Provided by: liblapack-doc_3.12.0-3build1.1_all bug

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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