# NAG Library Routine Document

## 1Purpose

f08jlf (dstegr) computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix.

## 2Specification

Fortran Interface
 Subroutine f08jlf ( jobz, n, d, e, vl, vu, il, iu, m, w, z, ldz, work, info)
 Integer, Intent (In) :: n, il, iu, ldz, lwork, liwork Integer, Intent (Inout) :: isuppz(*) Integer, Intent (Out) :: m, iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (In) :: vl, vu, abstol Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), w(*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobz, range
#include <nagmk26.h>
 void f08jlf_ (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, Integer isuppz[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_range)
The routine may be called by its LAPACK name dstegr.

## 3Description

f08jlf (dstegr) computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix $T$. That is, the routine computes the (partial) spectral factorization of $T$ given by
 $ZΛZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the selected eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the corresponding eigenvectors, ${z}_{i}$, of $T$. Thus
 $Tzi= λi zi , i = 1,2,…,m$
where $m$ is the number of selected eigenvalues computed.
The routine may also be used to compute selected eigenvalues and eigenvectors of a real symmetric matrix $A$ which has been reduced to tridiagonal form $T$:
 $QZ Λ QZT , where ​Q​ is orthogonal.$
In this case, the matrix $Q$ must be explicitly applied to the output matrix $Z$. The routines which must be called to perform the reduction to tridiagonal form and apply $Q$ are:
 full matrix f08fef (dsytrd) and f08fgf (dormtr) full matrix, packed storage f08gef (dsptrd) and f08ggf (dopmtr) band matrix f08hef (dsbtrd) with ${\mathbf{vect}}=\text{'V'}$ and f06yaf (dgemm).
This routine uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. f08jlf (dstegr) can usually compute all the eigenvalues and eigenvectors in $O\left({n}^{2}\right)$ floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal routines in this chapter when all the eigenvectors are required, particularly so compared to those routines that are based on the $QR$ algorithm.

## 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 http://www.netlib.org/lapack/lug
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

## 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: indicates which eigenvalues should be returned.
${\mathbf{range}}=\text{'A'}$
All eigenvalues will be found.
${\mathbf{range}}=\text{'V'}$
All eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\text{'I'}$
The ilth through iuth eigenvectors will be found.
Constraint: ${\mathbf{range}}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/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 $T$.
On exit: d is overwritten.
5:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: ${\mathbf{e}}\left(1:{\mathbf{n}}-1\right)$ contains the subdiagonal elements of the tridiagonal matrix $T$. ${\mathbf{e}}\left({\mathbf{n}}\right)$ need not be set.
On exit: e is overwritten.
6:     $\mathbf{vl}$ – Real (Kind=nag_wp)Input
7:     $\mathbf{vu}$ – Real (Kind=nag_wp)Input
On entry: if ${\mathbf{range}}=\text{'V'}$, vl and vu contain the lower and upper bounds respectively 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:     $\mathbf{il}$ – IntegerInput
9:     $\mathbf{iu}$ – IntegerInput
On entry: if ${\mathbf{range}}=\text{'I'}$, il and iu contains 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 ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.
10:   $\mathbf{abstol}$ – Real (Kind=nag_wp)Input
On entry: in earlier versions, this argument was the absolute error tolerance for the eigenvalues/eigenvectors. It is now deprecated, and only included for backwards-compatibility.
11:   $\mathbf{m}$ – IntegerOutput
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:   $\mathbf{w}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
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 eigenvalues in ascending order.
13:   $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayOutput
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 columns of z contain the orthonormal eigenvectors of the matrix $T$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left(i\right)$.
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:   $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08jlf (dstegr) 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:   $\mathbf{isuppz}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array isuppz must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$.
On exit: the support of the eigenvectors in $Z$, i.e., the indices indicating the nonzero elements in $Z$. The $i$th eigenvector is nonzero only in elements ${\mathbf{isuppz}}\left(2×i-1\right)$ through ${\mathbf{isuppz}}\left(2×i\right)$.
16:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ returns the minimum lwork.
17:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08jlf (dstegr) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,18×{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=-1$.
18:   $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$ – Integer arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ returns the minimum liwork.
19:   $\mathbf{liwork}$ – IntegerInput
On entry: the dimension of the array iwork as declared in the (sub)program from which f08jlf (dstegr) is called.
If ${\mathbf{liwork}}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraint: ${\mathbf{liwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,10×{\mathbf{n}}\right)$ or ${\mathbf{liwork}}=-1$.
20:   $\mathbf{info}$ – 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.
${\mathbf{info}}=1$
The $\mathrm{dqds}$ algorithm failed to converge.
${\mathbf{info}}=2$
Inverse iteration failed to converge.

## 7Accuracy

See the description for abstol. See also Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

## 8Parallelism and Performance

f08jlf (dstegr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jlf (dstegr) 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 required to compute all the eigenvalues and eigenvectors is approximately proportional to ${n}^{2}$.
The complex analogue of this routine is f08jyf (zstegr).

## 10Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .$
abstol is set to zero so that the default tolerance of $n\epsilon {‖T‖}_{1}$ is used.

### 10.1Program Text

Program Text (f08jlfe.f90)

### 10.2Program Data

Program Data (f08jlfe.d)

### 10.3Program Results

Program Results (f08jlfe.r)