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## strsna()

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

Download STRSNA + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  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).
      !>

  [in]

  HOWMNY

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

  [in]

  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.
      !>

  [in]

  N

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

  [in]

  T

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

  [in]

  LDT

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

  [in]

  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.
      !>

  [in]

  LDVL

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

  [in]

  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.
      !>

  [in]

  LDVR

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [in]

  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.
      !>

  [out]

  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.
      !>

  [out]

  WORK

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

  [in]

  LDWORK

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

  [out]

  IWORK

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

  [out]

  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)
      !>