LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sspevx.f
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00001 *> \brief <b> SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SSPEVX + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sspevx.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sspevx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sspevx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
00022 *                          ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
00023 *                          INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          JOBZ, RANGE, UPLO
00027 *       INTEGER            IL, INFO, IU, LDZ, M, N
00028 *       REAL               ABSTOL, VL, VU
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IFAIL( * ), IWORK( * )
00032 *       REAL               AP( * ), W( * ), WORK( * ), Z( LDZ, * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> SSPEVX computes selected eigenvalues and, optionally, eigenvectors
00042 *> of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
00043 *> can be selected by specifying either a range of values or a range of
00044 *> indices for the desired eigenvalues.
00045 *> \endverbatim
00046 *
00047 *  Arguments:
00048 *  ==========
00049 *
00050 *> \param[in] JOBZ
00051 *> \verbatim
00052 *>          JOBZ is CHARACTER*1
00053 *>          = 'N':  Compute eigenvalues only;
00054 *>          = 'V':  Compute eigenvalues and eigenvectors.
00055 *> \endverbatim
00056 *>
00057 *> \param[in] RANGE
00058 *> \verbatim
00059 *>          RANGE is CHARACTER*1
00060 *>          = 'A': all eigenvalues will be found;
00061 *>          = 'V': all eigenvalues in the half-open interval (VL,VU]
00062 *>                 will be found;
00063 *>          = 'I': the IL-th through IU-th eigenvalues will be found.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] UPLO
00067 *> \verbatim
00068 *>          UPLO is CHARACTER*1
00069 *>          = 'U':  Upper triangle of A is stored;
00070 *>          = 'L':  Lower triangle of A is stored.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] N
00074 *> \verbatim
00075 *>          N is INTEGER
00076 *>          The order of the matrix A.  N >= 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in,out] AP
00080 *> \verbatim
00081 *>          AP is REAL array, dimension (N*(N+1)/2)
00082 *>          On entry, the upper or lower triangle of the symmetric matrix
00083 *>          A, packed columnwise in a linear array.  The j-th column of A
00084 *>          is stored in the array AP as follows:
00085 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00086 *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00087 *>
00088 *>          On exit, AP is overwritten by values generated during the
00089 *>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
00090 *>          and first superdiagonal of the tridiagonal matrix T overwrite
00091 *>          the corresponding elements of A, and if UPLO = 'L', the
00092 *>          diagonal and first subdiagonal of T overwrite the
00093 *>          corresponding elements of A.
00094 *> \endverbatim
00095 *>
00096 *> \param[in] VL
00097 *> \verbatim
00098 *>          VL is REAL
00099 *> \endverbatim
00100 *>
00101 *> \param[in] VU
00102 *> \verbatim
00103 *>          VU is REAL
00104 *>          If RANGE='V', the lower and upper bounds of the interval to
00105 *>          be searched for eigenvalues. VL < VU.
00106 *>          Not referenced if RANGE = 'A' or 'I'.
00107 *> \endverbatim
00108 *>
00109 *> \param[in] IL
00110 *> \verbatim
00111 *>          IL is INTEGER
00112 *> \endverbatim
00113 *>
00114 *> \param[in] IU
00115 *> \verbatim
00116 *>          IU is INTEGER
00117 *>          If RANGE='I', the indices (in ascending order) of the
00118 *>          smallest and largest eigenvalues to be returned.
00119 *>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00120 *>          Not referenced if RANGE = 'A' or 'V'.
00121 *> \endverbatim
00122 *>
00123 *> \param[in] ABSTOL
00124 *> \verbatim
00125 *>          ABSTOL is REAL
00126 *>          The absolute error tolerance for the eigenvalues.
00127 *>          An approximate eigenvalue is accepted as converged
00128 *>          when it is determined to lie in an interval [a,b]
00129 *>          of width less than or equal to
00130 *>
00131 *>                  ABSTOL + EPS *   max( |a|,|b| ) ,
00132 *>
00133 *>          where EPS is the machine precision.  If ABSTOL is less than
00134 *>          or equal to zero, then  EPS*|T|  will be used in its place,
00135 *>          where |T| is the 1-norm of the tridiagonal matrix obtained
00136 *>          by reducing AP to tridiagonal form.
