LAPACK 3.3.1 Linear Algebra PACKage

# sgeev.f

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```00001       SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
00002      \$                  LDVR, WORK, LWORK, INFO )
00003 *
00004 *  -- LAPACK driver routine (version 3.3.1) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *  -- April 2011                                                      --
00008 *
00009 *     .. Scalar Arguments ..
00010       CHARACTER          JOBVL, JOBVR
00011       INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
00012 *     ..
00013 *     .. Array Arguments ..
00014       REAL               A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
00015      \$                   WI( * ), WORK( * ), WR( * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  SGEEV computes for an N-by-N real nonsymmetric matrix A, the
00022 *  eigenvalues and, optionally, the left and/or right eigenvectors.
00023 *
00024 *  The right eigenvector v(j) of A satisfies
00025 *                   A * v(j) = lambda(j) * v(j)
00026 *  where lambda(j) is its eigenvalue.
00027 *  The left eigenvector u(j) of A satisfies
00028 *                u(j)**T * A = lambda(j) * u(j)**T
00029 *  where u(j)**T denotes the transpose of u(j).
00030 *
00031 *  The computed eigenvectors are normalized to have Euclidean norm
00032 *  equal to 1 and largest component real.
00033 *
00034 *  Arguments
00035 *  =========
00036 *
00037 *  JOBVL   (input) CHARACTER*1
00038 *          = 'N': left eigenvectors of A are not computed;
00039 *          = 'V': left eigenvectors of A are computed.
00040 *
00041 *  JOBVR   (input) CHARACTER*1
00042 *          = 'N': right eigenvectors of A are not computed;
00043 *          = 'V': right eigenvectors of A are computed.
00044 *
00045 *  N       (input) INTEGER
00046 *          The order of the matrix A. N >= 0.
00047 *
00048 *  A       (input/output) REAL array, dimension (LDA,N)
00049 *          On entry, the N-by-N matrix A.
00050 *          On exit, A has been overwritten.
00051 *
00052 *  LDA     (input) INTEGER
00053 *          The leading dimension of the array A.  LDA >= max(1,N).
00054 *
00055 *  WR      (output) REAL array, dimension (N)
00056 *  WI      (output) REAL array, dimension (N)
00057 *          WR and WI contain the real and imaginary parts,
00058 *          respectively, of the computed eigenvalues.  Complex
00059 *          conjugate pairs of eigenvalues appear consecutively
00060 *          with the eigenvalue having the positive imaginary part
00061 *          first.
00062 *
00063 *  VL      (output) REAL array, dimension (LDVL,N)
00064 *          If JOBVL = 'V', the left eigenvectors u(j) are stored one
00065 *          after another in the columns of VL, in the same order
00066 *          as their eigenvalues.
00067 *          If JOBVL = 'N', VL is not referenced.
00068 *          If the j-th eigenvalue is real, then u(j) = VL(:,j),
00069 *          the j-th column of VL.
00070 *          If the j-th and (j+1)-st eigenvalues form a complex
00071 *          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
00072 *          u(j+1) = VL(:,j) - i*VL(:,j+1).
00073 *
00074 *  LDVL    (input) INTEGER
00075 *          The leading dimension of the array VL.  LDVL >= 1; if
00076 *          JOBVL = 'V', LDVL >= N.
00077 *
00078 *  VR      (output) REAL array, dimension (LDVR,N)
00079 *          If JOBVR = 'V', the right eigenvectors v(j) are stored one
00080 *          after another in the columns of VR, in the same order
00081 *          as their eigenvalues.
00082 *          If JOBVR = 'N', VR is not referenced.
00083 *          If the j-th eigenvalue is real, then v(j) = VR(:,j),
00084 *          the j-th column of VR.
00085 *          If the j-th and (j+1)-st eigenvalues form a complex
00086 *          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
00087 *          v(j+1) = VR(:,j) - i*VR(:,j+1).
00088 *
00089 *  LDVR    (input) INTEGER
00090 *          The leading dimension of the array VR.  LDVR >= 1; if
00091 *          JOBVR = 'V', LDVR >= N.
