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.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  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)