00137 *>
00138 *>          Eigenvalues will be computed most accurately when ABSTOL is
00139 *>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
00140 *>          If this routine returns with INFO>0, indicating that some
00141 *>          eigenvectors did not converge, try setting ABSTOL to
00142 *>          2*SLAMCH('S').
00143 *>
00144 *>          See "Computing Small Singular Values of Bidiagonal Matrices
00145 *>          with Guaranteed High Relative Accuracy," by Demmel and
00146 *>          Kahan, LAPACK Working Note #3.
00147 *> \endverbatim
00148 *>
00149 *> \param[out] M
00150 *> \verbatim
00151 *>          M is INTEGER
00152 *>          The total number of eigenvalues found.  0 <= M <= N.
00153 *>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00154 *> \endverbatim
00155 *>
00156 *> \param[out] W
00157 *> \verbatim
00158 *>          W is REAL array, dimension (N)
00159 *>          If INFO = 0, the selected eigenvalues in ascending order.
00160 *> \endverbatim
00161 *>
00162 *> \param[out] Z
00163 *> \verbatim
00164 *>          Z is REAL array, dimension (LDZ, max(1,M))
00165 *>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
00166 *>          contain the orthonormal eigenvectors of the matrix A
00167 *>          corresponding to the selected eigenvalues, with the i-th
00168 *>          column of Z holding the eigenvector associated with W(i).
00169 *>          If an eigenvector fails to converge, then that column of Z
00170 *>          contains the latest approximation to the eigenvector, and the
00171 *>          index of the eigenvector is returned in IFAIL.
00172 *>          If JOBZ = 'N', then Z is not referenced.
00173 *>          Note: the user must ensure that at least max(1,M) columns are
00174 *>          supplied in the array Z; if RANGE = 'V', the exact value of M
00175 *>          is not known in advance and an upper bound must be used.
00176 *> \endverbatim
00177 *>
00178 *> \param[in] LDZ
00179 *> \verbatim
00180 *>          LDZ is INTEGER
00181 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00182 *>          JOBZ = 'V', LDZ >= max(1,N).
00183 *> \endverbatim
00184 *>
00185 *> \param[out] WORK
00186 *> \verbatim
00187 *>          WORK is REAL array, dimension (8*N)
00188 *> \endverbatim
00189 *>
00190 *> \param[out] IWORK
00191 *> \verbatim
00192 *>          IWORK is INTEGER array, dimension (5*N)
00193 *> \endverbatim
00194 *>
00195 *> \param[out] IFAIL
00196 *> \verbatim
00197 *>          IFAIL is INTEGER array, dimension (N)
00198 *>          If JOBZ = 'V', then if INFO = 0, the first M elements of
00199 *>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00200 *>          indices of the eigenvectors that failed to converge.
00201 *>          If JOBZ = 'N', then IFAIL is not referenced.
00202 *> \endverbatim
00203 *>
00204 *> \param[out] INFO
00205 *> \verbatim
00206 *>          INFO is INTEGER
00207 *>          = 0:  successful exit
00208 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00209 *>          > 0:  if INFO = i, then i eigenvectors failed to converge.
00210 *>                Their indices are stored in array IFAIL.
00211 *> \endverbatim
00212 *
00213 *  Authors:
00214 *  ========
00215 *
00216 *> \author Univ. of Tennessee 
00217 *> \author Univ. of California Berkeley 
00218 *> \author Univ. of Colorado Denver 
00219 *> \author NAG Ltd. 
00220 *
00221 *> \date November 2011
00222 *
00223 *> \ingroup realOTHEReigen
00224 *
00225 *  =====================================================================
00226       SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
00227      $                   ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
00228      $                   INFO )
00229 *
00230 *  -- LAPACK driver routine (version 3.4.0) --
00231 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00232 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00233 *     November 2011
00234 *
00235 *     .. Scalar Arguments ..
00236       CHARACTER          JOBZ, RANGE, UPLO
00237       INTEGER            IL, INFO, IU, LDZ, M, N
00238       REAL               ABSTOL, VL, VU
00239 *     ..
00240 *     .. Array Arguments ..
