LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sgeevx.f
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00001 *> \brief <b> SGEEVX 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 SGEEVX + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgeevx.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgeevx.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgeevx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SGEEVX( 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 *       REAL               ABNRM
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IWORK( * )
00032 *       REAL               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 *> SGEEVX 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 REAL 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 REAL array, dimension (N)
00150 *> \endverbatim
00151 *>
00152 *> \param[out] WI
00153 *> \verbatim
00154 *>          WI is REAL 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 REAL 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 REAL 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 REAL 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 REAL
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 REAL 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 REAL 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 REAL 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 realGEeigen
00300 *
00301 *  =====================================================================
00302       SUBROUTINE SGEEVX( 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       REAL               ABNRM
00315 *     ..
00316 *     .. Array Arguments ..
00317       INTEGER            IWORK( * )
00318       REAL               A( LDA, * ), RCONDE( * ), RCONDV( * ),
00319      $                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
00320      $                   WI( * ), WORK( * ), WR( * )
00321 *     ..
00322 *
00323 *  =====================================================================
00324 *
00325 *     .. Parameters ..
00326       REAL               ZERO, ONE
00327       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
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       REAL               ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
00336      $                   SN
00337 *     ..
00338 *     .. Local Arrays ..
00339       LOGICAL            SELECT( 1 )
00340       REAL               DUM( 1 )
00341 *     ..
00342 *     .. External Subroutines ..
00343       EXTERNAL           SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD, SLACPY,
00344      $                   SLARTG, SLASCL, SORGHR, SROT, SSCAL, STREVC,
00345      $                   STRSNA, XERBLA
00346 *     ..
00347 *     .. External Functions ..
00348       LOGICAL            LSAME
00349       INTEGER            ILAENV, ISAMAX
00350       REAL               SLAMCH, SLANGE, SLAPY2, SNRM2
00351       EXTERNAL           LSAME, ILAENV, ISAMAX, SLAMCH, SLANGE, SLAPY2,
00352      $                   SNRM2
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, 'S' ) .OR.
00370      $    LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
00371          INFO = -1
00372       ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
00373          INFO = -2
00374       ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
00375          INFO = -3
00376       ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
00377      $         ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
00378      $         WANTVR ) ) ) THEN
00379          INFO = -4
00380       ELSE IF( N.LT.0 ) THEN
00381          INFO = -5
00382       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00383          INFO = -7
00384       ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
00385          INFO = -11
00386       ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
00387          INFO = -13
00388       END IF
00389 *
00390 *     Compute workspace
00391 *      (Note: Comments in the code beginning "Workspace:" describe the
00392 *       minimal amount of workspace needed at that point in the code,
00393 *       as well as the preferred amount for good performance.
00394 *       NB refers to the optimal block size for the immediately
00395 *       following subroutine, as returned by ILAENV.
00396 *       HSWORK refers to the workspace preferred by SHSEQR, as
00397 *       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
00398 *       the worst case.)
00399 *
00400       IF( INFO.EQ.0 ) THEN
00401          IF( N.EQ.0 ) THEN
00402             MINWRK = 1
00403             MAXWRK = 1
00404          ELSE
00405             MAXWRK = N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 )
00406 *
00407             IF( WANTVL ) THEN
00408                CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
00409      $                WORK, -1, INFO )
00410             ELSE IF( WANTVR ) THEN
00411                CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
00412      $                WORK, -1, INFO )
