.TH "trsna" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
trsna \- trsna: eig condition numbers
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBctrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)"
.br
.RI "\fBCTRSNA\fP "
.ti -1c
.RI "subroutine \fBdtrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)"
.br
.RI "\fBDTRSNA\fP "
.ti -1c
.RI "subroutine \fBstrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)"
.br
.RI "\fBSTRSNA\fP "
.ti -1c
.RI "subroutine \fBztrsna\fP (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)"
.br
.RI "\fBZTRSNA\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine ctrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, complex, dimension( ldwork, * ) work, integer ldwork, real, dimension( * ) rwork, integer info)"

.PP
\fBCTRSNA\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> CTRSNA estimates reciprocal condition numbers for specified
!> eigenvalues and/or right eigenvectors of a complex upper triangular
!> matrix T (or of any matrix Q*T*Q**H with Q unitary)\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          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 j-th eigenpair, SELECT(j) must be set to \&.TRUE\&.\&.
!>          If HOWMNY = 'A', SELECT is not referenced\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is COMPLEX array, dimension (LDT,N)
!>          The upper triangular matrix T\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          VL is COMPLEX array, dimension (LDVL,M)
!>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
!>          (or of any Q*T*Q**H with Q unitary), corresponding to the
!>          eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors
!>          must be stored in consecutive columns of VL, as returned by
!>          CHSEIN or CTREVC\&.
!>          If JOB = 'V', VL is not referenced\&.
!> 
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
!>          LDVL is INTEGER
!>          The leading dimension of the array VL\&.
!>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&.
!> 
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
!>          VR is COMPLEX array, dimension (LDVR,M)
!>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
!>          (or of any Q*T*Q**H with Q unitary), corresponding to the
!>          eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors
!>          must be stored in consecutive columns of VR, as returned by
!>          CHSEIN or CTREVC\&.
!>          If JOB = 'V', VR is not referenced\&.
!> 
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&.
!>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          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\&. 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\&.
!> 
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
!>          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\&.
!>          If JOB = 'E', SEP is not referenced\&.
!> 
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX array, dimension (LDWORK,N+6)
!>          If JOB = 'E', WORK is not referenced\&.
!> 
.fi
.PP
.br
\fILDWORK\fP 
.PP
.nf
!>          LDWORK is INTEGER
!>          The leading dimension of the array WORK\&.
!>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&.
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is REAL array, dimension (N)
!>          If JOB = 'E', RWORK is not referenced\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The reciprocal of the condition number of an eigenvalue lambda is
!>  defined as
!>
!>          S(lambda) = |v**H*u| / (norm(u)*norm(v))
!>
!>  where u and v are the right and left eigenvectors of T corresponding
!>  to lambda; v**H 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)
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB246\fP of file \fBctrsna\&.f\fP\&.
.SS "subroutine dtrsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, double precision, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)"

.PP
\fBDTRSNA\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is DOUBLE PRECISION array, dimension (LDT,N)
!>          The upper quasi-triangular matrix T, in Schur canonical form\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
!>          LDVL is INTEGER
!>          The leading dimension of the array VL\&.
!>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&.
!> 
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&.
!>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
!>          If JOB = 'E', WORK is not referenced\&.
!> 
.fi
.PP
.br
\fILDWORK\fP 
.PP
.nf
!>          LDWORK is INTEGER
!>          The leading dimension of the array WORK\&.
!>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (2*(N-1))
!>          If JOB = 'E', IWORK is not referenced\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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)
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB262\fP of file \fBdtrsna\&.f\fP\&.
.SS "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)"

.PP
\fBSTRSNA\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is REAL array, dimension (LDT,N)
!>          The upper quasi-triangular matrix T, in Schur canonical form\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
!>          LDVL is INTEGER
!>          The leading dimension of the array VL\&.
!>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&.
!> 
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&.
!>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (LDWORK,N+6)
!>          If JOB = 'E', WORK is not referenced\&.
!> 
.fi
.PP
.br
\fILDWORK\fP 
.PP
.nf
!>          LDWORK is INTEGER
!>          The leading dimension of the array WORK\&.
!>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (2*(N-1))
!>          If JOB = 'E', IWORK is not referenced\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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)
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB262\fP of file \fBstrsna\&.f\fP\&.
.SS "subroutine ztrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, complex*16, dimension( ldwork, * ) work, integer ldwork, double precision, dimension( * ) rwork, integer info)"

.PP
\fBZTRSNA\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZTRSNA estimates reciprocal condition numbers for specified
!> eigenvalues and/or right eigenvectors of a complex upper triangular
!> matrix T (or of any matrix Q*T*Q**H with Q unitary)\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          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 j-th eigenpair, SELECT(j) must be set to \&.TRUE\&.\&.
!>          If HOWMNY = 'A', SELECT is not referenced\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          T is COMPLEX*16 array, dimension (LDT,N)
!>          The upper triangular matrix T\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          VL is COMPLEX*16 array, dimension (LDVL,M)
!>          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
!>          (or of any Q*T*Q**H with Q unitary), corresponding to the
!>          eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors
!>          must be stored in consecutive columns of VL, as returned by
!>          ZHSEIN or ZTREVC\&.
!>          If JOB = 'V', VL is not referenced\&.
!> 
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
!>          LDVL is INTEGER
!>          The leading dimension of the array VL\&.
!>          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N\&.
!> 
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
!>          VR is COMPLEX*16 array, dimension (LDVR,M)
!>          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
!>          (or of any Q*T*Q**H with Q unitary), corresponding to the
!>          eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors
!>          must be stored in consecutive columns of VR, as returned by
!>          ZHSEIN or ZTREVC\&.
!>          If JOB = 'V', VR is not referenced\&.
!> 
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&.
!>          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          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\&. 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\&.
!> 
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
!>          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\&.
!>          If JOB = 'E', SEP is not referenced\&.
!> 
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension (LDWORK,N+6)
!>          If JOB = 'E', WORK is not referenced\&.
!> 
.fi
.PP
.br
\fILDWORK\fP 
.PP
.nf
!>          LDWORK is INTEGER
!>          The leading dimension of the array WORK\&.
!>          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N\&.
!> 
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
!>          RWORK is DOUBLE PRECISION array, dimension (N)
!>          If JOB = 'E', RWORK is not referenced\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0: successful exit
!>          < 0: if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The reciprocal of the condition number of an eigenvalue lambda is
!>  defined as
!>
!>          S(lambda) = |v**H*u| / (norm(u)*norm(v))
!>
!>  where u and v are the right and left eigenvectors of T corresponding
!>  to lambda; v**H 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)
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB246\fP of file \fBztrsna\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.