# NAG FL Interfacef08jbf (dstevx)

## 1Purpose

f08jbf computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix $A$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## 2Specification

Fortran Interface
 Subroutine f08jbf ( jobz, n, d, e, vl, vu, il, iu, m, w, z, ldz, work, info)
 Integer, Intent (In) :: n, il, iu, ldz Integer, Intent (Inout) :: jfail(*) Integer, Intent (Out) :: m, iwork(5*n), info Real (Kind=nag_wp), Intent (In) :: vl, vu, abstol Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: w(n), work(5*n) Character (1), Intent (In) :: jobz, range
#include <nag.h>
 void f08jbf_ (const char *jobz, const char *range, const Integer *n, double d[], double e[], const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], double z[], const Integer *ldz, double work[], Integer iwork[], Integer jfail[], Integer *info, const Charlen length_jobz, const Charlen length_range)
The routine may be called by the names f08jbf, nagf_lapackeig_dstevx or its LAPACK name dstevx.

## 3Description

f08jbf computes the required eigenvalues and eigenvectors of $A$ by reducing the tridiagonal matrix to diagonal form using the $QR$ algorithm. Bisection is used to determine selected eigenvalues.

## 4References

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 https://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

## 5Arguments

1: $\mathbf{jobz}$Character(1) Input
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 $\text{'V'}$.
2: $\mathbf{range}$Character(1) Input
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 $\text{'I'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ diagonal elements of the tridiagonal matrix $A$.
On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
5: $\mathbf{e}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
6: $\mathbf{vl}$Real (Kind=nag_wp) Input
7: $\mathbf{vu}$Real (Kind=nag_wp) Input
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 ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
8: $\mathbf{il}$Integer Input
9: $\mathbf{iu}$Integer Input
On entry: if ${\mathbf{range}}=\text{'I'}$, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
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 ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
10: $\mathbf{abstol}$Real (Kind=nag_wp) Input
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
 $abstol+ε maxa,b ,$
where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖A‖}_{1}$ will be used in its place. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this routine returns with ${\mathbf{info}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
11: $\mathbf{m}$Integer Output
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 ${\mathbf{range}}=\text{'I'}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
12: $\mathbf{w}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the first m elements contain the selected eigenvalues in ascending order.
13: $\mathbf{z}\left({\mathbf{ldz}},*\right)$Real (Kind=nag_wp) array Output
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 ${\mathbf{info}}={\mathbf{0}}$, 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 ${\mathbf{w}}\left(i\right)$;
• if an eigenvector fails to converge (${\mathbf{info}}>{\mathbf{0}}$), 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.
14: $\mathbf{ldz}$Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08jbf is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
15: $\mathbf{work}\left(5×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
16: $\mathbf{iwork}\left(5×{\mathbf{n}}\right)$Integer array Workspace
17: $\mathbf{jfail}\left(*\right)$Integer array Output
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 ${\mathbf{info}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if ${\mathbf{info}}>{\mathbf{0}}$, jfail contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
18: $\mathbf{info}$Integer Output
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.
${\mathbf{info}}>0$
The algorithm failed to converge; $〈\mathit{\text{value}}〉$ eigenvectors did not converge. Their indices are stored in array jfail.

## 7Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08jbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is proportional to ${n}^{2}$ if ${\mathbf{jobz}}=\text{'N'}$ and is proportional to ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{range}}=\text{'A'}$, otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.

## 10Example

This example finds the eigenvalues in the half-open interval $\left(0,5\right]$, and the corresponding eigenvectors, of the symmetric tridiagonal matrix
 $A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 .$

### 10.1Program Text

Program Text (f08jbfe.f90)

### 10.2Program Data

Program Data (f08jbfe.d)

### 10.3Program Results

Program Results (f08jbfe.r)