00413             ELSE
00414                IF( WNTSNN ) THEN
00415                   CALL SHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
00416      $                LDVR, WORK, -1, INFO )
00417                ELSE
00418                   CALL SHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
00419      $                LDVR, WORK, -1, INFO )
00420                END IF
00421             END IF
00422             HSWORK = WORK( 1 )
00423 *
00424             IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
00425                MINWRK = 2*N
00426                IF( .NOT.WNTSNN )
00427      $            MINWRK = MAX( MINWRK, N*N+6*N )
00428                MAXWRK = MAX( MAXWRK, HSWORK )
00429                IF( .NOT.WNTSNN )
00430      $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
00431             ELSE
00432                MINWRK = 3*N
00433                IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
00434      $            MINWRK = MAX( MINWRK, N*N + 6*N )
00435                MAXWRK = MAX( MAXWRK, HSWORK )
00436                MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'SORGHR',
00437      $                       ' ', N, 1, N, -1 ) )
00438                IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
00439      $            MAXWRK = MAX( MAXWRK, N*N + 6*N )
00440                MAXWRK = MAX( MAXWRK, 3*N )
00441             END IF
00442             MAXWRK = MAX( MAXWRK, MINWRK )
00443          END IF
00444          WORK( 1 ) = MAXWRK
00445 *
00446          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00447             INFO = -21
00448          END IF
00449       END IF
00450 *
00451       IF( INFO.NE.0 ) THEN
00452          CALL XERBLA( 'SGEEVX', -INFO )
00453          RETURN
00454       ELSE IF( LQUERY ) THEN
00455          RETURN
00456       END IF
00457 *
00458 *     Quick return if possible
00459 *
00460       IF( N.EQ.0 )
00461      $   RETURN
00462 *
00463 *     Get machine constants
00464 *
00465       EPS = SLAMCH( 'P' )
00466       SMLNUM = SLAMCH( 'S' )
00467       BIGNUM = ONE / SMLNUM
00468       CALL SLABAD( SMLNUM, BIGNUM )
00469       SMLNUM = SQRT( SMLNUM ) / EPS
00470       BIGNUM = ONE / SMLNUM
00471 *
00472 *     Scale A if max element outside range [SMLNUM,BIGNUM]
00473 *
00474       ICOND = 0
00475       ANRM = SLANGE( 'M', N, N, A, LDA, DUM )
00476       SCALEA = .FALSE.
00477       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00478          SCALEA = .TRUE.
00479          CSCALE = SMLNUM
00480       ELSE IF( ANRM.GT.BIGNUM ) THEN
00481          SCALEA = .TRUE.
00482          CSCALE = BIGNUM
00483       END IF
00484       IF( SCALEA )
00485      $   CALL SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
00486 *
00487 *     Balance the matrix and compute ABNRM
00488 *
00489       CALL SGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
00490       ABNRM = SLANGE( '1', N, N, A, LDA, DUM )
00491       IF( SCALEA ) THEN
00492          DUM( 1 ) = ABNRM
00493          CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
00494          ABNRM = DUM( 1 )
00495       END IF
00496 *
00497 *     Reduce to upper Hessenberg form
00498 *     (Workspace: need 2*N, prefer N+N*NB)
00499 *
00500       ITAU = 1
00501       IWRK = ITAU + N
00502       CALL SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
00503      $             LWORK-IWRK+1, IERR )
00504 *
00505       IF( WANTVL ) THEN
00506 *
00507 *        Want left eigenvectors
00508 *        Copy Householder vectors to VL
00509 *
00510          SIDE = 'L'
00511          CALL SLACPY( 'L', N, N, A, LDA, VL, LDVL )
00512 *
00513 *        Generate orthogonal matrix in VL
00514 *        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
00515 *
00516          CALL SORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
00517      $                LWORK-IWRK+1, IERR )
00518 *
00519 *        Perform QR iteration, accumulating Schur vectors in VL
00520 *        (Workspace: need 1, prefer HSWORK (see comments) )
00521 *
00522          IWRK = ITAU
00523          CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
00524      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00525 *
00526          IF( WANTVR ) THEN
00527 *
00528 *           Want left and right eigenvectors
00529 *           Copy Schur vectors to VR
00530 *
00531             SIDE = 'B'
00532             CALL SLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
00533          END IF
00534 *
00535       ELSE IF( WANTVR ) THEN
00536 *
00537 *        Want right eigenvectors
00538 *        Copy Householder vectors to VR
00539 *
00540          SIDE = 'R'
00541          CALL SLACPY( 'L', N, N, A, LDA, VR, LDVR )
00542 *
00543 *        Generate orthogonal matrix in VR
00544 *        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
00545 *
00546          CALL SORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
00547      $                LWORK-IWRK+1, IERR )
00548 *
00549 *        Perform QR iteration, accumulating Schur vectors in VR
00550 *        (Workspace: need 1, prefer HSWORK (see comments) )
