LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dgeevx.f
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00001 *> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE 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 DGEEVX + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
00022 *                          VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
00023 *                          RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
00027 *       INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
00028 *       DOUBLE PRECISION   ABNRM
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IWORK( * )
00032 *       DOUBLE PRECISION   A( LDA, * ), RCONDE( * ), RCONDV( * ),
00033 *      $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
00034 *      $                   WI( * ), WORK( * ), WR( * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
00044 *> eigenvalues and, optionally, the left and/or right eigenvectors.
00045 *>
00046 *> Optionally also, it computes a balancing transformation to improve
00047 *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
00048 *> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
00049 *> (RCONDE), and reciprocal condition numbers for the right
00050 *> eigenvectors (RCONDV).
00051 *>
00052 *> The right eigenvector v(j) of A satisfies
00053 *>                  A * v(j) = lambda(j) * v(j)
00054 *> where lambda(j) is its eigenvalue.
00055 *> The left eigenvector u(j) of A satisfies
00056 *>               u(j)**T * A = lambda(j) * u(j)**T
00057 *> where u(j)**T denotes the transpose of u(j).
00058 *>
00059 *> The computed eigenvectors are normalized to have Euclidean norm
00060 *> equal to 1 and largest component real.
00061 *>
00062 *> Balancing a matrix means permuting the rows and columns to make it
00063 *> more nearly upper triangular, and applying a diagonal similarity
00064 *> transformation D * A * D**(-1), where D is a diagonal matrix, to
00065 *> make its rows and columns closer in norm and the condition numbers
00066 *> of its eigenvalues and eigenvectors smaller.  The computed
00067 *> reciprocal condition numbers correspond to the balanced matrix.
00068 *> Permuting rows and columns will not change the condition numbers
00069 *> (in exact arithmetic) but diagonal scaling will.  For further
00070 *> explanation of balancing, see section 4.10.2 of the LAPACK
00071 *> Users' Guide.
00072 *> \endverbatim
00073 *
00074 *  Arguments:
00075 *  ==========
00076 *
00077 *> \param[in] BALANC
00078 *> \verbatim
00079 *>          BALANC is CHARACTER*1
00080 *>          Indicates how the input matrix should be diagonally scaled
00081 *>          and/or permuted to improve the conditioning of its
00082 *>          eigenvalues.
00083 *>          = 'N': Do not diagonally scale or permute;
00084 *>          = 'P': Perform permutations to make the matrix more nearly
00085 *>                 upper triangular. Do not diagonally scale;
00086 *>          = 'S': Diagonally scale the matrix, i.e. replace A by
00087 *>                 D*A*D**(-1), where D is a diagonal matrix chosen
00088 *>                 to make the rows and columns of A more equal in
00089 *>                 norm. Do not permute;
00090 *>          = 'B': Both diagonally scale and permute A.
00091 *>
00092 *>          Computed reciprocal condition numbers will be for the matrix
00093 *>          after balancing and/or permuting. Permuting does not change
00094 *>          condition numbers (in exact arithmetic), but balancing does.
00095 *> \endverbatim
00096 *>
00097 *> \param[in] JOBVL
00098 *> \verbatim
00099 *>          JOBVL is CHARACTER*1
00100 *>          = 'N': left eigenvectors of A are not computed;
00101 *>          = 'V': left eigenvectors of A are computed.
00102 *>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] JOBVR
00106 *> \verbatim
00107 *>          JOBVR is CHARACTER*1
00108 *>          = 'N': right eigenvectors of A are not computed;
00109 *>          = 'V': right eigenvectors of A are computed.
00110 *>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
00111 *> \endverbatim
00112 *>
00113 *> \param[in] SENSE
00114 *> \verbatim
00115 *>          SENSE is CHARACTER*1
00116 *>          Determines which reciprocal condition numbers are computed.
00117 *>          = 'N': None are computed;
00118 *>          = 'E': Computed for eigenvalues only;
00119 *>          = 'V': Computed for right eigenvectors only;
00120 *>          = 'B': Computed for eigenvalues and right eigenvectors.
