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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dstevr (f08jd)

## Purpose

nag_lapack_dstevr (f08jd) computes selected eigenvalues and, optionally, eigenvectors of a real n$n$ by n$n$ symmetric tridiagonal matrix T$T$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Syntax

[d, e, m, w, z, isuppz, info] = f08jd(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)
[d, e, m, w, z, isuppz, info] = nag_lapack_dstevr(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)

## Description

Whenever possible nag_lapack_dstevr (f08jd) computes the eigenspectrum using Relatively Robust Representations. nag_lapack_dstevr (f08jd) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ LDLT$LD{L}^{\mathrm{T}}$ representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i$i$th unreduced block of T$T$:
 (a) compute T − σi I = Li Di LiT $T-{\sigma }_{i}I={L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, such that Li Di LiT ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ is a relatively robust representation, (b) compute the eigenvalues, λj${\lambda }_{j}$, of Li Di LiT ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ to high relative accuracy by the dqds algorithm, (c) if there is a cluster of close eigenvalues, ‘choose’ σi${\sigma }_{i}$ close to the cluster, and go to (a), (d) given the approximate eigenvalue λj${\lambda }_{j}$ of Li Di LiT ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the parameter abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

## 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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new O(n2)$\mathit{O}\left({n}^{2}\right)$ algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

## Parameters

### Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz = 'N'${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
jobz = 'V'${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: jobz = 'N'${\mathbf{jobz}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     range – string (length ≥ 1)
If range = 'A'${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If range = 'V'${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval (vl,vu]$\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If range = 'I'${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'${\mathbf{range}}=\text{'A'}$, 'V'$\text{'V'}$ or 'I'$\text{'I'}$.
3:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ diagonal elements of the tridiagonal matrix T$T$.
4:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix T$T$.
5:     vl – double scalar
6:     vu – double scalar
If range = 'V'${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'I'$\text{'I'}$, vl and vu are not referenced.
Constraint: if range = 'V'${\mathbf{range}}=\text{'V'}$, vl < vu${\mathbf{vl}}<{\mathbf{vu}}$.
7:     il – int64int32nag_int scalar
8:     iu – int64int32nag_int scalar
If range = 'I'${\mathbf{range}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'V'$\text{'V'}$, il and iu are not referenced.
Constraints:
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n = 0${\mathbf{n}}=0$, il = 1${\mathbf{il}}=1$ and iu = 0${\mathbf{iu}}=0$;
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n > 0${\mathbf{n}}>0$, 1 il iu n $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
9:     abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] $\left[a,b\right]$ of width less than or equal to
 abstol + ε max (|a|,|b|) , $abstol+ε max(|a|,|b|) ,$
where ε $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then ε T1 $\epsilon {‖T‖}_{1}$ will be used in its place. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to x02am(   ) , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array d.
n$n$, the order of the matrix.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldz work lwork iwork liwork

### Output Parameters

1:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
2:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
3:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn$0\le {\mathbf{m}}\le {\mathbf{n}}$.
If range = 'A'${\mathbf{range}}=\text{'A'}$, m = n${\mathbf{m}}={\mathbf{n}}$.
If range = 'I'${\mathbf{range}}=\text{'I'}$, m = iuil + 1${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
4:     w( : $:$) – double array
Note: the dimension of the array w must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The first m elements contain the selected eigenvalues in ascending order.
5:     z(ldz, : $:$) – double array
The first dimension, ldz, of the array z will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldz1$\mathit{ldz}\ge 1$.
The second dimension of the array will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, the first m columns of Z$Z$ contain the orthonormal eigenvectors of the matrix A$A$ corresponding to the selected eigenvalues, with the i$i$th column of Z$Z$ holding the eigenvector associated with w(i)${\mathbf{w}}\left(i\right)$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
6:     isuppz( : $:$) – int64int32nag_int array
Note: the dimension of the array isuppz must be at least max (1,2 × m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$.
The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The i$i$th eigenvector is nonzero only in elements isuppz(2 × i1)${\mathbf{isuppz}}\left(2×i-1\right)$ through isuppz(2 × i)${\mathbf{isuppz}}\left(2×i\right)$. Implemented only for range = 'A'${\mathbf{range}}=\text{'A'}$ or 'I'$\text{'I'}$ and iuil = n1${\mathbf{iu}}-{\mathbf{il}}={\mathbf{n}}-1$.
7:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: range, 3: n, 4: d, 5: e, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: isuppz, 16: work, 17: lwork, 18: iwork, 19: liwork, 20: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
INFO > 0${\mathbf{INFO}}>0$
An internal error has occurred in this function. Please refer to info in nag_lapack_dstebz (f08jj).

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

The total number of floating point operations is proportional to n2${n}^{2}$ if jobz = 'N'${\mathbf{jobz}}=\text{'N'}$ and is proportional to n3${n}^{3}$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$ and range = 'A'${\mathbf{range}}=\text{'A'}$, otherwise the number of floating point operations will depend upon the number of computed eigenvectors.

## Example

```function nag_lapack_dstevr_example
jobz = 'Vectors';
range = 'Indices';
d = [1;
4;
9;
16];
e = [1;
2;
3];
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[dOut, eOut, m, w, z, isuppz, info] = nag_lapack_dstevr(jobz, range, d, e, vl, vu, il, iu, abstol)
```
```

dOut =

1
4
9
16

eOut =

1
2
3

m =

2

w =

3.5470
8.6578
0
0

z =

0.3388    0.0494
0.8628    0.3781
-0.3648    0.8558
0.0879   -0.3497

isuppz =

0
0
0
0
0
0
0
0

info =

0

```
```function f08jd_example
jobz = 'Vectors';
range = 'Indices';
d = [1;
4;
9;
16];
e = [1;
2;
3];
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[dOut, eOut, m, w, z, isuppz, info] = f08jd(jobz, range, d, e, vl, vu, il, iu, abstol)
```
```

dOut =

1
4
9
16

eOut =

1
2
3

m =

2

w =

3.5470
8.6578
0
0

z =

0.3388    0.0494
0.8628    0.3781
-0.3648    0.8558
0.0879   -0.3497

isuppz =

0
0
0
0
0
0
0
0

info =

0

```