Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  computes  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( * )

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)**T * A = lambda(j) * u(j)**T
        where u(j)**T denotes the 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     (output) INTEGER
                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 (MAX(1,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.

 LAPACK driver routine (version 3.3.1)      April 2011                            DGEEVX(3lapack)