00121 *>
00122 *>          If SENSE = 'E' or 'B', both left and right eigenvectors
00123 *>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
00124 *> \endverbatim
00125 *>
00126 *> \param[in] N
00127 *> \verbatim
00128 *>          N is INTEGER
00129 *>          The order of the matrix A. N >= 0.
00130 *> \endverbatim
00131 *>
00132 *> \param[in,out] A
00133 *> \verbatim
00134 *>          A is DOUBLE PRECISION array, dimension (LDA,N)
00135 *>          On entry, the N-by-N matrix A.
00136 *>          On exit, A has been overwritten.  If JOBVL = 'V' or
00137 *>          JOBVR = 'V', A contains the real Schur form of the balanced
00138 *>          version of the input matrix A.
00139 *> \endverbatim
00140 *>
00141 *> \param[in] LDA
00142 *> \verbatim
00143 *>          LDA is INTEGER
00144 *>          The leading dimension of the array A.  LDA >= max(1,N).
00145 *> \endverbatim
00146 *>
00147 *> \param[out] WR
00148 *> \verbatim
00149 *>          WR is DOUBLE PRECISION array, dimension (N)
00150 *> \endverbatim
00151 *>
00152 *> \param[out] WI
00153 *> \verbatim
00154 *>          WI is DOUBLE PRECISION array, dimension (N)
00155 *>          WR and WI contain the real and imaginary parts,
00156 *>          respectively, of the computed eigenvalues.  Complex
00157 *>          conjugate pairs of eigenvalues will appear consecutively
00158 *>          with the eigenvalue having the positive imaginary part
00159 *>          first.
00160 *> \endverbatim
00161 *>
00162 *> \param[out] VL
00163 *> \verbatim
00164 *>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
00165 *>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
00166 *>          after another in the columns of VL, in the same order
00167 *>          as their eigenvalues.
00168 *>          If JOBVL = 'N', VL is not referenced.
00169 *>          If the j-th eigenvalue is real, then u(j) = VL(:,j),
00170 *>          the j-th column of VL.
00171 *>          If the j-th and (j+1)-st eigenvalues form a complex
00172 *>          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
00173 *>          u(j+1) = VL(:,j) - i*VL(:,j+1).
00174 *> \endverbatim
00175 *>
00176 *> \param[in] LDVL
00177 *> \verbatim
00178 *>          LDVL is INTEGER
00179 *>          The leading dimension of the array VL.  LDVL >= 1; if
00180 *>          JOBVL = 'V', LDVL >= N.
00181 *> \endverbatim
00182 *>
00183 *> \param[out] VR
00184 *> \verbatim
00185 *>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
00186 *>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
00187 *>          after another in the columns of VR, in the same order
00188 *>          as their eigenvalues.
00189 *>          If JOBVR = 'N', VR is not referenced.
00190 *>          If the j-th eigenvalue is real, then v(j) = VR(:,j),
00191 *>          the j-th column of VR.
00192 *>          If the j-th and (j+1)-st eigenvalues form a complex
00193 *>          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
00194 *>          v(j+1) = VR(:,j) - i*VR(:,j+1).
00195 *> \endverbatim
00196 *>
00197 *> \param[in] LDVR
00198 *> \verbatim
00199 *>          LDVR is INTEGER
00200 *>          The leading dimension of the array VR.  LDVR >= 1, and if
00201 *>          JOBVR = 'V', LDVR >= N.
00202 *> \endverbatim
00203 *>
00204 *> \param[out] ILO
00205 *> \verbatim
00206 *>          ILO is INTEGER
00207 *> \endverbatim
00208 *>
00209 *> \param[out] IHI
00210 *> \verbatim
00211 *>          IHI is INTEGER
00212 *>          ILO and IHI are integer values determined when A was
00213 *>          balanced.  The balanced A(i,j) = 0 if I > J and
00214 *>          J = 1,...,ILO-1 or I = IHI+1,...,N.
