ggev3(3)

NAME
ggev3 - ggev3: eig

SYNOPSIS

  Functions
subroutine cggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
subroutine dggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
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)
subroutine zggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Detailed Description

Function Documentation

  subroutine cggev3 (character jobvl, character jobvr, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

     CGGEV3 computes for a pair of N-by-N complex 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 generalized eigenvector v(j) corresponding to the
     generalized eigenvalue lambda(j) of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
     The left generalized eigenvector u(j) corresponding to the
     generalized eigenvalues 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 COMPLEX 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 COMPLEX 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).

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.
              Note: the quotients ALPHA(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, ALPHA 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 COMPLEX array, dimension (LDVL,N)
              If JOBVL = 'V', the left generalized eigenvectors u(j) are
              stored one after another in the columns of VL, in the same
              order as their eigenvalues.
              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 COMPLEX array, dimension (LDVR,N)
              If JOBVR = 'V', the right generalized eigenvectors v(j) are
              stored one after another in the columns of VR, in the same
              order as their eigenvalues.
              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 COMPLEX 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 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.

    RWORK

              RWORK is REAL array, dimension (8*N)

    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 ALPHA(j) and BETA(j) should be
                    correct for j=INFO+1,...,N.
              > N:  =N+1: other then QZ iteration failed in CHGEQZ,
                    =N+2: error return from CTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 214 of file cggev3.f.

  subroutine dggev3 (character jobvl, character jobvr, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) work, integer lwork, integer info)

DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

     DGGEV3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

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

    LWORK

              LWORK is INTEGER
              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 DLAQZ0.
                    =N+2: error return from DTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 223 of file dggev3.f.

  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
              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 223 of file sggev3.f.

  subroutine zggev3 (character jobvl, character jobvr, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

     ZGGEV3 computes for a pair of N-by-N complex 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 generalized eigenvector v(j) corresponding to the
     generalized eigenvalue lambda(j) of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
     The left generalized eigenvector u(j) corresponding to the
     generalized eigenvalues 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 COMPLEX*16 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 COMPLEX*16 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).

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)

    BETA

              BETA is COMPLEX*16 array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.
              Note: the quotients ALPHA(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, ALPHA 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 COMPLEX*16 array, dimension (LDVL,N)
              If JOBVL = 'V', the left generalized eigenvectors u(j) are
              stored one after another in the columns of VL, in the same
              order as their eigenvalues.
              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 COMPLEX*16 array, dimension (LDVR,N)
              If JOBVR = 'V', the right generalized eigenvectors v(j) are
              stored one after another in the columns of VR, in the same
              order as their eigenvalues.
              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 COMPLEX*16 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 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.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (8*N)

    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 ALPHA(j) and BETA(j) should be
                    correct for j=INFO+1,...,N.
              > N:  =N+1: other then QZ iteration failed in ZHGEQZ,
                    =N+2: error return from ZTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 214 of file zggev3.f.

Author

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