LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
chbgvx.f
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00001 *> \brief \b CHBGST
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CHBGVX + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chbgvx.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chbgvx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chbgvx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
00022 *                          LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
00023 *                          LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE, UPLO
00027 *       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
00028 *      $                   N
00029 *       REAL               ABSTOL, VL, VU
00030 *       ..
00031 *       .. Array Arguments ..
00032 *       INTEGER            IFAIL( * ), IWORK( * )
00033 *       REAL               RWORK( * ), W( * )
00034 *       COMPLEX            AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
00035 *      $                   WORK( * ), Z( LDZ, * )
00036 *       ..
00037 *  
00038 *
00039 *> \par Purpose:
00040 *  =============
00041 *>
00042 *> \verbatim
00043 *>
00044 *> CHBGVX computes all the eigenvalues, and optionally, the eigenvectors
00045 *> of a complex generalized Hermitian-definite banded eigenproblem, of
00046 *> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
00047 *> and banded, and B is also positive definite.  Eigenvalues and
00048 *> eigenvectors can be selected by specifying either all eigenvalues,
00049 *> a range of values or a range of indices for the desired eigenvalues.
00050 *> \endverbatim
00051 *
00052 *  Arguments:
00053 *  ==========
00054 *
00055 *> \param[in] JOBZ
00056 *> \verbatim
00057 *>          JOBZ is CHARACTER*1
00058 *>          = 'N':  Compute eigenvalues only;
00059 *>          = 'V':  Compute eigenvalues and eigenvectors.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] RANGE
00063 *> \verbatim
00064 *>          RANGE is CHARACTER*1
00065 *>          = 'A': all eigenvalues will be found;
00066 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00067 *>                 will be found;
00068 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00069 *> \endverbatim
00070 *>
00071 *> \param[in] UPLO
00072 *> \verbatim
00073 *>          UPLO is CHARACTER*1
00074 *>          = 'U':  Upper triangles of A and B are stored;
00075 *>          = 'L':  Lower triangles of A and B are stored.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] N
00079 *> \verbatim
00080 *>          N is INTEGER
00081 *>          The order of the matrices A and B.  N >= 0.
00082 *> \endverbatim
00083 *>
00084 *> \param[in] KA
00085 *> \verbatim
00086 *>          KA is INTEGER
00087 *>          The number of superdiagonals of the matrix A if UPLO = 'U',
00088 *>          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
00089 *> \endverbatim
00090 *>
00091 *> \param[in] KB
00092 *> \verbatim
00093 *>          KB is INTEGER
00094 *>          The number of superdiagonals of the matrix B if UPLO = 'U',
00095 *>          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
00096 *> \endverbatim
00097 *>
00098 *> \param[in,out] AB
00099 *> \verbatim
00100 *>          AB is COMPLEX array, dimension (LDAB, N)
00101 *>          On entry, the upper or lower triangle of the Hermitian band
00102 *>          matrix A, stored in the first ka+1 rows of the array.  The
00103 *>          j-th column of A is stored in the j-th column of the array AB
00104 *>          as follows:
00105 *>          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
00106 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
00107 *>
00108 *>          On exit, the contents of AB are destroyed.
00109 *> \endverbatim
00110 *>
00111 *> \param[in] LDAB
00112 *> \verbatim
00113 *>          LDAB is INTEGER
00114 *>          The leading dimension of the array AB.  LDAB >= KA+1.
00115 *> \endverbatim
00116 *>
00117 *> \param[in,out] BB
00118 *> \verbatim
00119 *>          BB is COMPLEX array, dimension (LDBB, N)
00120 *>          On entry, the upper or lower triangle of the Hermitian band
00121 *>          matrix B, stored in the first kb+1 rows of the array.  The
00122 *>          j-th column of B is stored in the j-th column of the array BB
00123 *>          as follows:
00124 *>          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
00125 *>          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
00126 *>
00127 *>          On exit, the factor S from the split Cholesky factorization
00128 *>          B = S**H*S, as returned by CPBSTF.
00129 *> \endverbatim
00130 *>
00131 *> \param[in] LDBB
00132 *> \verbatim
00133 *>          LDBB is INTEGER
00134 *>          The leading dimension of the array BB.  LDBB >= KB+1.
00135 *> \endverbatim
00136 *>
00137 *> \param[out] Q
00138 *> \verbatim
00139 *>          Q is COMPLEX array, dimension (LDQ, N)
00140 *>          If JOBZ = 'V', the n-by-n matrix used in the reduction of
00141 *>          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
00142 *>          and consequently C to tridiagonal form.