00215 *> \endverbatim
00216 *>
00217 *> \param[out] SCALE
00218 *> \verbatim
00219 *>          SCALE is DOUBLE PRECISION array, dimension (N)
00220 *>          Details of the permutations and scaling factors applied
00221 *>          when balancing A.  If P(j) is the index of the row and column
00222 *>          interchanged with row and column j, and D(j) is the scaling
00223 *>          factor applied to row and column j, then
00224 *>          SCALE(J) = P(J),    for J = 1,...,ILO-1
00225 *>                   = D(J),    for J = ILO,...,IHI
00226 *>                   = P(J)     for J = IHI+1,...,N.
00227 *>          The order in which the interchanges are made is N to IHI+1,
00228 *>          then 1 to ILO-1.
00229 *> \endverbatim
00230 *>
00231 *> \param[out] ABNRM
00232 *> \verbatim
00233 *>          ABNRM is DOUBLE PRECISION
00234 *>          The one-norm of the balanced matrix (the maximum
00235 *>          of the sum of absolute values of elements of any column).
00236 *> \endverbatim
00237 *>
00238 *> \param[out] RCONDE
00239 *> \verbatim
00240 *>          RCONDE is DOUBLE PRECISION array, dimension (N)
00241 *>          RCONDE(j) is the reciprocal condition number of the j-th
00242 *>          eigenvalue.
00243 *> \endverbatim
00244 *>
00245 *> \param[out] RCONDV
00246 *> \verbatim
00247 *>          RCONDV is DOUBLE PRECISION array, dimension (N)
00248 *>          RCONDV(j) is the reciprocal condition number of the j-th
00249 *>          right eigenvector.
00250 *> \endverbatim
00251 *>
00252 *> \param[out] WORK
00253 *> \verbatim
00254 *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00255 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00256 *> \endverbatim
00257 *>
00258 *> \param[in] LWORK
00259 *> \verbatim
00260 *>          LWORK is INTEGER
00261 *>          The dimension of the array WORK.   If SENSE = 'N' or 'E',
00262 *>          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
00263 *>          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
00264 *>          For good performance, LWORK must generally be larger.
00265 *>
00266 *>          If LWORK = -1, then a workspace query is assumed; the routine
00267 *>          only calculates the optimal size of the WORK array, returns
00268 *>          this value as the first entry of the WORK array, and no error
00269 *>          message related to LWORK is issued by XERBLA.
00270 *> \endverbatim
00271 *>
00272 *> \param[out] IWORK
00273 *> \verbatim
00274 *>          IWORK is INTEGER array, dimension (2*N-2)
00275 *>          If SENSE = 'N' or 'E', not referenced.
00276 *> \endverbatim
00277 *>
00278 *> \param[out] INFO
00279 *> \verbatim
00280 *>          INFO is INTEGER
00281 *>          = 0:  successful exit
00282 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00283 *>          > 0:  if INFO = i, the QR algorithm failed to compute all the
00284 *>                eigenvalues, and no eigenvectors or condition numbers
00285 *>                have been computed; elements 1:ILO-1 and i+1:N of WR
00286 *>                and WI contain eigenvalues which have converged.
00287 *> \endverbatim
00288 *
00289 *  Authors:
00290 *  ========
00291 *
00292 *> \author Univ. of Tennessee 
00293 *> \author Univ. of California Berkeley 
00294 *> \author Univ. of Colorado Denver 
00295 *> \author NAG Ltd. 
00296 *
00297 *> \date November 2011
00298 *
00299 *> \ingroup doubleGEeigen
00300 *
00301 *  =====================================================================
00302       SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
00303      $                   VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
00304      $                   RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
00305 *
00306 *  -- LAPACK driver routine (version 3.4.0) --
00307 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00308 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00309 *     November 2011
00310 *
00311 *     .. Scalar Arguments ..
00312       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
00313       INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
00314       DOUBLE PRECISION   ABNRM
00315 *     ..
