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)

Definition at line 246 of file ctrsna.f.

  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)

Definition at line 262 of file dtrsna.f.

  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)

Definition at line 262 of file strsna.f.

  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)

Definition at line 246 of file ztrsna.f.

Author

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