SRC/sggev3.f(3)

NAME
SRC/sggev3.f

SYNOPSIS

  Functions/Subroutines
subroutine sggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Function/Subroutine Documentation

  subroutine sggev3 (character jobvl, character jobvr, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work, integer lwork, integer info)

SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

    !>
    !> SGGEV3 computes for a pair of N-by-N real nonsymmetric matrices (A,B)
    !> the generalized eigenvalues, and optionally, the left and/or right
    !> generalized eigenvectors.
    !>
    !> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
    !> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
    !> singular. It is usually represented as the pair (alpha,beta), as
    !> there is a reasonable interpretation for beta=0, and even for both
    !> being zero.
    !>
    !> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>
    !>                  A * v(j) = lambda(j) * B * v(j).
    !>
    !> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>
    !>                  u(j)**H * A  = lambda(j) * u(j)**H * B .
    !>
    !> where u(j)**H is the conjugate-transpose of u(j).
    !>
    !>

Parameters
    JOBVL

    !>          JOBVL is CHARACTER*1
    !>          = 'N':  do not compute the left generalized eigenvectors;
    !>          = 'V':  compute the left generalized eigenvectors.
    !>

    JOBVR

    !>          JOBVR is CHARACTER*1
    !>          = 'N':  do not compute the right generalized eigenvectors;
    !>          = 'V':  compute the right generalized eigenvectors.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A, B, VL, and VR.  N >= 0.
    !>

    A

    !>          A is REAL array, dimension (LDA, N)
    !>          On entry, the matrix A in the pair (A,B).
    !>          On exit, A has been overwritten.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          On entry, the matrix B in the pair (A,B).
    !>          On exit, B has been overwritten.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of B.  LDB >= max(1,N).
    !>

    ALPHAR

    !>          ALPHAR is REAL array, dimension (N)
    !>

    ALPHAI

    !>          ALPHAI is REAL array, dimension (N)
    !>

    BETA

    !>          BETA is REAL array, dimension (N)
    !>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
    !>          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
    !>          the j-th eigenvalue is real; if positive, then the j-th and
    !>          (j+1)-st eigenvalues are a complex conjugate pair, with
    !>          ALPHAI(j+1) negative.
    !>
    !>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
    !>          may easily over- or underflow, and BETA(j) may even be zero.
    !>          Thus, the user should avoid naively computing the ratio
    !>          alpha/beta.  However, ALPHAR and ALPHAI will be always less
    !>          than and usually comparable with norm(A) in magnitude, and
    !>          BETA always less than and usually comparable with norm(B).
    !>

    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 the j-th eigenvalue is real, then
    !>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
    !>          (j+1)-th 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).
    !>          Each eigenvector is scaled so the largest component has
    !>          abs(real part)+abs(imag. part)=1.
    !>          Not referenced if JOBVL = 'N'.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the matrix VL. LDVL >= 1, and
    !>          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 the j-th eigenvalue is real, then
    !>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
    !>          (j+1)-th 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).
    !>          Each eigenvector is scaled so the largest component has
    !>          abs(real part)+abs(imag. part)=1.
    !>          Not referenced if JOBVR = 'N'.
    !>

    LDVR

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

    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. LWORK >= MAX(1,8*N).
    !>          For good performance, LWORK should 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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          = 1,...,N:
    !>                The QZ iteration failed.  No eigenvectors have been
    !>                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
    !>                should be correct for j=INFO+1,...,N.
    !>          > N:  =N+1: other than QZ iteration failed in SLAQZ0.
    !>                =N+2: error return from STGEVC.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 225 of file sggev3.f.

Author

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