F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08FLF (DDISNA)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

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 $A$.

## 2  Specification

 SUBROUTINE F08FLF ( JOB, M, N, D, SEP, INFO)
 INTEGER M, N, INFO REAL (KIND=nag_wp) D(*), SEP(*) CHARACTER(1) JOB
The routine may be called by its LAPACK name ddisna.

## 3  Description

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.

## 4  References

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

## 5  Parameters

1:     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:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
3:     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:     D($*$) – 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:     SEP($*$) – 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:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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