f08fb computes selected eigenvalues and, optionally, eigenvectors of a real $n$ by $n$ symmetric matrix $A$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

# Syntax

C#
```public static void f08fb(
string jobz,
string range,
string uplo,
int n,
double[,] a,
double vl,
double vu,
int il,
int iu,
double abstol,
out int m,
double[] w,
double[,] z,
int[] jfail,
out int info
)```
Visual Basic
```Public Shared Sub f08fb ( _
jobz As String, _
range As String, _
uplo As String, _
n As Integer, _
a As Double(,), _
vl As Double, _
vu As Double, _
il As Integer, _
iu As Integer, _
abstol As Double, _
<OutAttribute> ByRef m As Integer, _
w As Double(), _
z As Double(,), _
jfail As Integer(), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f08fb(
String^ jobz,
String^ range,
String^ uplo,
int n,
array<double,2>^ a,
double vl,
double vu,
int il,
int iu,
double abstol,
[OutAttribute] int% m,
array<double>^ w,
array<double,2>^ z,
array<int>^ jfail,
[OutAttribute] int% info
)```
F#
```static member f08fb :
jobz : string *
range : string *
uplo : string *
n : int *
a : float[,] *
vl : float *
vu : float *
il : int *
iu : int *
abstol : float *
m : int byref *
w : float[] *
z : float[,] *
jfail : int[] *
info : int byref -> unit
```

#### Parameters

jobz
Type: System..::..String
On entry: indicates whether eigenvectors are computed.
${\mathbf{jobz}}=\text{"N"}$
Only eigenvalues are computed.
${\mathbf{jobz}}=\text{"V"}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{jobz}}=\text{"N"}$ or $\text{"V"}$.
range
Type: System..::..String
On entry: if ${\mathbf{range}}=\text{"A"}$, all eigenvalues will be found.
If ${\mathbf{range}}=\text{"V"}$, all eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If ${\mathbf{range}}=\text{"I"}$, the ilth to iuth eigenvalues will be found.
Constraint: ${\mathbf{range}}=\text{"A"}$, $\text{"V"}$ or $\text{"I"}$.
uplo
Type: System..::..String
On entry: if ${\mathbf{uplo}}=\text{"U"}$, the upper triangular part of $A$ is stored.
If ${\mathbf{uplo}}=\text{"L"}$, the lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{"U"}$ or $\text{"L"}$.
n
Type: System..::..Int32
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
a
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ symmetric matrix $A$.
• If ${\mathbf{uplo}}=\text{"U"}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{"L"}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the lower triangle (if ${\mathbf{uplo}}=\text{"L"}$) or the upper triangle (if ${\mathbf{uplo}}=\text{"U"}$) of a, including the diagonal, is overwritten.
vl
Type: System..::..Double
On entry: if ${\mathbf{range}}=\text{"V"}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\text{"A"}$ or $\text{"I"}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\text{"V"}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
vu
Type: System..::..Double
On entry: if ${\mathbf{range}}=\text{"V"}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\text{"A"}$ or $\text{"I"}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\text{"V"}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
il
Type: System..::..Int32
On entry: if ${\mathbf{range}}=\text{"I"}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If ${\mathbf{range}}=\text{"A"}$ or $\text{"V"}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{"I"}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\text{"I"}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
iu
Type: System..::..Int32
On entry: if ${\mathbf{range}}=\text{"I"}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If ${\mathbf{range}}=\text{"A"}$ or $\text{"V"}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{"I"}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\text{"I"}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
abstol
Type: System..::..Double
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to
 $abstol+εmaxa,b,$
where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $A$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this method returns with ${\mathbf{info}}>{0}$, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
m
Type: System..::..Int32%
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\text{"A"}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\text{"I"}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
w
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the first m elements contain the selected eigenvalues in ascending order.
z
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint:
• if ${\mathbf{jobz}}=\text{"V"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathrm{dim1}\ge 1$.
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobz}}=\text{"V"}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobz}}=\text{"V"}$, then
• if ${\mathbf{info}}={0}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left[i-1\right]$;
• if an eigenvector fails to converge (${\mathbf{info}}>{0}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If ${\mathbf{jobz}}=\text{"N"}$, z is not referenced.
Note:  you must ensure that at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ columns are supplied in the array z; if ${\mathbf{range}}=\text{"V"}$, the exact value of m is not known in advance and an upper bound of at least n must be used.
jfail
Type: array<System..::..Int32>[]()[][]
An array of size [dim1]
Note: the dimension of the array jfail must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: if ${\mathbf{jobz}}=\text{"V"}$, then
• if ${\mathbf{info}}={0}$, the first m elements of jfail are zero;
• if ${\mathbf{info}}>{0}$, jfail contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{jobz}}=\text{"N"}$, jfail is not referenced.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# Description

The symmetric matrix $A$ is first reduced to tridiagonal form, using orthogonal similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

# References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

# Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDZ) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
If ${\mathbf{info}}=i$, then $i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
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 computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2=OεA2,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

# Parallelism and Performance

None.

The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this method is f08fp.

# Example

This example finds the eigenvalues in the half-open interval $\left(-1,1\right]$, and the corresponding eigenvectors, of the symmetric matrix
 $A=1234223433344444.$

Example program (C#): f08fbe.cs

Example program data: f08fbe.d

Example program results: f08fbe.r