trsna(3)

NAME
trsna - trsna: eig condition numbers

SYNOPSIS

  Functions

subroutine ctrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)
CTRSNA
subroutine dtrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
DTRSNA
subroutine strsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
STRSNA
subroutine ztrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)
ZTRSNA

Detailed Description

Function Documentation

  subroutine ctrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, complex, dimension( ldwork, * ) work, integer ldwork, real, dimension( * ) rwork, integer info)

CTRSNA

Purpose:

    !>
    !> CTRSNA estimates reciprocal condition numbers for specified
    !> eigenvalues and/or right eigenvectors of a complex upper triangular
    !> matrix T (or of any matrix Q*T*Q**H with Q unitary).
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies whether condition numbers are required for
    !>          eigenvalues (S) or eigenvectors (SEP):
    !>          = 'E': for eigenvalues only (S);
    !>          = 'V': for eigenvectors only (SEP);
    !>          = 'B': for both eigenvalues and eigenvectors (S and SEP).
    !>

    HOWMNY

    !>          HOWMNY is CHARACTER*1
    !>          = 'A': compute condition numbers for all eigenpairs;
    !>          = 'S': compute condition numbers for selected eigenpairs
    !>                 specified by the array SELECT.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    !>          condition numbers are required. To select condition numbers
    !>          for the j-th eigenpair, SELECT(j) must be set to .TRUE..
    !>          If HOWMNY = 'A', SELECT is not referenced.
    !>

    N

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

    T

    !>          T is COMPLEX array, dimension (LDT,N)
    !>          The upper triangular matrix T.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T. LDT >= max(1,N).
    !>

    VL

    !>          VL is COMPLEX array, dimension (LDVL,M)
    !>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
    !>          (or of any Q*T*Q**H with Q unitary), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VL, as returned by
    !>          CHSEIN or CTREVC.
    !>          If JOB = 'V', VL is not referenced.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.
    !>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
    !>

    VR

    !>          VR is COMPLEX array, dimension (LDVR,M)
    !>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
    !>          (or of any Q*T*Q**H with Q unitary), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VR, as returned by
    !>          CHSEIN or CTREVC.
    !>          If JOB = 'V', VR is not referenced.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.
    !>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
    !>

    S

    !>          S is REAL array, dimension (MM)
    !>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    !>          selected eigenvalues, stored in consecutive elements of the
    !>          array. Thus S(j), SEP(j), and the j-th columns of VL and VR
    !>          all correspond to the same eigenpair (but not in general the
    !>          j-th eigenpair, unless all eigenpairs are selected).
    !>          If JOB = 'V', S is not referenced.
    !>

    SEP

    !>          SEP is REAL array, dimension (MM)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the selected eigenvectors, stored in consecutive
    !>          elements of the array.
    !>          If JOB = 'E', SEP is not referenced.
    !>

    MM

    !>          MM is INTEGER
    !>          The number of elements in the arrays S (if JOB = 'E' or 'B')
    !>           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
    !>

    M

    !>          M is INTEGER
    !>          The number of elements of the arrays S and/or SEP actually
    !>          used to store the estimated condition numbers.
    !>          If HOWMNY = 'A', M is set to N.
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (LDWORK,N+6)
    !>          If JOB = 'E', WORK is not referenced.
    !>

    LDWORK

    !>          LDWORK is INTEGER
    !>          The leading dimension of the array WORK.
    !>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (N)
    !>          If JOB = 'E', RWORK is not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The reciprocal of the condition number of an eigenvalue lambda is
    !>  defined as
    !>
    !>          S(lambda) = |v**H*u| / (norm(u)*norm(v))
    !>
    !>  where u and v are the right and left eigenvectors of T corresponding
    !>  to lambda; v**H denotes the conjugate transpose of v, and norm(u)
    !>  denotes the Euclidean norm. These reciprocal condition numbers always
    !>  lie between zero (very badly conditioned) and one (very well
    !>  conditioned). If n = 1, S(lambda) is defined to be 1.
    !>
    !>  An approximate error bound for a computed eigenvalue W(i) is given by
    !>
    !>                      EPS * norm(T) / S(i)
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal of the condition number of the right eigenvector u
    !>  corresponding to lambda is defined as follows. Suppose
    !>
    !>              T = ( lambda  c  )
    !>                  (   0    T22 )
    !>
    !>  Then the reciprocal condition number is
    !>
    !>          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
    !>
    !>  where sigma-min denotes the smallest singular value. We approximate
    !>  the smallest singular value by the reciprocal of an estimate of the
    !>  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
    !>  defined to be abs(T(1,1)).
    !>
    !>  An approximate error bound for a computed right eigenvector VR(i)
    !>  is given by
    !>
    !>                      EPS * norm(T) / SEP(i)
    !>

  subroutine dtrsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, double precision, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)