00143 *>          If JOBZ = 'N', the array Q is not referenced.
00144 *> \endverbatim
00145 *>
00146 *> \param[in] LDQ
00147 *> \verbatim
00148 *>          LDQ is INTEGER
00149 *>          The leading dimension of the array Q.  If JOBZ = 'N',
00150 *>          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
00151 *> \endverbatim
00152 *>
00153 *> \param[in] VL
00154 *> \verbatim
00155 *>          VL is REAL
00156 *> \endverbatim
00157 *>
00158 *> \param[in] VU
00159 *> \verbatim
00160 *>          VU is REAL
00161 *>
00162 *>          If RANGE='V', the lower and upper bounds of the interval to
00163 *>          be searched for eigenvalues. VL < VU.
00164 *>          Not referenced if RANGE = 'A' or 'I'.
00165 *> \endverbatim
00166 *>
00167 *> \param[in] IL
00168 *> \verbatim
00169 *>          IL is INTEGER
00170 *> \endverbatim
00171 *>
00172 *> \param[in] IU
00173 *> \verbatim
00174 *>          IU is INTEGER
00175 *>
00176 *>          If RANGE='I', the indices (in ascending order) of the
00177 *>          smallest and largest eigenvalues to be returned.
00178 *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00179 *>          Not referenced if RANGE = 'A' or 'V'.
00180 *> \endverbatim
00181 *>
00182 *> \param[in] ABSTOL
00183 *> \verbatim
00184 *>          ABSTOL is REAL
00185 *>          The absolute error tolerance for the eigenvalues.
00186 *>          An approximate eigenvalue is accepted as converged
00187 *>          when it is determined to lie in an interval [a,b]
00188 *>          of width less than or equal to
00189 *>
00190 *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
00191 *>
00192 *>          where EPS is the machine precision.  If ABSTOL is less than
00193 *>          or equal to zero, then  EPS*|T|  will be used in its place,
00194 *>          where |T| is the 1-norm of the tridiagonal matrix obtained
00195 *>          by reducing AP to tridiagonal form.
00196 *>
00197 *>          Eigenvalues will be computed most accurately when ABSTOL is
00198 *>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
00199 *>          If this routine returns with INFO>0, indicating that some
00200 *>          eigenvectors did not converge, try setting ABSTOL to
00201 *>          2*SLAMCH('S').
00202 *> \endverbatim
00203 *>
00204 *> \param[out] M
00205 *> \verbatim
00206 *>          M is INTEGER
00207 *>          The total number of eigenvalues found.  0 <= M <= N.
00208 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00209 *> \endverbatim
00210 *>
00211 *> \param[out] W
00212 *> \verbatim
00213 *>          W is REAL array, dimension (N)
00214 *>          If INFO = 0, the eigenvalues in ascending order.
00215 *> \endverbatim
00216 *>
00217 *> \param[out] Z
00218 *> \verbatim
00219 *>          Z is COMPLEX array, dimension (LDZ, N)
00220 *>          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
00221 *>          eigenvectors, with the i-th column of Z holding the
00222 *>          eigenvector associated with W(i). The eigenvectors are
00223 *>          normalized so that Z**H*B*Z = I.
00224 *>          If JOBZ = 'N', then Z is not referenced.
00225 *> \endverbatim
00226 *>
00227 *> \param[in] LDZ
00228 *> \verbatim
00229 *>          LDZ is INTEGER
00230 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00231 *>          JOBZ = 'V', LDZ >= N.
00232 *> \endverbatim
00233 *>
00234 *> \param[out] WORK
00235 *> \verbatim
00236 *>          WORK is COMPLEX array, dimension (N)
00237 *> \endverbatim
00238 *>
00239 *> \param[out] RWORK
00240 *> \verbatim
00241 *>          RWORK is REAL array, dimension (7*N)
00242 *> \endverbatim
00243 *>
00244 *> \param[out] IWORK
00245 *> \verbatim
00246 *>          IWORK is INTEGER array, dimension (5*N)
00247 *> \endverbatim
00248 *>
00249 *> \param[out] IFAIL
00250 *> \verbatim
00251 *>          IFAIL is INTEGER array, dimension (N)
00252 *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
00253 *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00254 *>          indices of the eigenvectors that failed to converge.
00255 *>          If JOBZ = 'N', then IFAIL is not referenced.