00316 *     .. Array Arguments ..
00317       INTEGER            IWORK( * )
00318       DOUBLE PRECISION   A( LDA, * ), RCONDE( * ), RCONDV( * ),
00319      $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
00320      $                   WI( * ), WORK( * ), WR( * )
00321 *     ..
00322 *
00323 *  =====================================================================
00324 *
00325 *     .. Parameters ..
00326       DOUBLE PRECISION   ZERO, ONE
00327       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
00328 *     ..
00329 *     .. Local Scalars ..
00330       LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
00331      $                   WNTSNN, WNTSNV
00332       CHARACTER          JOB, SIDE
00333       INTEGER            HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
00334      $                   MINWRK, NOUT
00335       DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
00336      $                   SN
00337 *     ..
00338 *     .. Local Arrays ..
00339       LOGICAL            SELECT( 1 )
00340       DOUBLE PRECISION   DUM( 1 )
00341 *     ..
00342 *     .. External Subroutines ..
00343       EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
00344      $                   DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
00345      $                   DTRSNA, XERBLA
00346 *     ..
00347 *     .. External Functions ..
00348       LOGICAL            LSAME
00349       INTEGER            IDAMAX, ILAENV
00350       DOUBLE PRECISION   DLAMCH, DLANGE, DLAPY2, DNRM2
00351       EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
00352      $                   DNRM2
00353 *     ..
00354 *     .. Intrinsic Functions ..
00355       INTRINSIC          MAX, SQRT
00356 *     ..
00357 *     .. Executable Statements ..
00358 *
00359 *     Test the input arguments
00360 *
00361       INFO = 0
00362       LQUERY = ( LWORK.EQ.-1 )
00363       WANTVL = LSAME( JOBVL, 'V' )
00364       WANTVR = LSAME( JOBVR, 'V' )
00365       WNTSNN = LSAME( SENSE, 'N' )
00366       WNTSNE = LSAME( SENSE, 'E' )
00367       WNTSNV = LSAME( SENSE, 'V' )
00368       WNTSNB = LSAME( SENSE, 'B' )
00369       IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
00370      $    'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
00371      $     THEN
00372          INFO = -1
00373       ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
00374          INFO = -2
00375       ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
00376          INFO = -3
00377       ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
00378      $         ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
00379      $         WANTVR ) ) ) THEN
00380          INFO = -4
00381       ELSE IF( N.LT.0 ) THEN
00382          INFO = -5
00383       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00384          INFO = -7
00385       ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
00386          INFO = -11
00387       ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
00388          INFO = -13
00389       END IF
00390 *
00391 *     Compute workspace
00392 *      (Note: Comments in the code beginning "Workspace:" describe the
00393 *       minimal amount of workspace needed at that point in the code,
00394 *       as well as the preferred amount for good performance.
00395 *       NB refers to the optimal block size for the immediately
00396 *       following subroutine, as returned by ILAENV.
00397 *       HSWORK refers to the workspace preferred by DHSEQR, as
00398 *       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
00399 *       the worst case.)