00092 *
00093 *  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
00094 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00095 *
00096 *  LWORK   (input) INTEGER
00097 *          The dimension of the array WORK.  LWORK >= max(1,3*N), and
00098 *          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good
00099 *          performance, LWORK must generally be larger.
00100 *
00101 *          If LWORK = -1, then a workspace query is assumed; the routine
00102 *          only calculates the optimal size of the WORK array, returns
00103 *          this value as the first entry of the WORK array, and no error
00104 *          message related to LWORK is issued by XERBLA.
00105 *
00106 *  INFO    (output) INTEGER
00107 *          = 0:  successful exit
00108 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00109 *          > 0:  if INFO = i, the QR algorithm failed to compute all the
00110 *                eigenvalues, and no eigenvectors have been computed;
00111 *                elements i+1:N of WR and WI contain eigenvalues which
00112 *                have converged.
00113 *
00114 *  =====================================================================
00115 *
00116 *     .. Parameters ..
00117       REAL               ZERO, ONE
00118       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
00119 *     ..
00120 *     .. Local Scalars ..
00121       LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
00122       CHARACTER          SIDE
00123       INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
00124      \$                   MAXWRK, MINWRK, NOUT
00125       REAL               ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
00126      \$                   SN
00127 *     ..
00128 *     .. Local Arrays ..
00129       LOGICAL            SELECT( 1 )
00130       REAL               DUM( 1 )
00131 *     ..
00132 *     .. External Subroutines ..
00133       EXTERNAL           SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD, SLACPY,
00134      \$                   SLARTG, SLASCL, SORGHR, SROT, SSCAL, STREVC,
00135      \$                   XERBLA
00136 *     ..
00137 *     .. External Functions ..
00138       LOGICAL            LSAME
00139       INTEGER            ILAENV, ISAMAX
00140       REAL               SLAMCH, SLANGE, SLAPY2, SNRM2
00141       EXTERNAL           LSAME, ILAENV, ISAMAX, SLAMCH, SLANGE, SLAPY2,
00142      \$                   SNRM2
00143 *     ..
00144 *     .. Intrinsic Functions ..
00145       INTRINSIC          MAX, SQRT
00146 *     ..
00147 *     .. Executable Statements ..
00148 *
00149 *     Test the input arguments
00150 *
00151       INFO = 0
00152       LQUERY = ( LWORK.EQ.-1 )
00153       WANTVL = LSAME( JOBVL, 'V' )
00154       WANTVR = LSAME( JOBVR, 'V' )
00155       IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
00156          INFO = -1
00157       ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
00158          INFO = -2
00159       ELSE IF( N.LT.0 ) THEN
00160          INFO = -3
00161       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00162          INFO = -5
00163       ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
00164          INFO = -9
00165       ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
00166          INFO = -11
00167       END IF
00168 *
00169 *     Compute workspace
00170 *      (Note: Comments in the code beginning "Workspace:" describe the
00171 *       minimal amount of workspace needed at that point in the code,
00172 *       as well as the preferred amount for good performance.
00173 *       NB refers to the optimal block size for the immediately
00174 *       following subroutine, as returned by ILAENV.
00175 *       HSWORK refers to the workspace preferred by SHSEQR, as
00176 *       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
00177 *       the worst case.)
