---
name: "dtrsna"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/dtrsna/3lapack"
---

# dtrsna(3lapack)

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)

## NAME

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)

## SYNOPSIS

- SUBROUTINE DTRSNA(

  JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO )

- CHARACTER

  HOWMNY, JOB

- INTEGER

  INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N

- LOGICAL

  SELECT( \* )

- INTEGER

  IWORK( \* )

- DOUBLE

  PRECISION S( \* ), SEP( \* ), T( LDT, \* ), VL( LDVL, \* ), VR( LDVR, \* ), WORK( LDWORK, \* )

## 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.\

## ARGUMENTS

- JOB (input) 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 (input) CHARACTER\*1

  \= 'A': compute condition numbers for all eigenpairs;\
   \= 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.

- SELECT (input) 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 (input) INTEGER

  The order of the matrix T. N \>= 0.

- T (input) DOUBLE PRECISION array, dimension (LDT,N)

  The upper quasi-triangular matrix T, in Schur canonical form.

- LDT (input) INTEGER

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

- VL (input) 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 (input) INTEGER

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

- VR (input) 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 (input) INTEGER

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

- S (output) 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 (output) 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 (input) 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 (output) 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 (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+6)

  If JOB = 'E', WORK is not referenced.

- LDWORK (input) INTEGER

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

- IWORK (workspace) INTEGER array, dimension (2\*(N-1))

  If JOB = 'E', IWORK is not referenced.

- INFO (output) INTEGER

  \= 0: successful exit\
   \< 0: if INFO = -i, the i-th argument had an illegal value

## FURTHER DETAILS

The reciprocal of the condition number of an eigenvalue lambda is defined as\
\
         S(lambda) = \|v'\*u\| / (norm(u)\*norm(v))\
 where u and v are the right and left eigenvectors of T corresponding to lambda; v' 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)\