00400 *
00401       IF( INFO.EQ.0 ) THEN
00402          IF( N.EQ.0 ) THEN
00403             MINWRK = 1
00404             MAXWRK = 1
00405          ELSE
00406             MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
00407 *
00408             IF( WANTVL ) THEN
00409                CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
00410      $                WORK, -1, INFO )
00411             ELSE IF( WANTVR ) THEN
00412                CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
00413      $                WORK, -1, INFO )
00414             ELSE
00415                IF( WNTSNN ) THEN
00416                   CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
00417      $                LDVR, WORK, -1, INFO )
00418                ELSE
00419                   CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
00420      $                LDVR, WORK, -1, INFO )
00421                END IF
00422             END IF
00423             HSWORK = WORK( 1 )
00424 *
00425             IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
00426                MINWRK = 2*N
00427                IF( .NOT.WNTSNN )
00428      $            MINWRK = MAX( MINWRK, N*N+6*N )
00429                MAXWRK = MAX( MAXWRK, HSWORK )
00430                IF( .NOT.WNTSNN )
00431      $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
00432             ELSE
00433                MINWRK = 3*N
00434                IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
00435      $            MINWRK = MAX( MINWRK, N*N + 6*N )
00436                MAXWRK = MAX( MAXWRK, HSWORK )
00437                MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'DORGHR',
00438      $                       ' ', N, 1, N, -1 ) )
00439                IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
00440      $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
00441                MAXWRK = MAX( MAXWRK, 3*N )
00442             END IF
00443             MAXWRK = MAX( MAXWRK, MINWRK )
00444          END IF
00445          WORK( 1 ) = MAXWRK
00446 *
00447          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00448             INFO = -21
00449          END IF
00450       END IF
00451 *
00452       IF( INFO.NE.0 ) THEN
00453          CALL XERBLA( 'DGEEVX', -INFO )
00454          RETURN
00455       ELSE IF( LQUERY ) THEN
00456          RETURN
00457       END IF
00458 *
00459 *     Quick return if possible
00460 *
00461       IF( N.EQ.0 )
00462      $   RETURN
00463 *
00464 *     Get machine constants
00465 *
00466       EPS = DLAMCH( 'P' )
00467       SMLNUM = DLAMCH( 'S' )
00468       BIGNUM = ONE / SMLNUM
00469       CALL DLABAD( SMLNUM, BIGNUM )
00470       SMLNUM = SQRT( SMLNUM ) / EPS
00471       BIGNUM = ONE / SMLNUM
00472 *
00473 *     Scale A if max element outside range [SMLNUM,BIGNUM]
00474 *
00475       ICOND = 0
00476       ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
00477       SCALEA = .FALSE.
00478       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00479          SCALEA = .TRUE.
00480          CSCALE = SMLNUM
00481       ELSE IF( ANRM.GT.BIGNUM ) THEN
00482          SCALEA = .TRUE.
00483          CSCALE = BIGNUM
00484       END IF
00485       IF( SCALEA )
00486      $   CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
00487 *
00488 *     Balance the matrix and compute ABNRM
00489 *
00490       CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
00491       ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
00492       IF( SCALEA ) THEN
00493          DUM( 1 ) = ABNRM
00494          CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
00495          ABNRM = DUM( 1 )
00496       END IF
00497 *
00498 *     Reduce to upper Hessenberg form
00499 *     (Workspace: need 2*N, prefer N+N*NB)
00500 *
00501       ITAU = 1
00502       IWRK = ITAU + N
00503       CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
00504      $             LWORK-IWRK+1, IERR )
00505 *
00506       IF( WANTVL ) THEN
00507 *
00508 *        Want left eigenvectors
00509 *        Copy Householder vectors to VL
00510 *
00511          SIDE = 'L'
00512          CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
00513 *
00514 *        Generate orthogonal matrix in VL
00515 *        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
00516 *
00517          CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
00518      $                LWORK-IWRK+1, IERR )
00519 *
00520 *        Perform QR iteration, accumulating Schur vectors in VL
00521 *        (Workspace: need 1, prefer HSWORK (see comments) )
00522 *
00523          IWRK = ITAU
00524          CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
00525      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00526 *
00527          IF( WANTVR ) THEN
00528 *
00529 *           Want left and right eigenvectors
00530 *           Copy Schur vectors to VR
00531 *
00532             SIDE = 'B'
00533             CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
00534          END IF
00535 *
00536       ELSE IF( WANTVR ) THEN
00537 *
00538 *        Want right eigenvectors
00539 *        Copy Householder vectors to VR
00540 *
00541          SIDE = 'R'
00542          CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
00543 *
00544 *        Generate orthogonal matrix in VR
00545 *        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
