M       (output) INTEGER
00118 *          The total number of eigenvalues found.  0 <= M <= N.
00119 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00120 *
00121 *  W       (output) DOUBLE PRECISION array, dimension (N)
00122 *          On normal exit, the first M elements contain the selected
00123 *          eigenvalues in ascending order.
00124 *
00125 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
00126 *          If JOBZ = 'N', then Z is not referenced.
00127 *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00128 *          contain the orthonormal eigenvectors of the matrix A
00129 *          corresponding to the selected eigenvalues, with the i-th
00130 *          column of Z holding the eigenvector associated with W(i).
00131 *          The eigenvectors are normalized as follows:
00132 *          if ITYPE = 1 or 2, Z**T*B*Z = I;
00133 *          if ITYPE = 3, Z**T*inv(B)*Z = I.
00134 *
00135 *          If an eigenvector fails to converge, then that column of Z
00136 *          contains the latest approximation to the eigenvector, and the
00137 *          index of the eigenvector is returned in IFAIL.
00138 *          Note: the user must ensure that at least max(1,M) columns are
00139 *          supplied in the array Z; if RANGE = 'V', the exact value of M
00140 *          is not known in advance and an upper bound must be used.
00141 *
00142 *  LDZ     (input) INTEGER
00143 *          The leading dimension of the array Z.  LDZ >= 1, and if
00144 *          JOBZ = 'V', LDZ >= max(1,N).
00145 *
00146 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00147 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00148 *
00149 *  LWORK   (input) INTEGER
00150 *          The length of the array WORK.  LWORK >= max(1,8*N).
00151 *          For optimal efficiency, LWORK >= (NB+3)*N,
00152 *          where NB is the blocksize for DSYTRD returned by ILAENV.
00153 *
00154 *          If LWORK = -1, then a workspace query is assumed; the routine
00155 *          only calculates the optimal size of the WORK array, returns
00156 *          this value as the first entry of the WORK array, and no error
00157 *          message related to LWORK is issued by XERBLA.
00158 *
00159 *  IWORK   (workspace) INTEGER array, dimension (5*N)
00160 *
00161 *  IFAIL   (output) INTEGER array, dimension (N)
00162 *          If JOBZ = 'V', then if INFO = 0, the first M elements of
00163 *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00164 *          indices of the eigenvectors that failed to converge.
00165 *          If JOBZ = 'N', then IFAIL is not referenced.
00166 *
00167 *  INFO    (output) INTEGER
00168 *          = 0:  successful exit
00169 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00170 *          > 0:  DPOTRF or DSYEVX returned an error code:
00171 *             <= N:  if INFO = i, DSYEVX failed to converge;
00172 *                    i eigenvectors failed to converge.  Their indices
00173 *                    are stored in array IFAIL.
00174 *             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
00175 *                    minor of order i of B is not positive definite.
00176 *                    The factorization of B could not be completed and
00177 *                    no eigenvalues or eigenvectors were computed.
00178 *
00179 *  Further Details
00180 *  ===============
00181 *
00182 *  Based on contributions by
00183 *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
00184 *
00185 * =====================================================================
00186 *
00187 *     .. Parameters ..
00188       DOUBLE PRECISION   ONE
00189       PARAMETER          ( ONE = 1.0D+0 )
00190 *     ..
00191 *     .. Local Scalars ..
00192       LOGICAL            ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
00193       CHARACTER          TRANS
00194       INTEGER            LWKMIN, LWKOPT, NB
00195 *     ..
00196 *     .. External Functions ..
00197       LOGICAL            LSAME
00198       INTEGER            ILAENV
00199       EXTERNAL           LSAME, ILAENV
00200 *     ..
00201 *     .. External Subroutines ..
00202       EXTERNAL           DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM, XERBLA
00203 *     ..
00204 *     .. Intrinsic Functions ..
00205       INTRINSIC          MAX, MIN
00206 *     ..
00207 *     .. Executable Statements ..
00208 *
00209 *     Test the input parameters.
00210 *
00211       UPPER = LSAME( UPLO, 'U' )
00212       WANTZ = LSAME( JOBZ, 'V' )
00213       ALLEIG = LSAME( RANGE, 'A' )
00214       VALEIG = LSAME( RANGE, 'V' )
00215       INDEIG = LSAME( RANGE, 'I' )
00216       LQUERY = ( LWORK.EQ.-1 )
00217 *
00218       INFO = 0
00219       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
00220          INFO = -1
00221       ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00222          INFO = -2
00223       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00224          INFO = -3
00225       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
00226          INFO = -4
00227       ELSE IF( N.LT.0 ) THEN
00228          INFO = -5
00229       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00230          INFO = -7
00231       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00232          INFO = -9
00233       ELSE
00234          IF( VALEIG ) THEN
00235             IF( N.GT.0 .AND. VU.LE.VL )
00236      $         INFO = -11
00237          ELSE IF( INDEIG ) THEN
00238             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00239                INFO = -12
00240             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00241                INFO = -13
00242             END IF
00243          END IF
00244       END IF
00245       IF (INFO.EQ.0) THEN
00246          IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
00247             INFO = -18
00248          END IF
00249       END IF
00250 *
00251       IF( INFO.EQ.0 ) THEN
00252          LWKMIN = MAX( 1, 8*N )
00253          NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
00254          LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
00255          WORK( 1 ) = LWKOPT
00256 *
00257          IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
00258             INFO = -20
00259          END IF
00260       END IF
00261 *
00262       IF( INFO.NE.0 ) THEN
00263          CALL XERBLA( 'DSYGVX', -INFO )
00264          RETURN
00265       ELSE IF( LQUERY ) THEN
00266          RETURN
00267       END IF
00268 *
00269 *     Quick return if possible
00270 *
00271       M = 0
00272       IF( N.EQ.0 ) THEN
00273          RETURN
00274       END IF
00275 *
00276 *     Form a Cholesky factorization of B.
00277 *
00278       CALL DPOTRF( UPLO, N, B, LDB, INFO )
00279       IF( INFO.NE.0 ) THEN
00280          INFO = N + INFO
00281          RETURN
00282       END IF
00283 *
00284 *     Transform problem to standard eigenvalue problem and solve.
00285 *
00286       CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
00287       CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
00288      $             M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
00289 *
00290       IF( WANTZ ) THEN
00291 *
00292 *        Backtransform eigenvectors to the original problem.
00293 *
00294          IF( INFO.GT.0 )
00295      $      M = INFO - 1
00296          IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
00297 *
00298 *           For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
00299 *           backtransform eigenvectors: x = inv(L)'*y or inv(U)*y
00300 *
00301             IF( UPPER ) THEN
00302                TRANS = 'N'
00303             ELSE
00304                TRANS = 'T'
00305             END IF
00306 *
00307             CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
00308      $                  LDB, Z, LDZ )
00309 *
00310          ELSE IF( ITYPE.EQ.3 ) THEN
00311 *
00312 *           For B*A*x=(lambda)*x;
00313 *           backtransform eigenvectors: x = L*y or U'*y
00314 *
00315             IF( UPPER ) THEN
00316                TRANS = 'T'
00317             ELSE
00318                TRANS = 'N'
00319             END IF
00320 *
00321             CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B,
00322      $                  LDB, Z, LDZ )
00323          END IF
00324       END IF
00325 *
00326 *     Set WORK(1) to optimal workspace size.
00327 *
00328       WORK( 1 ) = LWKOPT
00329 *
00330       RETURN
00331 *
00332 *     End of DSYGVX
00333 *
00334       END
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