DTRSNA

Purpose:

    !>
    !> DTRSNA estimates reciprocal condition numbers for specified
    !> eigenvalues and/or right eigenvectors of a real upper
    !> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
    !> orthogonal).
    !>
    !> T must be in Schur canonical form (as returned by DHSEQR), that is,
    !> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
    !> 2-by-2 diagonal block has its diagonal elements equal and its
    !> off-diagonal elements of opposite sign.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies whether condition numbers are required for
    !>          eigenvalues (S) or eigenvectors (SEP):
    !>          = 'E': for eigenvalues only (S);
    !>          = 'V': for eigenvectors only (SEP);
    !>          = 'B': for both eigenvalues and eigenvectors (S and SEP).
    !>

    HOWMNY

    !>          HOWMNY is CHARACTER*1
    !>          = 'A': compute condition numbers for all eigenpairs;
    !>          = 'S': compute condition numbers for selected eigenpairs
    !>                 specified by the array SELECT.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    !>          condition numbers are required. To select condition numbers
    !>          for the eigenpair corresponding to a real eigenvalue w(j),
    !>          SELECT(j) must be set to .TRUE.. To select condition numbers
    !>          corresponding to a complex conjugate pair of eigenvalues w(j)
    !>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
    !>          set to .TRUE..
    !>          If HOWMNY = 'A', SELECT is not referenced.
    !>

    N

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

    T

    !>          T is DOUBLE PRECISION array, dimension (LDT,N)
    !>          The upper quasi-triangular matrix T, in Schur canonical form.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T. LDT >= max(1,N).
    !>

    VL

    !>          VL is DOUBLE PRECISION array, dimension (LDVL,M)
    !>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
    !>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VL, as returned by
    !>          DHSEIN or DTREVC.
    !>          If JOB = 'V', VL is not referenced.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.
    !>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
    !>

    VR

    !>          VR is DOUBLE PRECISION array, dimension (LDVR,M)
    !>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
    !>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VR, as returned by
    !>          DHSEIN or DTREVC.
    !>          If JOB = 'V', VR is not referenced.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.
    !>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    !>          selected eigenvalues, stored in consecutive elements of the
    !>          array. For a complex conjugate pair of eigenvalues two
    !>          consecutive elements of S are set to the same value. Thus
    !>          S(j), SEP(j), and the j-th columns of VL and VR all
    !>          correspond to the same eigenpair (but not in general the
    !>          j-th eigenpair, unless all eigenpairs are selected).
    !>          If JOB = 'V', S is not referenced.
    !>

    SEP

    !>          SEP is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the selected eigenvectors, stored in consecutive
    !>          elements of the array. For a complex eigenvector two
    !>          consecutive elements of SEP are set to the same value. If
    !>          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
    !>          is set to 0; this can only occur when the true value would be
    !>          very small anyway.
    !>          If JOB = 'E', SEP is not referenced.
    !>

    MM

    !>          MM is INTEGER
    !>          The number of elements in the arrays S (if JOB = 'E' or 'B')
    !>           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
    !>

    M

    !>          M is INTEGER
    !>          The number of elements of the arrays S and/or SEP actually
    !>          used to store the estimated condition numbers.
    !>          If HOWMNY = 'A', M is set to N.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
    !>          If JOB = 'E', WORK is not referenced.
    !>

    LDWORK

    !>          LDWORK is INTEGER
    !>          The leading dimension of the array WORK.
    !>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (2*(N-1))
    !>          If JOB = 'E', IWORK is not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The reciprocal of the condition number of an eigenvalue lambda is
    !>  defined as
    !>
    !>          S(lambda) = |v**T*u| / (norm(u)*norm(v))
    !>
    !>  where u and v are the right and left eigenvectors of T corresponding
    !>  to lambda; v**T denotes the transpose of v, and norm(u)
    !>  denotes the Euclidean norm. These reciprocal condition numbers always
    !>  lie between zero (very badly conditioned) and one (very well
    !>  conditioned). If n = 1, S(lambda) is defined to be 1.
    !>
    !>  An approximate error bound for a computed eigenvalue W(i) is given by
    !>
    !>                      EPS * norm(T) / S(i)
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal of the condition number of the right eigenvector u
    !>  corresponding to lambda is defined as follows. Suppose
    !>
    !>              T = ( lambda  c  )
    !>                  (   0    T22 )
    !>
    !>  Then the reciprocal condition number is
    !>
    !>          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
    !>
    !>  where sigma-min denotes the smallest singular value. We approximate
    !>  the smallest singular value by the reciprocal of an estimate of the
    !>  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
    !>  defined to be abs(T(1,1)).
    !>
    !>  An approximate error bound for a computed right eigenvector VR(i)
    !>  is given by
    !>
    !>                      EPS * norm(T) / SEP(i)
    !>

  subroutine strsna (character job, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, real, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)

