• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS
• FURTHER DETAILS

NAME

dtrsna - estimate 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, WORK1, INFO)
  CHARACTER * 1 JOB, HOWMNY
  INTEGER N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
  INTEGER WORK1(*)
  LOGICAL SELECT(*)
  DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), S(*), SEP(*), WORK(LDWORK,*)

  SUBROUTINE DTRSNA_64( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, 
 *      LDVR, S, SEP, MM, M, WORK, LDWORK, WORK1, INFO)
  CHARACTER * 1 JOB, HOWMNY
  INTEGER*8 N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
  INTEGER*8 WORK1(*)
  LOGICAL*8 SELECT(*)
  DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), S(*), SEP(*), WORK(LDWORK,*)

F95 INTERFACE

  SUBROUTINE TRSNA( JOB, HOWMNY, SELECT, [N], T, [LDT], VL, [LDVL], 
 *       VR, [LDVR], S, SEP, MM, M, [WORK], [LDWORK], [WORK1], [INFO])
  CHARACTER(LEN=1) :: JOB, HOWMNY
  INTEGER :: N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
  INTEGER, DIMENSION(:) :: WORK1
  LOGICAL, DIMENSION(:) :: SELECT
  REAL(8), DIMENSION(:) :: S, SEP
  REAL(8), DIMENSION(:,:) :: T, VL, VR, WORK

  SUBROUTINE TRSNA_64( JOB, HOWMNY, SELECT, [N], T, [LDT], VL, [LDVL], 
 *       VR, [LDVR], S, SEP, MM, M, [WORK], [LDWORK], [WORK1], [INFO])
  CHARACTER(LEN=1) :: JOB, HOWMNY
  INTEGER(8) :: N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
  INTEGER(8), DIMENSION(:) :: WORK1
  LOGICAL(8), DIMENSION(:) :: SELECT
  REAL(8), DIMENSION(:) :: S, SEP
  REAL(8), DIMENSION(:,:) :: T, VL, VR, WORK

C INTERFACE

#include <sunperf.h>

void dtrsna(char job, char howmny, logical *select, int n, double *t, int ldt, double *vl, int ldvl, double *vr, int ldvr, double *s, double *sep, int mm, int *m, int ldwork, int *info);

void dtrsna_64(char job, char howmny, logical *select, long n, double *t, long ldt, double *vl, long ldvl, double *vr, long ldvr, double *s, double *sep, long mm, long *m, long ldwork, long *info);

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 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.

ARGUMENTS

• JOB (input)
  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)
  

   = 'A': compute condition numbers for all eigenpairs;

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

• SELECT (input)
  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)
  The order of the matrix T. N > = 0.

• T (input)
  The upper quasi-triangular matrix T, in Schur canonical form.

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

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

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

• S (output)
  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)
  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)
  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)
  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)
  dimension(LDWORK,N+1) If JOB = 'E', WORK is not referenced.

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

• WORK1 (workspace)
  dimension(N) If JOB = 'E', WORK1 is not referenced.

• INFO (output)
  

   = 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)