00256 *> \endverbatim
00257 *>
00258 *> \param[out] INFO
00259 *> \verbatim
00260 *>          INFO is INTEGER
00261 *>          = 0:  successful exit
00262 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00263 *>          > 0:  if INFO = i, and i is:
00264 *>             <= N:  then i eigenvectors failed to converge.  Their
00265 *>                    indices are stored in array IFAIL.
00266 *>             > N:   if INFO = N + i, for 1 <= i <= N, then CPBSTF
00267 *>                    returned INFO = i: B is not positive definite.
00268 *>                    The factorization of B could not be completed and
00269 *>                    no eigenvalues or eigenvectors were computed.
00270 *> \endverbatim
00271 *
00272 *  Authors:
00273 *  ========
00274 *
00275 *> \author Univ. of Tennessee 
00276 *> \author Univ. of California Berkeley 
00277 *> \author Univ. of Colorado Denver 
00278 *> \author NAG Ltd. 
00279 *
00280 *> \date November 2011
00281 *
00282 *> \ingroup complexOTHEReigen
00283 *
00284 *> \par Contributors:
00285 *  ==================
00286 *>
00287 *>     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
00288 *
00289 *  =====================================================================
00290       SUBROUTINE CHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
00291      $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
00292      $                   LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
00293 *
00294 *  -- LAPACK driver routine (version 3.4.0) --
00295 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00296 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00297 *     November 2011
00298 *
00299 *     .. Scalar Arguments ..
00300       CHARACTER          JOBZ, RANGE, UPLO
00301       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
00302      $                   N
00303       REAL               ABSTOL, VL, VU
00304 *     ..
00305 *     .. Array Arguments ..
00306       INTEGER            IFAIL( * ), IWORK( * )
00307       REAL               RWORK( * ), W( * )
00308       COMPLEX            AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
00309      $                   WORK( * ), Z( LDZ, * )
00310 *     ..
00311 *
00312 *  =====================================================================
00313 *
00314 *     .. Parameters ..
00315       REAL               ZERO
00316       PARAMETER          ( ZERO = 0.0E+0 )
00317       COMPLEX            CZERO, CONE
00318       PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ),
00319      $                   CONE = ( 1.0E+0, 0.0E+0 ) )
00320 *     ..
00321 *     .. Local Scalars ..
00322       LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
00323       CHARACTER          ORDER, VECT
00324       INTEGER            I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
00325      $                   INDIWK, INDRWK, INDWRK, ITMP1, J, JJ, NSPLIT
00326       REAL               TMP1
00327 *     ..
00328 *     .. External Functions ..
00329       LOGICAL            LSAME
00330       EXTERNAL           LSAME
00331 *     ..
00332 *     .. External Subroutines ..
00333       EXTERNAL           CCOPY, CGEMV, CHBGST, CHBTRD, CLACPY, CPBSTF,
00334      $                   CSTEIN, CSTEQR, CSWAP, SCOPY, SSTEBZ, SSTERF,
00335      $                   XERBLA
00336 *     ..
00337 *     .. Intrinsic Functions ..
00338       INTRINSIC          MIN
00339 *     ..
00340 *     .. Executable Statements ..
00341 *
00342 *     Test the input parameters.
00343 *
00344       WANTZ = LSAME( JOBZ, 'V' )
00345       UPPER = LSAME( UPLO, 'U' )
00346       ALLEIG = LSAME( RANGE, 'A' )
00347       VALEIG = LSAME( RANGE, 'V' )
00348       INDEIG = LSAME( RANGE, 'I' )
00349 *
00350       INFO = 0
00351       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00352          INFO = -1
00353       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00354          INFO = -2
00355       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
00356          INFO = -3
00357       ELSE IF( N.LT.0 ) THEN
00358          INFO = -4
00359       ELSE IF( KA.LT.0 ) THEN
00360          INFO = -5
00361       ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
00362          INFO = -6
00363       ELSE IF( LDAB.LT.KA+1 ) THEN
00364          INFO = -8
00365       ELSE IF( LDBB.LT.KB+1 ) THEN
00366          INFO = -10
00367       ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
00368          INFO = -12
00369       ELSE
00370          IF( VALEIG ) THEN
00371             IF( N.GT.0 .AND. VU.LE.VL )
00372      $         INFO = -14
00373          ELSE IF( INDEIG ) THEN
00374             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00375                INFO = -15
00376             ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00377                INFO = -16
00378             END IF
00379          END IF
00380       END IF
00381       IF( INFO.EQ.0) THEN
00382          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00383             INFO = -21
00384          END IF
00385       END IF
00386 *
00387       IF( INFO.NE.0 ) THEN
00388          CALL XERBLA( 'CHBGVX', -INFO )
00389          RETURN
00390       END IF
00391 *
00392 *     Quick return if possible
00393 *
00394       M = 0
00395       IF( N.EQ.0 )
00396      $   RETURN
00397 *
00398 *     Form a split Cholesky factorization of B.