00241       INTEGER            IFAIL( * ), IWORK( * )
00242       REAL               AP( * ), W( * ), WORK( * ), Z( LDZ, * )
00243 *     ..
00244 *
00245 *  =====================================================================
00246 *
00247 *     .. Parameters ..
00248       REAL               ZERO, ONE
00249       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
00250 *     ..
00251 *     .. Local Scalars ..
00252       LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
00253       CHARACTER          ORDER
00254       INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
00255      $                   INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
00256      $                   J, JJ, NSPLIT
00257       REAL               ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
00258      $                   SIGMA, SMLNUM, TMP1, VLL, VUU
00259 *     ..
00260 *     .. External Functions ..
00261       LOGICAL            LSAME
00262       REAL               SLAMCH, SLANSP
00263       EXTERNAL           LSAME, SLAMCH, SLANSP
00264 *     ..
00265 *     .. External Subroutines ..
00266       EXTERNAL           SCOPY, SOPGTR, SOPMTR, SSCAL, SSPTRD, SSTEBZ,
00267      $                   SSTEIN, SSTEQR, SSTERF, SSWAP, XERBLA
00268 *     ..
00269 *     .. Intrinsic Functions ..
00270       INTRINSIC          MAX, MIN, SQRT
00271 *     ..
00272 *     .. Executable Statements ..
00273 *
00274 *     Test the input parameters.
00275 *
00276       WANTZ = LSAME( JOBZ, 'V' )
00277       ALLEIG = LSAME( RANGE, 'A' )
00278       VALEIG = LSAME( RANGE, 'V' )
00279       INDEIG = LSAME( RANGE, 'I' )
00280 *
00281       INFO = 0
00282       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00283          INFO = -1
00284       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00285          INFO = -2
00286       ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
00287      $          THEN
00288          INFO = -3
00289       ELSE IF( N.LT.0 ) THEN
00290          INFO = -4
00291       ELSE
00292          IF( VALEIG ) THEN
00293             IF( N.GT.0 .AND. VU.LE.VL )
00294      $         INFO = -7
00295          ELSE IF( INDEIG ) THEN
00296             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00297                INFO = -8
00298             ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00299                INFO = -9
00300             END IF
00301          END IF
00302       END IF
00303       IF( INFO.EQ.0 ) THEN
00304          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
00305      $      INFO = -14
00306       END IF
00307 *
00308       IF( INFO.NE.0 ) THEN
00309          CALL XERBLA( 'SSPEVX', -INFO )
00310          RETURN
00311       END IF
00312 *
00313 *     Quick return if possible
00314 *
00315       M = 0
00316       IF( N.EQ.0 )
00317      $   RETURN
00318 *
00319       IF( N.EQ.1 ) THEN
00320          IF( ALLEIG .OR. INDEIG ) THEN
00321             M = 1
00322             W( 1 ) = AP( 1 )
00323          ELSE
00324             IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
00325                M = 1
00326                W( 1 ) = AP( 1 )
00327             END IF
00328          END IF
00329          IF( WANTZ )
00330      $      Z( 1, 1 ) = ONE
00331          RETURN
00332       END IF
00333 *
00334 *     Get machine constants.
00335 *
00336       SAFMIN = SLAMCH( 'Safe minimum' )
00337       EPS = SLAMCH( 'Precision' )
00338       SMLNUM = SAFMIN / EPS
00339       BIGNUM = ONE / SMLNUM
00340       RMIN = SQRT( SMLNUM )
00341       RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
00342 *
00343 *     Scale matrix to allowable range, if necessary.
00344 *
00345       ISCALE = 0
00346       ABSTLL = ABSTOL
00347       IF ( VALEIG ) THEN
00348          VLL = VL
00349          VUU = VU
00350       ELSE
00351          VLL = ZERO
00352          VUU = ZERO
00353       ENDIF
00354       ANRM = SLANSP( 'M', UPLO, N, AP, WORK )
00355       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00356          ISCALE = 1
00357          SIGMA = RMIN / ANRM
00358       ELSE IF( ANRM.GT.RMAX ) THEN
00359          ISCALE = 1
00360          SIGMA = RMAX / ANRM
00361       END IF
00362       IF( ISCALE.EQ.1 ) THEN
00363          CALL SSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
00364          IF( ABSTOL.GT.0 )
00365      $      ABSTLL = ABSTOL*SIGMA
00366          IF( VALEIG ) THEN
00367             VLL = VL*SIGMA
00368             VUU = VU*SIGMA
00369          END IF
00370       END IF
00371 *
00372 *     Call SSPTRD to reduce symmetric packed matrix to tridiagonal form.