00178 *
00179       IF( INFO.EQ.0 ) THEN
00180          IF( N.EQ.0 ) THEN
00181             MINWRK = 1
00182             MAXWRK = 1
00183          ELSE
00184             MAXWRK = 2*N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 )
00185             IF( WANTVL ) THEN
00186                MINWRK = 4*N
00187                MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
00188      \$                       'SORGHR', ' ', N, 1, N, -1 ) )
00189                CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
00190      \$                WORK, -1, INFO )
00191                HSWORK = WORK( 1 )
00192                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
00193                MAXWRK = MAX( MAXWRK, 4*N )
00194             ELSE IF( WANTVR ) THEN
00195                MINWRK = 4*N
00196                MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
00197      \$                       'SORGHR', ' ', N, 1, N, -1 ) )
00198                CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
00199      \$                WORK, -1, INFO )
00200                HSWORK = WORK( 1 )
00201                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
00202                MAXWRK = MAX( MAXWRK, 4*N )
00203             ELSE
00204                MINWRK = 3*N
00205                CALL SHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
00206      \$                WORK, -1, INFO )
00207                HSWORK = WORK( 1 )
00208                MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
00209             END IF
00210             MAXWRK = MAX( MAXWRK, MINWRK )
00211          END IF
00212          WORK( 1 ) = MAXWRK
00213 *
00214          IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
00215             INFO = -13
00216          END IF
00217       END IF
00218 *
00219       IF( INFO.NE.0 ) THEN
00220          CALL XERBLA( 'SGEEV ', -INFO )
00221          RETURN
00222       ELSE IF( LQUERY ) THEN
00223          RETURN
00224       END IF
00225 *
00226 *     Quick return if possible
00227 *
00228       IF( N.EQ.0 )
00229      \$   RETURN
00230 *
00231 *     Get machine constants
00232 *
00233       EPS = SLAMCH( 'P' )
00234       SMLNUM = SLAMCH( 'S' )
00235       BIGNUM = ONE / SMLNUM
00236       CALL SLABAD( SMLNUM, BIGNUM )
00237       SMLNUM = SQRT( SMLNUM ) / EPS
00238       BIGNUM = ONE / SMLNUM
00239 *
00240 *     Scale A if max element outside range [SMLNUM,BIGNUM]
00241 *
00242       ANRM = SLANGE( 'M', N, N, A, LDA, DUM )
00243       SCALEA = .FALSE.
00244       IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
00245          SCALEA = .TRUE.
00246          CSCALE = SMLNUM
00247       ELSE IF( ANRM.GT.BIGNUM ) THEN
00248          SCALEA = .TRUE.
00249          CSCALE = BIGNUM
00250       END IF
00251       IF( SCALEA )
00252      \$   CALL SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
00253 *
00254 *     Balance the matrix
00255 *     (Workspace: need N)
00256 *
00257       IBAL = 1
00258       CALL SGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
00259 *
00260 *     Reduce to upper Hessenberg form
00261 *     (Workspace: need 3*N, prefer 2*N+N*NB)
00262 *
00263       ITAU = IBAL + N
00264       IWRK = ITAU + N
00265       CALL SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
00266      \$             LWORK-IWRK+1, IERR )
00267 *
00268       IF( WANTVL ) THEN
00269 *
00270 *        Want left eigenvectors
00271 *        Copy Householder vectors to VL
00272 *
00273          SIDE = 'L'
00274          CALL SLACPY( 'L', N, N, A, LDA, VL, LDVL )
00275 *
00276 *        Generate orthogonal matrix in VL
00277 *        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
00278 *
00279          CALL SORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
00280      \$                LWORK-IWRK+1, IERR )
00281 *
00282 *        Perform QR iteration, accumulating Schur vectors in VL
00283 *        (Workspace: need N+1, prefer N+HSWORK (see comments) )
00284 *
00285          IWRK = ITAU
00286          CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
00287      \$                WORK( IWRK ), LWORK-IWRK+1, INFO )
00288 *
00289          IF( WANTVR ) THEN
00290 *
00291 *           Want left and right eigenvectors
00292 *           Copy Schur vectors to VR
00293 *
00294             SIDE = 'B'
00295             CALL SLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
00296          END IF
00297 *
00298       ELSE IF( WANTVR ) THEN
00299 *
00300 *        Want right eigenvectors
00301 *        Copy Householder vectors to VR
00302 *
00303          SIDE = 'R'
00304          CALL SLACPY( 'L', N, N, A, LDA, VR, LDVR )
00305 *
