# NAG Library Routine Document

## 1Purpose

f08flf (ddisna) computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian $m$ by $m$ matrix $A$, or for the left or right singular vectors of a general $m$ by $n$ matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f08flf ( job, m, n, d, sep, info)
 Integer, Intent (In) :: m, n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*) Real (Kind=nag_wp), Intent (Inout) :: sep(*) Character (1), Intent (In) :: job
#include <nagmk26.h>
 void f08flf_ (const char *job, const Integer *m, const Integer *n, const double d[], double sep[], Integer *info, const Charlen length_job)
The routine may be called by its LAPACK name ddisna.

## 3Description

The bound on the error, measured by the angle in radians, for the $i$th computed vector is given by $\epsilon {‖A‖}_{2}/{\mathrm{sep}}_{i}$, where $\epsilon$ is the machine precision and ${\mathrm{sep}}_{i}$ is the reciprocal condition number for the vectors, returned in the array element ${\mathbf{sep}}\left(i\right)$. ${\mathbf{sep}}\left(i\right)$ is restricted to be at least $\epsilon {‖A‖}_{2}$ in order to limit the size of the error bound.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{job}$ – Character(1)Input
On entry: specifies for which problem the reciprocal condition number should be computed.
${\mathbf{job}}=\text{'E'}$
The eigenvectors of a symmetric or Hermitian matrix.
${\mathbf{job}}=\text{'L'}$
The left singular vectors of a general matrix.
${\mathbf{job}}=\text{'R'}$
The right singular vectors of a general matrix.
Constraint: ${\mathbf{job}}=\text{'E'}$, $\text{'L'}$ or $\text{'R'}$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix when ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$.
If ${\mathbf{job}}=\text{'E'}$, n is not referenced.
Constraint: if ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$, ${\mathbf{n}}\ge 0$.
4:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{job}}=\text{'E'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ if ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$.
On entry: the eigenvalues if ${\mathbf{job}}=\text{'E'}$, or singular values if ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$ of the matrix $A$.
Constraints:
• the elements of the array d must be in either increasing or decreasing order;
• if ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$ the elements of d must be non-negative.
5:     $\mathbf{sep}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array sep must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{job}}=\text{'E'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ if ${\mathbf{job}}=\text{'L'}$ or $\text{'R'}$.
On exit: the reciprocal condition numbers of the vectors.
6:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The reciprocal condition numbers are computed to machine precision relative to the size of the eigenvalues, or singular values.

## 8Parallelism and Performance

f08flf (ddisna) is not threaded in any implementation.