00373 *
00374       INDTAU = 1
00375       INDE = INDTAU + N
00376       INDD = INDE + N
00377       INDWRK = INDD + N
00378       CALL SSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
00379      $             WORK( INDTAU ), IINFO )
00380 *
00381 *     If all eigenvalues are desired and ABSTOL is less than or equal
00382 *     to zero, then call SSTERF or SOPGTR and SSTEQR.  If this fails
00383 *     for some eigenvalue, then try SSTEBZ.
00384 *
00385       TEST = .FALSE.
00386       IF (INDEIG) THEN
00387          IF (IL.EQ.1 .AND. IU.EQ.N) THEN
00388             TEST = .TRUE.
00389          END IF
00390       END IF
00391       IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
00392          CALL SCOPY( N, WORK( INDD ), 1, W, 1 )
00393          INDEE = INDWRK + 2*N
00394          IF( .NOT.WANTZ ) THEN
00395             CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
00396             CALL SSTERF( N, W, WORK( INDEE ), INFO )
00397          ELSE
00398             CALL SOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
00399      $                   WORK( INDWRK ), IINFO )
00400             CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
00401             CALL SSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
00402      $                   WORK( INDWRK ), INFO )
00403             IF( INFO.EQ.0 ) THEN
00404                DO 10 I = 1, N
00405                   IFAIL( I ) = 0
00406    10          CONTINUE
00407             END IF
00408          END IF
00409          IF( INFO.EQ.0 ) THEN
00410             M = N
00411             GO TO 20
00412          END IF
00413          INFO = 0
00414       END IF
00415 *
00416 *     Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
00417 *
00418       IF( WANTZ ) THEN
00419          ORDER = 'B'
00420       ELSE
00421          ORDER = 'E'
00422       END IF
00423       INDIBL = 1
00424       INDISP = INDIBL + N
00425       INDIWO = INDISP + N
00426       CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
00427      $             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
00428      $             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
00429      $             IWORK( INDIWO ), INFO )
00430 *
00431       IF( WANTZ ) THEN
00432          CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
00433      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00434      $                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
00435 *
00436 *        Apply orthogonal matrix used in reduction to tridiagonal
00437 *        form to eigenvectors returned by SSTEIN.
00438 *
00439          CALL SOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
00440      $                WORK( INDWRK ), IINFO )
00441       END IF
00442 *
00443 *     If matrix was scaled, then rescale eigenvalues appropriately.
00444 *
00445    20 CONTINUE
00446       IF( ISCALE.EQ.1 ) THEN
00447          IF( INFO.EQ.0 ) THEN
00448             IMAX = M
00449          ELSE
00450             IMAX = INFO - 1
00451          END IF
00452          CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
00453       END IF
00454 *
00455 *     If eigenvalues are not in order, then sort them, along with
00456 *     eigenvectors.
00457 *
00458       IF( WANTZ ) THEN
00459          DO 40 J = 1, M - 1
00460             I = 0
00461             TMP1 = W( J )
00462             DO 30 JJ = J + 1, M
00463                IF( W( JJ ).LT.TMP1 ) THEN
00464                   I = JJ
00465                   TMP1 = W( JJ )
00466                END IF
00467    30       CONTINUE
00468 *
00469             IF( I.NE.0 ) THEN
00470                ITMP1 = IWORK( INDIBL+I-1 )
00471                W( I ) = W( J )
00472                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00473                W( J ) = TMP1
00474                IWORK( INDIBL+J-1 ) = ITMP1
00475                CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00476                IF( INFO.NE.0 ) THEN
00477                   ITMP1 = IFAIL( I )
00478                   IFAIL( I ) = IFAIL( J )
00479                   IFAIL( J ) = ITMP1
00480                END IF
00481             END IF
00482    40    CONTINUE
00483       END IF
00484 *
00485       RETURN
00486 *
00487 *     End of SSPEVX
00488 *
00489       END
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