00399 *
00400       CALL CPBSTF( UPLO, N, KB, BB, LDBB, INFO )
00401       IF( INFO.NE.0 ) THEN
00402          INFO = N + INFO
00403          RETURN
00404       END IF
00405 *
00406 *     Transform problem to standard eigenvalue problem.
00407 *
00408       CALL CHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
00409      $             WORK, RWORK, IINFO )
00410 *
00411 *     Solve the standard eigenvalue problem.
00412 *     Reduce Hermitian band matrix to tridiagonal form.
00413 *
00414       INDD = 1
00415       INDE = INDD + N
00416       INDRWK = INDE + N
00417       INDWRK = 1
00418       IF( WANTZ ) THEN
00419          VECT = 'U'
00420       ELSE
00421          VECT = 'N'
00422       END IF
00423       CALL CHBTRD( VECT, UPLO, N, KA, AB, LDAB, RWORK( INDD ),
00424      $             RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
00425 *
00426 *     If all eigenvalues are desired and ABSTOL is less than or equal
00427 *     to zero, then call SSTERF or CSTEQR.  If this fails for some
00428 *     eigenvalue, then try SSTEBZ.
00429 *
00430       TEST = .FALSE.
00431       IF( INDEIG ) THEN
00432          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00433             TEST = .TRUE.
00434          END IF
00435       END IF
00436       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
00437          CALL SCOPY( N, RWORK( INDD ), 1, W, 1 )
00438          INDEE = INDRWK + 2*N
00439          CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00440          IF( .NOT.WANTZ ) THEN
00441             CALL SSTERF( N, W, RWORK( INDEE ), INFO )
00442          ELSE
00443             CALL CLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
00444             CALL CSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
00445      $                   RWORK( INDRWK ), INFO )
00446             IF( INFO.EQ.0 ) THEN
00447                DO 10 I = 1, N
00448                   IFAIL( I ) = 0
00449    10          CONTINUE
00450             END IF
00451          END IF
00452          IF( INFO.EQ.0 ) THEN
00453             M = N
00454             GO TO 30
00455          END IF
00456          INFO = 0
00457       END IF
00458 *
00459 *     Otherwise, call SSTEBZ and, if eigenvectors are desired,
00460 *     call CSTEIN.
00461 *
00462       IF( WANTZ ) THEN
00463          ORDER = 'B'
00464       ELSE
00465          ORDER = 'E'
00466       END IF
00467       INDIBL = 1
00468       INDISP = INDIBL + N
00469       INDIWK = INDISP + N
00470       CALL SSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
00471      $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
00472      $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
00473      $             IWORK( INDIWK ), INFO )
00474 *
00475       IF( WANTZ ) THEN
00476          CALL CSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
00477      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00478      $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
00479 *
00480 *        Apply unitary matrix used in reduction to tridiagonal
00481 *        form to eigenvectors returned by CSTEIN.
00482 *
00483          DO 20 J = 1, M
00484             CALL CCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
00485             CALL CGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
00486      $                  Z( 1, J ), 1 )
00487    20    CONTINUE
00488       END IF
00489 *
00490    30 CONTINUE
00491 *
00492 *     If eigenvalues are not in order, then sort them, along with
00493 *     eigenvectors.
00494 *
00495       IF( WANTZ ) THEN
00496          DO 50 J = 1, M - 1
00497             I = 0
00498             TMP1 = W( J )
00499             DO 40 JJ = J + 1, M
00500                IF( W( JJ ).LT.TMP1 ) THEN
00501                   I = JJ
00502                   TMP1 = W( JJ )
00503                END IF
00504    40       CONTINUE
00505 *
00506             IF( I.NE.0 ) THEN
00507                ITMP1 = IWORK( INDIBL+I-1 )
00508                W( I ) = W( J )
00509                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00510                W( J ) = TMP1
00511                IWORK( INDIBL+J-1 ) = ITMP1
00512                CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00513                IF( INFO.NE.0 ) THEN
00514                   ITMP1 = IFAIL( I )
00515                   IFAIL( I ) = IFAIL( J )
00516                   IFAIL( J ) = ITMP1
00517                END IF
00518             END IF
00519    50    CONTINUE
00520       END IF
00521 *
00522       RETURN
00523 *
00524 *     End of CHBGVX
00525 *
00526       END
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