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