﻿ f08fl Method
f08fl 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 $A$.

# Syntax

C#
```public static void f08fl(
string job,
int m,
int n,
double[] d,
double[] sep,
out int info
)```
Visual Basic
```Public Shared Sub f08fl ( _
job As String, _
m As Integer, _
n As Integer, _
d As Double(), _
sep As Double(), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f08fl(
String^ job,
int m,
int n,
array<double>^ d,
array<double>^ sep,
[OutAttribute] int% info
)```
F#
```static member f08fl :
job : string *
m : int *
n : int *
d : float[] *
sep : float[] *
info : int byref -> unit
```

#### Parameters

job
Type: System..::..String
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 $\text{"R"}$.
m
Type: System..::..Int32
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
n
Type: System..::..Int32
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 $\text{"R"}$, ${\mathbf{n}}\ge 0$.
d
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
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.
sep
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
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.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# 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 ${\mathbf{sep}}\left[i-1\right]$. ${\mathbf{sep}}\left[i-1\right]$ is restricted to be at least $\epsilon {‖A‖}_{2}$ in order to limit the size of the error bound.

# References

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

# Error 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.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

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

None.