LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zlaein.f
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00001 *> \brief \b ZLAEIN
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
00022 *                          EPS3, SMLNUM, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       LOGICAL            NOINIT, RIGHTV
00026 *       INTEGER            INFO, LDB, LDH, N
00027 *       DOUBLE PRECISION   EPS3, SMLNUM
00028 *       COMPLEX*16         W
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       DOUBLE PRECISION   RWORK( * )
00032 *       COMPLEX*16         B( LDB, * ), H( LDH, * ), V( * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> ZLAEIN uses inverse iteration to find a right or left eigenvector
00042 *> corresponding to the eigenvalue W of a complex upper Hessenberg
00043 *> matrix H.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] RIGHTV
00050 *> \verbatim
00051 *>          RIGHTV is LOGICAL
00052 *>          = .TRUE. : compute right eigenvector;
00053 *>          = .FALSE.: compute left eigenvector.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] NOINIT
00057 *> \verbatim
00058 *>          NOINIT is LOGICAL
00059 *>          = .TRUE. : no initial vector supplied in V
00060 *>          = .FALSE.: initial vector supplied in V.
00061 *> \endverbatim
00062 *>
00063 *> \param[in] N
00064 *> \verbatim
00065 *>          N is INTEGER
00066 *>          The order of the matrix H.  N >= 0.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] H
00070 *> \verbatim
00071 *>          H is COMPLEX*16 array, dimension (LDH,N)
00072 *>          The upper Hessenberg matrix H.
00073 *> \endverbatim
00074 *>
00075 *> \param[in] LDH
00076 *> \verbatim
00077 *>          LDH is INTEGER
00078 *>          The leading dimension of the array H.  LDH >= max(1,N).
00079 *> \endverbatim
00080 *>
00081 *> \param[in] W
00082 *> \verbatim
00083 *>          W is COMPLEX*16
00084 *>          The eigenvalue of H whose corresponding right or left
00085 *>          eigenvector is to be computed.
00086 *> \endverbatim
00087 *>
00088 *> \param[in,out] V
00089 *> \verbatim
00090 *>          V is COMPLEX*16 array, dimension (N)
00091 *>          On entry, if NOINIT = .FALSE., V must contain a starting
00092 *>          vector for inverse iteration; otherwise V need not be set.
00093 *>          On exit, V contains the computed eigenvector, normalized so
00094 *>          that the component of largest magnitude has magnitude 1; here
00095 *>          the magnitude of a complex number (x,y) is taken to be
00096 *>          |x| + |y|.
00097 *> \endverbatim
00098 *>
00099 *> \param[out] B
00100 *> \verbatim
00101 *>          B is COMPLEX*16 array, dimension (LDB,N)
00102 *> \endverbatim
00103 *>
00104 *> \param[in] LDB
00105 *> \verbatim
00106 *>          LDB is INTEGER
00107 *>          The leading dimension of the array B.  LDB >= max(1,N).
00108 *> \endverbatim
00109 *>
00110 *> \param[out] RWORK
00111 *> \verbatim
00112 *>          RWORK is DOUBLE PRECISION array, dimension (N)
00113 *> \endverbatim
00114 *>
00115 *> \param[in] EPS3
00116 *> \verbatim
00117 *>          EPS3 is DOUBLE PRECISION
00118 *>          A small machine-dependent value which is used to perturb
00119 *>          close eigenvalues, and to replace zero pivots.
00120 *> \endverbatim
00121 *>
00122 *> \param[in] SMLNUM
00123 *> \verbatim
00124 *>          SMLNUM is DOUBLE PRECISION
00125 *>          A machine-dependent value close to the underflow threshold.
00126 *> \endverbatim
00127 *>
00128 *> \param[out] INFO
00129 *> \verbatim
00130 *>          INFO is INTEGER
00131 *>          = 0:  successful exit
00132 *>          = 1:  inverse iteration did not converge; V is set to the
00133 *>                last iterate.
