dgeevx(3S)

NAME
     DGEEVX - compute for an N-by-N real nonsymmetric matrix A, the
     eigenvalues and, optionally, the left and/or right eigenvectors

SYNOPSIS
     SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL,
                        LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
                        RCONDV, WORK, LWORK, IWORK, INFO )

         CHARACTER      BALANC, JOBVL, JOBVR, SENSE

         INTEGER        IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N

         DOUBLE         PRECISION ABNRM

         INTEGER        IWORK( * )

         DOUBLE         PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
                        SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
                        WORK( * ), WR( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues
     and, optionally, the left and/or right eigenvectors. Optionally also, it
     computes a balancing transformation to improve the conditioning of the
     eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal
     condition numbers for the eigenvalues (RCONDE), and reciprocal condition
     numbers for the right
     eigenvectors (RCONDV).

     The right eigenvector v(j) of A satisfies
                      A * v(j) = lambda(j) * v(j)
     where lambda(j) is its eigenvalue.
     The left eigenvector u(j) of A satisfies
                   u(j)**H * A = lambda(j) * u(j)**H
     where u(j)**H denotes the conjugate transpose of u(j).

     The computed eigenvectors are normalized to have Euclidean norm equal to
     1 and largest component real.
     Balancing a matrix means permuting the rows and columns to make it more
     nearly upper triangular, and applying a diagonal similarity
     transformation D * A * D**(-1), where D is a diagonal matrix, to make its
     rows and columns closer in norm and the condition numbers of its
     eigenvalues and eigenvectors smaller.  The computed reciprocal condition
     numbers correspond to the balanced matrix.  Permuting rows and columns
     will not change the condition numbers (in exact arithmetic) but diagonal
     scaling will.  For further explanation of balancing, see section 4.10.2
     of the LAPACK Users' Guide.


ARGUMENTS
     BALANC  (input) CHARACTER*1
             Indicates how the input matrix should be diagonally scaled and/or
             permuted to improve the conditioning of its eigenvalues.  = 'N':
             Do not diagonally scale or permute;
             = 'P': Perform permutations to make the matrix more nearly upper
             triangular. Do not diagonally scale; = 'S': Diagonally scale the
             matrix, i.e. replace A by D*A*D**(-1), where D is a diagonal
             matrix chosen to make the rows and columns of A more equal in
             norm. Do not permute; = 'B': Both diagonally scale and permute A.

             Computed reciprocal condition numbers will be for the matrix
             after balancing and/or permuting. Permuting does not change
             condition numbers (in exact arithmetic), but balancing does.

     JOBVL   (input) CHARACTER*1
             = 'N': left eigenvectors of A are not computed;
             = 'V': left eigenvectors of A are computed.  If SENSE = 'E' or
             'B', JOBVL must = 'V'.

     JOBVR   (input) CHARACTER*1
             = 'N': right eigenvectors of A are not computed;
             = 'V': right eigenvectors of A are computed.  If SENSE = 'E' or
             'B', JOBVR must = 'V'.

     SENSE   (input) CHARACTER*1
             Determines which reciprocal condition numbers are computed.  =
             'N': None are computed;
             = 'E': Computed for eigenvalues only;
             = 'V': Computed for right eigenvectors only;
             = 'B': Computed for eigenvalues and right eigenvectors.

             If SENSE = 'E' or 'B', both left and right eigenvectors must also
             be computed (JOBVL = 'V' and JOBVR = 'V').

     N       (input) INTEGER
             The order of the matrix A. N >= 0.

     A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
             On entry, the N-by-N matrix A.  On exit, A has been overwritten.
             If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur form of
             the balanced version of the input matrix A.

     LDA     (input) INTEGER
             The leading dimension of the array A.  LDA >= max(1,N).

     WR      (output) DOUBLE PRECISION array, dimension (N)
             WI      (output) DOUBLE PRECISION array, dimension (N) WR and WI
             contain the real and imaginary parts, respectively, of the
             computed eigenvalues.  Complex conjugate pairs of eigenvalues
             will appear consecutively with the eigenvalue having the positive
             imaginary part first.

     VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
             If JOBVL = 'V', the left eigenvectors u(j) are stored one after
             another in the columns of VL, in the same order as their
             eigenvalues.  If JOBVL = 'N', VL is not referenced.  If the j-th
             eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL.
             If the j-th and (j+1)-st eigenvalues form a complex conjugate
             pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
             u(j+1) = VL(:,j) - i*VL(:,j+1).

     LDVL    (input) INTEGER
             The leading dimension of the array VL.  LDVL >= 1; if JOBVL =
             'V', LDVL >= N.

     VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
             If JOBVR = 'V', the right eigenvectors v(j) are stored one after
             another in the columns of VR, in the same order as their
             eigenvalues.  If JOBVR = 'N', VR is not referenced.  If the j-th
             eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR.
             If the j-th and (j+1)-st eigenvalues form a complex conjugate
             pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
             v(j+1) = VR(:,j) - i*VR(:,j+1).

     LDVR    (input) INTEGER
             The leading dimension of the array VR.  LDVR >= 1, and if JOBVR =
             'V', LDVR >= N.

             ILO,IHI (output) INTEGER ILO and IHI are integer values
             determined when A was balanced.  The balanced A(i,j) = 0 if I > J
             and J = 1,...,ILO-1 or I = IHI+1,...,N.

     SCALE   (output) DOUBLE PRECISION array, dimension (N)
             Details of the permutations and scaling factors applied when
             balancing A.  If P(j) is the index of the row and column
             interchanged with row and column j, and D(j) is the scaling
             factor applied to row and column j, then SCALE(J) = P(J),    for
             J = 1,...,ILO-1 = D(J),    for J = ILO,...,IHI = P(J)     for J =
             IHI+1,...,N.  The order in which the interchanges are made is N
             to IHI+1, then 1 to ILO-1.

     ABNRM   (output) DOUBLE PRECISION
             The one-norm of the balanced matrix (the maximum of the sum of
             absolute values of elements of any column).

     RCONDE  (output) DOUBLE PRECISION array, dimension (N)
             RCONDE(j) is the reciprocal condition number of the j-th
             eigenvalue.

     RCONDV  (output) DOUBLE PRECISION array, dimension (N)
             RCONDV(j) is the reciprocal condition number of the j-th right
             eigenvector.

     WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.   If SENSE = 'N' or 'E', LWORK
             >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.
             If SENSE = 'V' or 'B', LWORK >= N*(N+6).  For good performance,
             LWORK must generally be larger.

             If LWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the WORK array, returns this
             value as the first entry of the WORK array, and no error message
             related to LWORK is issued by XERBLA.

     IWORK   (workspace) INTEGER array, dimension (2*N-2)
             If SENSE = 'N' or 'E', not referenced.

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  if INFO = i, the QR algorithm failed to compute all the
             eigenvalues, and no eigenvectors or condition numbers have been
             computed; elements 1:ILO-1 and i+1:N of WR and WI contain
             eigenvalues which have converged.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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