sgeevx.f(3)

NAME
sgeevx.f -

SYNOPSIS

  Functions/Subroutines
subroutine sgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Function/Subroutine Documentation

  subroutine sgeevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( * )WR, real, dimension( * )WI, real, dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, real, dimension( * )SCALE, realABNRM, real, dimension( * )RCONDE, real, dimension( * )RCONDV, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     SGEEVX 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.

Parameters:
    BALANC

              BALANC is 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

              JOBVL is 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

              JOBVR is 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

              SENSE is 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

              N is INTEGER
              The order of the matrix A. N >= 0.

    A

              A is REAL 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

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

    WR

              WR is REAL array, dimension (N)

    WI

              WI is REAL 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

              VL is REAL 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

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

    VR

              VR is REAL 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

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

    ILO

              ILO is INTEGER

    IHI

              IHI is 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

              SCALE is REAL 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

              ABNRM is REAL
              The one-norm of the balanced matrix (the maximum
              of the sum of absolute values of elements of any column).

    RCONDE

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

    RCONDV

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

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is 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

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

    INFO

              INFO is 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    September 2012

Definition at line 302 of file sgeevx.f.

Author

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