00546 *
00547          CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
00548      $                LWORK-IWRK+1, IERR )
00549 *
00550 *        Perform QR iteration, accumulating Schur vectors in VR
00551 *        (Workspace: need 1, prefer HSWORK (see comments) )
00552 *
00553          IWRK = ITAU
00554          CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00555      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00556 *
00557       ELSE
00558 *
00559 *        Compute eigenvalues only
00560 *        If condition numbers desired, compute Schur form
00561 *
00562          IF( WNTSNN ) THEN
00563             JOB = 'E'
00564          ELSE
00565             JOB = 'S'
00566          END IF
00567 *
00568 *        (Workspace: need 1, prefer HSWORK (see comments) )
00569 *
00570          IWRK = ITAU
00571          CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00572      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00573       END IF
00574 *
00575 *     If INFO > 0 from DHSEQR, then quit
00576 *
00577       IF( INFO.GT.0 )
00578      $   GO TO 50
00579 *
00580       IF( WANTVL .OR. WANTVR ) THEN
00581 *
00582 *        Compute left and/or right eigenvectors
00583 *        (Workspace: need 3*N)
00584 *
00585          CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
00586      $                N, NOUT, WORK( IWRK ), IERR )
00587       END IF
00588 *
00589 *     Compute condition numbers if desired
00590 *     (Workspace: need N*N+6*N unless SENSE = 'E')
00591 *
00592       IF( .NOT.WNTSNN ) THEN
00593          CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
00594      $                RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
00595      $                ICOND )
00596       END IF
00597 *
00598       IF( WANTVL ) THEN
00599 *
00600 *        Undo balancing of left eigenvectors
00601 *
00602          CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
00603      $                IERR )
00604 *
00605 *        Normalize left eigenvectors and make largest component real
00606 *
00607          DO 20 I = 1, N
00608             IF( WI( I ).EQ.ZERO ) THEN
00609                SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
00610                CALL DSCAL( N, SCL, VL( 1, I ), 1 )
00611             ELSE IF( WI( I ).GT.ZERO ) THEN
00612                SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
00613      $               DNRM2( N, VL( 1, I+1 ), 1 ) )
00614                CALL DSCAL( N, SCL, VL( 1, I ), 1 )
00615                CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
00616                DO 10 K = 1, N
00617                   WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
00618    10          CONTINUE
00619                K = IDAMAX( N, WORK, 1 )
00620                CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
00621                CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
00622                VL( K, I+1 ) = ZERO
00623             END IF
00624    20    CONTINUE
00625       END IF
00626 *
00627       IF( WANTVR ) THEN
00628 *
00629 *        Undo balancing of right eigenvectors
00630 *
00631          CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
00632      $                IERR )
00633 *
00634 *        Normalize right eigenvectors and make largest component real
00635 *
00636          DO 40 I = 1, N
00637             IF( WI( I ).EQ.ZERO ) THEN
00638                SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
00639                CALL DSCAL( N, SCL, VR( 1, I ), 1 )
00640             ELSE IF( WI( I ).GT.ZERO ) THEN
00641                SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
00642      $               DNRM2( N, VR( 1, I+1 ), 1 ) )
00643                CALL DSCAL( N, SCL, VR( 1, I ), 1 )
00644                CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
00645                DO 30 K = 1, N
00646                   WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
00647    30          CONTINUE
00648                K = IDAMAX( N, WORK, 1 )
00649                CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
00650                CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
00651                VR( K, I+1 ) = ZERO
00652             END IF
00653    40    CONTINUE
00654       END IF
00655 *
00656 *     Undo scaling if necessary
00657 *
00658    50 CONTINUE
00659       IF( SCALEA ) THEN
00660          CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
00661      $                MAX( N-INFO, 1 ), IERR )
00662          CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
00663      $                MAX( N-INFO, 1 ), IERR )
00664          IF( INFO.EQ.0 ) THEN
00665             IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
00666      $         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
00667      $                      IERR )
00668          ELSE
00669             CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
00670      $                   IERR )
00671             CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
00672      $                   IERR )
00673          END IF
00674       END IF
00675 *
00676       WORK( 1 ) = MAXWRK
00677       RETURN
00678 *
00679 *     End of DGEEVX
00680 *
00681       END
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