00551 *
00552          IWRK = ITAU
00553          CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00554      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00555 *
00556       ELSE
00557 *
00558 *        Compute eigenvalues only
00559 *        If condition numbers desired, compute Schur form
00560 *
00561          IF( WNTSNN ) THEN
00562             JOB = 'E'
00563          ELSE
00564             JOB = 'S'
00565          END IF
00566 *
00567 *        (Workspace: need 1, prefer HSWORK (see comments) )
00568 *
00569          IWRK = ITAU
00570          CALL SHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00571      $                WORK( IWRK ), LWORK-IWRK+1, INFO )
00572       END IF
00573 *
00574 *     If INFO > 0 from SHSEQR, then quit
00575 *
00576       IF( INFO.GT.0 )
00577      $   GO TO 50
00578 *
00579       IF( WANTVL .OR. WANTVR ) THEN
00580 *
00581 *        Compute left and/or right eigenvectors
00582 *        (Workspace: need 3*N)
00583 *
00584          CALL STREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
00585      $                N, NOUT, WORK( IWRK ), IERR )
00586       END IF
00587 *
00588 *     Compute condition numbers if desired
00589 *     (Workspace: need N*N+6*N unless SENSE = 'E')
00590 *
00591       IF( .NOT.WNTSNN ) THEN
00592          CALL STRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
00593      $                RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
00594      $                ICOND )
00595       END IF
00596 *
00597       IF( WANTVL ) THEN
00598 *
00599 *        Undo balancing of left eigenvectors
00600 *
00601          CALL SGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
00602      $                IERR )
00603 *
00604 *        Normalize left eigenvectors and make largest component real
00605 *
00606          DO 20 I = 1, N
00607             IF( WI( I ).EQ.ZERO ) THEN
00608                SCL = ONE / SNRM2( N, VL( 1, I ), 1 )
00609                CALL SSCAL( N, SCL, VL( 1, I ), 1 )
00610             ELSE IF( WI( I ).GT.ZERO ) THEN
00611                SCL = ONE / SLAPY2( SNRM2( N, VL( 1, I ), 1 ),
00612      $               SNRM2( N, VL( 1, I+1 ), 1 ) )
00613                CALL SSCAL( N, SCL, VL( 1, I ), 1 )
00614                CALL SSCAL( N, SCL, VL( 1, I+1 ), 1 )
00615                DO 10 K = 1, N
00616                   WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
00617    10          CONTINUE
00618                K = ISAMAX( N, WORK, 1 )
00619                CALL SLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
00620                CALL SROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
00621                VL( K, I+1 ) = ZERO
00622             END IF
00623    20    CONTINUE
00624       END IF
00625 *
00626       IF( WANTVR ) THEN
00627 *
00628 *        Undo balancing of right eigenvectors
00629 *
00630          CALL SGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
00631      $                IERR )
00632 *
00633 *        Normalize right eigenvectors and make largest component real
00634 *
00635          DO 40 I = 1, N
00636             IF( WI( I ).EQ.ZERO ) THEN
00637                SCL = ONE / SNRM2( N, VR( 1, I ), 1 )
00638                CALL SSCAL( N, SCL, VR( 1, I ), 1 )
00639             ELSE IF( WI( I ).GT.ZERO ) THEN
00640                SCL = ONE / SLAPY2( SNRM2( N, VR( 1, I ), 1 ),
00641      $               SNRM2( N, VR( 1, I+1 ), 1 ) )
00642                CALL SSCAL( N, SCL, VR( 1, I ), 1 )
00643                CALL SSCAL( N, SCL, VR( 1, I+1 ), 1 )
00644                DO 30 K = 1, N
00645                   WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
00646    30          CONTINUE
00647                K = ISAMAX( N, WORK, 1 )
00648                CALL SLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
00649                CALL SROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
00650                VR( K, I+1 ) = ZERO
00651             END IF
00652    40    CONTINUE
00653       END IF
00654 *
00655 *     Undo scaling if necessary
00656 *
00657    50 CONTINUE
00658       IF( SCALEA ) THEN
00659          CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
00660      $                MAX( N-INFO, 1 ), IERR )
00661          CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
00662      $                MAX( N-INFO, 1 ), IERR )
00663          IF( INFO.EQ.0 ) THEN
00664             IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
00665      $         CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
00666      $                      IERR )
00667          ELSE
00668             CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
00669      $                   IERR )
00670             CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
00671      $                   IERR )
00672          END IF
00673       END IF
00674 *
00675       WORK( 1 ) = MAXWRK
00676       RETURN
00677 *
00678 *     End of SGEEVX
00679 *
00680       END
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