00134 *> \endverbatim
00135 *
00136 *  Authors:
00137 *  ========
00138 *
00139 *> \author Univ. of Tennessee 
00140 *> \author Univ. of California Berkeley 
00141 *> \author Univ. of Colorado Denver 
00142 *> \author NAG Ltd. 
00143 *
00144 *> \date November 2011
00145 *
00146 *> \ingroup complex16OTHERauxiliary
00147 *
00148 *  =====================================================================
00149       SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
00150      $                   EPS3, SMLNUM, INFO )
00151 *
00152 *  -- LAPACK auxiliary routine (version 3.4.0) --
00153 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00154 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00155 *     November 2011
00156 *
00157 *     .. Scalar Arguments ..
00158       LOGICAL            NOINIT, RIGHTV
00159       INTEGER            INFO, LDB, LDH, N
00160       DOUBLE PRECISION   EPS3, SMLNUM
00161       COMPLEX*16         W
00162 *     ..
00163 *     .. Array Arguments ..
00164       DOUBLE PRECISION   RWORK( * )
00165       COMPLEX*16         B( LDB, * ), H( LDH, * ), V( * )
00166 *     ..
00167 *
00168 *  =====================================================================
00169 *
00170 *     .. Parameters ..
00171       DOUBLE PRECISION   ONE, TENTH
00172       PARAMETER          ( ONE = 1.0D+0, TENTH = 1.0D-1 )
00173       COMPLEX*16         ZERO
00174       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ) )
00175 *     ..
00176 *     .. Local Scalars ..
00177       CHARACTER          NORMIN, TRANS
00178       INTEGER            I, IERR, ITS, J
00179       DOUBLE PRECISION   GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
00180       COMPLEX*16         CDUM, EI, EJ, TEMP, X
00181 *     ..
00182 *     .. External Functions ..
00183       INTEGER            IZAMAX
00184       DOUBLE PRECISION   DZASUM, DZNRM2
00185       COMPLEX*16         ZLADIV
00186       EXTERNAL           IZAMAX, DZASUM, DZNRM2, ZLADIV
00187 *     ..
00188 *     .. External Subroutines ..
00189       EXTERNAL           ZDSCAL, ZLATRS
00190 *     ..
00191 *     .. Intrinsic Functions ..
00192       INTRINSIC          ABS, DBLE, DIMAG, MAX, SQRT
00193 *     ..
00194 *     .. Statement Functions ..
00195       DOUBLE PRECISION   CABS1
00196 *     ..
00197 *     .. Statement Function definitions ..
00198       CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
00199 *     ..
00200 *     .. Executable Statements ..
00201 *
00202       INFO = 0
00203 *
00204 *     GROWTO is the threshold used in the acceptance test for an
00205 *     eigenvector.
00206 *
00207       ROOTN = SQRT( DBLE( N ) )
00208       GROWTO = TENTH / ROOTN
00209       NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
00210 *
00211 *     Form B = H - W*I (except that the subdiagonal elements are not
00212 *     stored).
00213 *
00214       DO 20 J = 1, N
00215          DO 10 I = 1, J - 1
00216             B( I, J ) = H( I, J )
00217    10    CONTINUE
00218          B( J, J ) = H( J, J ) - W
00219    20 CONTINUE
00220 *
00221       IF( NOINIT ) THEN
00222 *
00223 *        Initialize V.
00224 *
00225          DO 30 I = 1, N
00226             V( I ) = EPS3
00227    30    CONTINUE
00228       ELSE
00229 *
00230 *        Scale supplied initial vector.
00231 *
00232          VNORM = DZNRM2( N, V, 1 )
00233          CALL ZDSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
00234       END IF
00235 *
00236       IF( RIGHTV ) THEN
00237 *
00238 *        LU decomposition with partial pivoting of B, replacing zero
00239 *        pivots by EPS3.
