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.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  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)