00306 *        Generate orthogonal matrix in VR
00307 *        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
00308 *
00309          CALL SORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
00310      \$                LWORK-IWRK+1, IERR )
00311 *
00312 *        Perform QR iteration, accumulating Schur vectors in VR
00313 *        (Workspace: need N+1, prefer N+HSWORK (see comments) )
00314 *
00315          IWRK = ITAU
00316          CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00317      \$                WORK( IWRK ), LWORK-IWRK+1, INFO )
00318 *
00319       ELSE
00320 *
00321 *        Compute eigenvalues only
00322 *        (Workspace: need N+1, prefer N+HSWORK (see comments) )
00323 *
00324          IWRK = ITAU
00325          CALL SHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
00326      \$                WORK( IWRK ), LWORK-IWRK+1, INFO )
00327       END IF
00328 *
00329 *     If INFO > 0 from SHSEQR, then quit
00330 *
00331       IF( INFO.GT.0 )
00332      \$   GO TO 50
00333 *
00334       IF( WANTVL .OR. WANTVR ) THEN
00335 *
00336 *        Compute left and/or right eigenvectors
00337 *        (Workspace: need 4*N)
00338 *
00339          CALL STREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
00340      \$                N, NOUT, WORK( IWRK ), IERR )
00341       END IF
00342 *
00343       IF( WANTVL ) THEN
00344 *
00345 *        Undo balancing of left eigenvectors
00346 *        (Workspace: need N)
00347 *
00348          CALL SGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL,
00349      \$                IERR )
00350 *
00351 *        Normalize left eigenvectors and make largest component real
00352 *
00353          DO 20 I = 1, N
00354             IF( WI( I ).EQ.ZERO ) THEN
00355                SCL = ONE / SNRM2( N, VL( 1, I ), 1 )
00356                CALL SSCAL( N, SCL, VL( 1, I ), 1 )
00357             ELSE IF( WI( I ).GT.ZERO ) THEN
00358                SCL = ONE / SLAPY2( SNRM2( N, VL( 1, I ), 1 ),
00359      \$               SNRM2( N, VL( 1, I+1 ), 1 ) )
00360                CALL SSCAL( N, SCL, VL( 1, I ), 1 )
00361                CALL SSCAL( N, SCL, VL( 1, I+1 ), 1 )
00362                DO 10 K = 1, N
00363                   WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
00364    10          CONTINUE
00365                K = ISAMAX( N, WORK( IWRK ), 1 )
00366                CALL SLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
00367                CALL SROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
00368                VL( K, I+1 ) = ZERO
00369             END IF
00370    20    CONTINUE
00371       END IF
00372 *
00373       IF( WANTVR ) THEN
00374 *
00375 *        Undo balancing of right eigenvectors
00376 *        (Workspace: need N)
00377 *
00378          CALL SGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR,
00379      \$                IERR )
00380 *
00381 *        Normalize right eigenvectors and make largest component real
00382 *
00383          DO 40 I = 1, N
00384             IF( WI( I ).EQ.ZERO ) THEN
00385                SCL = ONE / SNRM2( N, VR( 1, I ), 1 )
00386                CALL SSCAL( N, SCL, VR( 1, I ), 1 )
00387             ELSE IF( WI( I ).GT.ZERO ) THEN
00388                SCL = ONE / SLAPY2( SNRM2( N, VR( 1, I ), 1 ),
00389      \$               SNRM2( N, VR( 1, I+1 ), 1 ) )
00390                CALL SSCAL( N, SCL, VR( 1, I ), 1 )
00391                CALL SSCAL( N, SCL, VR( 1, I+1 ), 1 )
00392                DO 30 K = 1, N
00393                   WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
00394    30          CONTINUE
00395                K = ISAMAX( N, WORK( IWRK ), 1 )
00396                CALL SLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
00397                CALL SROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
00398                VR( K, I+1 ) = ZERO
00399             END IF
00400    40    CONTINUE
00401       END IF
00402 *
00403 *     Undo scaling if necessary
00404 *
00405    50 CONTINUE
00406       IF( SCALEA ) THEN
00407          CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
00408      \$                MAX( N-INFO, 1 ), IERR )
00409          CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
00410      \$                MAX( N-INFO, 1 ), IERR )
00411          IF( INFO.GT.0 ) THEN
00412             CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
00413      \$                   IERR )
00414             CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
00415      \$                   IERR )
00416          END IF
00417       END IF
00418 *
00419       WORK( 1 ) = MAXWRK
00420       RETURN
00421 *
00422 *     End of SGEEV
00423 *
00424       END
```