00240 *
00241          DO 60 I = 1, N - 1
00242             EI = H( I+1, I )
00243             IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
00244 *
00245 *              Interchange rows and eliminate.
00246 *
00247                X = ZLADIV( B( I, I ), EI )
00248                B( I, I ) = EI
00249                DO 40 J = I + 1, N
00250                   TEMP = B( I+1, J )
00251                   B( I+1, J ) = B( I, J ) - X*TEMP
00252                   B( I, J ) = TEMP
00253    40          CONTINUE
00254             ELSE
00255 *
00256 *              Eliminate without interchange.
00257 *
00258                IF( B( I, I ).EQ.ZERO )
00259      $            B( I, I ) = EPS3
00260                X = ZLADIV( EI, B( I, I ) )
00261                IF( X.NE.ZERO ) THEN
00262                   DO 50 J = I + 1, N
00263                      B( I+1, J ) = B( I+1, J ) - X*B( I, J )
00264    50             CONTINUE
00265                END IF
00266             END IF
00267    60    CONTINUE
00268          IF( B( N, N ).EQ.ZERO )
00269      $      B( N, N ) = EPS3
00270 *
00271          TRANS = 'N'
00272 *
00273       ELSE
00274 *
00275 *        UL decomposition with partial pivoting of B, replacing zero
00276 *        pivots by EPS3.
00277 *
00278          DO 90 J = N, 2, -1
00279             EJ = H( J, J-1 )
00280             IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
00281 *
00282 *              Interchange columns and eliminate.
00283 *
00284                X = ZLADIV( B( J, J ), EJ )
00285                B( J, J ) = EJ
00286                DO 70 I = 1, J - 1
00287                   TEMP = B( I, J-1 )
00288                   B( I, J-1 ) = B( I, J ) - X*TEMP
00289                   B( I, J ) = TEMP
00290    70          CONTINUE
00291             ELSE
00292 *
00293 *              Eliminate without interchange.
00294 *
00295                IF( B( J, J ).EQ.ZERO )
00296      $            B( J, J ) = EPS3
00297                X = ZLADIV( EJ, B( J, J ) )
00298                IF( X.NE.ZERO ) THEN
00299                   DO 80 I = 1, J - 1
00300                      B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
00301    80             CONTINUE
00302                END IF
00303             END IF
00304    90    CONTINUE
00305          IF( B( 1, 1 ).EQ.ZERO )
00306      $      B( 1, 1 ) = EPS3
00307 *
00308          TRANS = 'C'
00309 *
00310       END IF
00311 *
00312       NORMIN = 'N'
00313       DO 110 ITS = 1, N
00314 *
00315 *        Solve U*x = scale*v for a right eigenvector
00316 *          or U**H *x = scale*v for a left eigenvector,
00317 *        overwriting x on v.
00318 *
00319          CALL ZLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
00320      $                SCALE, RWORK, IERR )
00321          NORMIN = 'Y'
00322 *
00323 *        Test for sufficient growth in the norm of v.
00324 *
00325          VNORM = DZASUM( N, V, 1 )
00326          IF( VNORM.GE.GROWTO*SCALE )
00327      $      GO TO 120
00328 *
00329 *        Choose new orthogonal starting vector and try again.
00330 *
00331          RTEMP = EPS3 / ( ROOTN+ONE )
00332          V( 1 ) = EPS3
00333          DO 100 I = 2, N
00334             V( I ) = RTEMP
00335   100    CONTINUE
00336          V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
00337   110 CONTINUE
00338 *
00339 *     Failure to find eigenvector in N iterations.
00340 *
00341       INFO = 1
00342 *
00343   120 CONTINUE
00344 *
00345 *     Normalize eigenvector.
00346 *
00347       I = IZAMAX( N, V, 1 )
00348       CALL ZDSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
00349 *
00350       RETURN
00351 *
00352 *     End of ZLAEIN
00353 *
00354       END
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