STRSNA

Purpose:

    !>
    !> STRSNA estimates reciprocal condition numbers for specified
    !> eigenvalues and/or right eigenvectors of a real upper
    !> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
    !> orthogonal).
    !>
    !> T must be in Schur canonical form (as returned by SHSEQR), that is,
    !> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
    !> 2-by-2 diagonal block has its diagonal elements equal and its
    !> off-diagonal elements of opposite sign.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies whether condition numbers are required for
    !>          eigenvalues (S) or eigenvectors (SEP):
    !>          = 'E': for eigenvalues only (S);
    !>          = 'V': for eigenvectors only (SEP);
    !>          = 'B': for both eigenvalues and eigenvectors (S and SEP).
    !>

    HOWMNY

    !>          HOWMNY is CHARACTER*1
    !>          = 'A': compute condition numbers for all eigenpairs;
    !>          = 'S': compute condition numbers for selected eigenpairs
    !>                 specified by the array SELECT.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    !>          condition numbers are required. To select condition numbers
    !>          for the eigenpair corresponding to a real eigenvalue w(j),
    !>          SELECT(j) must be set to .TRUE.. To select condition numbers
    !>          corresponding to a complex conjugate pair of eigenvalues w(j)
    !>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
    !>          set to .TRUE..
    !>          If HOWMNY = 'A', SELECT is not referenced.
    !>

    N

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

    T

    !>          T is REAL array, dimension (LDT,N)
    !>          The upper quasi-triangular matrix T, in Schur canonical form.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T. LDT >= max(1,N).
    !>

    VL

    !>          VL is REAL array, dimension (LDVL,M)
    !>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
    !>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VL, as returned by
    !>          SHSEIN or STREVC.
    !>          If JOB = 'V', VL is not referenced.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.
    !>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
    !>

    VR

    !>          VR is REAL array, dimension (LDVR,M)
    !>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
    !>          (or of any Q*T*Q**T with Q orthogonal), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VR, as returned by
    !>          SHSEIN or STREVC.
    !>          If JOB = 'V', VR is not referenced.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.
    !>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
    !>

    S

    !>          S is REAL array, dimension (MM)
    !>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    !>          selected eigenvalues, stored in consecutive elements of the
    !>          array. For a complex conjugate pair of eigenvalues two
    !>          consecutive elements of S are set to the same value. Thus
    !>          S(j), SEP(j), and the j-th columns of VL and VR all
    !>          correspond to the same eigenpair (but not in general the
    !>          j-th eigenpair, unless all eigenpairs are selected).
    !>          If JOB = 'V', S is not referenced.
    !>

    SEP

    !>          SEP is REAL array, dimension (MM)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the selected eigenvectors, stored in consecutive
    !>          elements of the array. For a complex eigenvector two
    !>          consecutive elements of SEP are set to the same value. If
    !>          the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
    !>          is set to 0; this can only occur when the true value would be
    !>          very small anyway.
    !>          If JOB = 'E', SEP is not referenced.
    !>

    MM

    !>          MM is INTEGER
    !>          The number of elements in the arrays S (if JOB = 'E' or 'B')
    !>           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
    !>

    M

    !>          M is INTEGER
    !>          The number of elements of the arrays S and/or SEP actually
    !>          used to store the estimated condition numbers.
    !>          If HOWMNY = 'A', M is set to N.
    !>

    WORK

    !>          WORK is REAL array, dimension (LDWORK,N+6)
    !>          If JOB = 'E', WORK is not referenced.
    !>

    LDWORK

    !>          LDWORK is INTEGER
    !>          The leading dimension of the array WORK.
    !>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (2*(N-1))
    !>          If JOB = 'E', IWORK is not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The reciprocal of the condition number of an eigenvalue lambda is
    !>  defined as
    !>
    !>          S(lambda) = |v**T*u| / (norm(u)*norm(v))
    !>
    !>  where u and v are the right and left eigenvectors of T corresponding
    !>  to lambda; v**T denotes the transpose of v, and norm(u)
    !>  denotes the Euclidean norm. These reciprocal condition numbers always
    !>  lie between zero (very badly conditioned) and one (very well
    !>  conditioned). If n = 1, S(lambda) is defined to be 1.
    !>
    !>  An approximate error bound for a computed eigenvalue W(i) is given by
    !>
    !>                      EPS * norm(T) / S(i)
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal of the condition number of the right eigenvector u
    !>  corresponding to lambda is defined as follows. Suppose
    !>
    !>              T = ( lambda  c  )
    !>                  (   0    T22 )
    !>
    !>  Then the reciprocal condition number is
    !>
    !>          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
    !>
    !>  where sigma-min denotes the smallest singular value. We approximate
    !>  the smallest singular value by the reciprocal of an estimate of the
    !>  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
    !>  defined to be abs(T(1,1)).
    !>
    !>  An approximate error bound for a computed right eigenvector VR(i)
    !>  is given by
    !>
    !>                      EPS * norm(T) / SEP(i)
    !>

  subroutine ztrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, complex*16, dimension( ldwork, * ) work, integer ldwork, double precision, dimension( * ) rwork, integer info)

