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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsbevx (f08hb)

## Purpose

nag_lapack_dsbevx (f08hb) computes selected eigenvalues and, optionally, eigenvectors of a real n$n$ by n$n$ symmetric band matrix A$A$ of bandwidth (2kd + 1) $\left(2{k}_{d}+1\right)$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Syntax

[ab, q, m, w, z, jfail, info] = f08hb(jobz, range, uplo, kd, ab, vl, vu, il, iu, abstol, 'n', n)
[ab, q, m, w, z, jfail, info] = nag_lapack_dsbevx(jobz, range, uplo, kd, ab, vl, vu, il, iu, abstol, 'n', n)

## Description

The symmetric band matrix A$A$ 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

## 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:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of A$A$ is stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
4:     kd – int64int32nag_int scalar
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, kd${k}_{d}$, of the matrix A$A$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, kd${k}_{d}$, of the matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
5:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The upper or lower triangle of the n$n$ by n$n$ symmetric band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
6:     vl – double scalar
7:     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${\mathbf{vl}}<{\mathbf{vu}}$.
8:     il – int64int32nag_int scalar
9:     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 $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
10:   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, where T$T$ is the tridiagonal matrix obtained by reducing A$A$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am (   ) , not zero. If this function returns with ${\mathbf{INFO}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am (   ) . See Demmel and Kahan (1990).

### Optional Input Parameters

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

### Input Parameters Omitted from the MATLAB Interface

ldab ldq ldz work iwork

### Output Parameters

1:     ab(ldab, : $:$) – double array
The first dimension of the array ab will be kd + 1${\mathbf{kd}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldabkd + 1$\mathit{ldab}\ge {\mathbf{kd}}+1$.
ab stores values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T$T$ are returned in ab using the same storage format as described above.
2:     q(ldq, : $:$) – double array
The first dimension, ldq, of the array q will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldq max (1,n) $\mathit{ldq}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldq1$\mathit{ldq}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, the n$n$ by n$n$ orthogonal matrix used in the reduction to tridiagonal form.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, q is not referenced.
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(n) – double array
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'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, 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 an eigenvector fails to converge (${\mathbf{INFO}}>{\mathbf{0}}$), then that column of Z$Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
6:     jfail( : $:$) – int64int32nag_int array
Note: the dimension of the array jfail must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, 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 jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: uplo, 4: n, 5: kd, 6: ab, 7: ldab, 8: q, 9: ldq, 10: vl, 11: vu, 12: il, 13: iu, 14: abstol, 15: m, 16: w, 17: z, 18: ldz, 19: work, 20: iwork, 21: jfail, 22: 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.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, then i$i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.

## 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 kd n2 ${k}_{d}{n}^{2}$ 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.
The complex analogue of this function is nag_lapack_zhbevx (f08hp).

## Example

```function nag_lapack_dsbevx_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
0, 2, 3, 4, 5;
1, 2, 3, 4, 5];
vl = -3;
vu = 3;
il = int64(0);
iu = int64(0);
abstol = 0;
[abOut, q, m, w, z, jfail, info] = ...
nag_lapack_dsbevx(jobz, range, uplo, kd, ab, vl, vu, il, iu, abstol)
```
```

0         0    3.0000    6.9338    1.5841
0    3.6056    6.9682   -2.3328   -0.2640
1.0000    5.4615    8.9115    2.8591   -3.2322

q =

1.0000         0         0         0         0
0    0.5547    0.0827    0.6078    0.5622
0    0.8321   -0.0551   -0.4052   -0.3748
0         0    0.7960    0.3491   -0.4944
0         0    0.5970   -0.5870    0.5468

m =

2

w =

-2.6633
1.7511
0
0
0

z =

-0.6238   -0.5635
0.2575    0.3896
0.5900   -0.4008
-0.4308    0.5581
-0.1039   -0.2421

jfail =

0
0
0
0
0

info =

0

```
```function f08hb_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
0, 2, 3, 4, 5;
1, 2, 3, 4, 5];
vl = -3;
vu = 3;
il = int64(0);
iu = int64(0);
abstol = 0;
[abOut, q, m, w, z, jfail, info] = ...
f08hb(jobz, range, uplo, kd, ab, vl, vu, il, iu, abstol)
```
```

0         0    3.0000    6.9338    1.5841
0    3.6056    6.9682   -2.3328   -0.2640
1.0000    5.4615    8.9115    2.8591   -3.2322

q =

1.0000         0         0         0         0
0    0.5547    0.0827    0.6078    0.5622
0    0.8321   -0.0551   -0.4052   -0.3748
0         0    0.7960    0.3491   -0.4944
0         0    0.5970   -0.5870    0.5468

m =

2

w =

-2.6633
1.7511
0
0
0

z =

-0.6238   -0.5635
0.2575    0.3896
0.5900   -0.4008
-0.4308    0.5581
-0.1039   -0.2421

jfail =

0
0
0
0
0

info =

0

```