---
name: "ctrsna.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/ctrsna.f"
---

# ctrsna.f(3)

## NAME

SRC/ctrsna.f

## SYNOPSIS

### Functions/Subroutines

subroutine **ctrsna** (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)\
 **CTRSNA**

## Function/Subroutine Documentation

### 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)

**CTRSNA**

**Purpose:**

```
!>
!> 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).
!>
```

**Parameters**

*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 j-th eigenpair, SELECT(j) 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 COMPLEX array, dimension (LDT,N)
!>          The upper triangular matrix T.
!>
```

*LDT*

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

*VL*

```
!>          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.
!>
```

*LDVL*

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

*VR*

```
!>          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.
!>
```

*LDVR*

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

*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. 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 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.
!>
```

*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 COMPLEX 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.
!>
```

*RWORK*

```
!>          RWORK is REAL array, dimension (N)
!>          If JOB = 'E', RWORK is not referenced.
!>
```

*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**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)
!>
```

Definition at line **246** of file **ctrsna.f**.

## Author

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