ZTRSNA

Purpose:

    !>
    !> ZTRSNA estimates reciprocal condition numbers for specified
    !> eigenvalues and/or right eigenvectors of a complex upper triangular
    !> matrix T (or of any matrix Q*T*Q**H with Q unitary).
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies whether condition numbers are required for
    !>          eigenvalues (S) or eigenvectors (SEP):
    !>          = 'E': for eigenvalues only (S);
    !>          = 'V': for eigenvectors only (SEP);
    !>          = 'B': for both eigenvalues and eigenvectors (S and SEP).
    !>

    HOWMNY

    !>          HOWMNY is CHARACTER*1
    !>          = 'A': compute condition numbers for all eigenpairs;
    !>          = 'S': compute condition numbers for selected eigenpairs
    !>                 specified by the array SELECT.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    !>          condition numbers are required. To select condition numbers
    !>          for the j-th eigenpair, SELECT(j) must be set to .TRUE..
    !>          If HOWMNY = 'A', SELECT is not referenced.
    !>

    N

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

    T

    !>          T is COMPLEX*16 array, dimension (LDT,N)
    !>          The upper triangular matrix T.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T. LDT >= max(1,N).
    !>

    VL

    !>          VL is COMPLEX*16 array, dimension (LDVL,M)
    !>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
    !>          (or of any Q*T*Q**H with Q unitary), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VL, as returned by
    !>          ZHSEIN or ZTREVC.
    !>          If JOB = 'V', VL is not referenced.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL.
    !>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
    !>

    VR

    !>          VR is COMPLEX*16 array, dimension (LDVR,M)
    !>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
    !>          (or of any Q*T*Q**H with Q unitary), corresponding to the
    !>          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
    !>          must be stored in consecutive columns of VR, as returned by
    !>          ZHSEIN or ZTREVC.
    !>          If JOB = 'V', VR is not referenced.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR.
    !>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    !>          selected eigenvalues, stored in consecutive elements of the
    !>          array. Thus S(j), SEP(j), and the j-th columns of VL and VR
    !>          all correspond to the same eigenpair (but not in general the
    !>          j-th eigenpair, unless all eigenpairs are selected).
    !>          If JOB = 'V', S is not referenced.
    !>

    SEP

    !>          SEP is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the selected eigenvectors, stored in consecutive
    !>          elements of the array.
    !>          If JOB = 'E', SEP is not referenced.
    !>

    MM

    !>          MM is INTEGER
    !>          The number of elements in the arrays S (if JOB = 'E' or 'B')
    !>           and/or SEP (if JOB = 'V' or 'B'). MM >= M.
    !>

    M

    !>          M is INTEGER
    !>          The number of elements of the arrays S and/or SEP actually
    !>          used to store the estimated condition numbers.
    !>          If HOWMNY = 'A', M is set to N.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (LDWORK,N+6)
    !>          If JOB = 'E', WORK is not referenced.
    !>

    LDWORK

    !>          LDWORK is INTEGER
    !>          The leading dimension of the array WORK.
    !>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (N)
    !>          If JOB = 'E', RWORK is not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The reciprocal of the condition number of an eigenvalue lambda is
    !>  defined as
    !>
    !>          S(lambda) = |v**H*u| / (norm(u)*norm(v))
    !>
    !>  where u and v are the right and left eigenvectors of T corresponding
    !>  to lambda; v**H denotes the conjugate transpose of v, and norm(u)
    !>  denotes the Euclidean norm. These reciprocal condition numbers always
    !>  lie between zero (very badly conditioned) and one (very well
    !>  conditioned). If n = 1, S(lambda) is defined to be 1.
    !>
    !>  An approximate error bound for a computed eigenvalue W(i) is given by
    !>
    !>                      EPS * norm(T) / S(i)
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal of the condition number of the right eigenvector u
    !>  corresponding to lambda is defined as follows. Suppose
    !>
    !>              T = ( lambda  c  )
    !>                  (   0    T22 )
    !>
    !>  Then the reciprocal condition number is
    !>
    !>          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
    !>
    !>  where sigma-min denotes the smallest singular value. We approximate
    !>  the smallest singular value by the reciprocal of an estimate of the
    !>  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
    !>  defined to be abs(T(1,1)).
    !>
    !>  An approximate error bound for a computed right eigenvector VR(i)
    !>  is given by
    !>
    !>                      EPS * norm(T) / SEP(i)
    !>

Author

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