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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsbev (f08ha)

## Purpose

nag_lapack_dsbev (f08ha) computes all the eigenvalues and, optionally, all the eigenvectors of a real n$n$ by n$n$ symmetric band matrix A$A$ of bandwidth (2kd + 1) $\left(2{k}_{d}+1\right)$.

## Syntax

[ab, w, z, info] = f08ha(jobz, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab, 'n', n)

## Description

The symmetric band matrix A$A$ is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QR$QR$ algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

## 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
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:     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'$\text{'L'}$.
3:     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${\mathbf{kd}}\ge 0$.
4:     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).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$

### 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${\mathbf{n}}\ge 0$.

ldab ldz work

### 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:     w(n) – double array
The eigenvalues in ascending order.
3:     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,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'${\mathbf{jobz}}=\text{'V'}$, z contains the orthonormal eigenvectors of the matrix A$A$, 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.
4:     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: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: 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${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the algorithm failed to converge; i$i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

## 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 n3${n}^{3}$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$ and is proportional to kd n2 ${k}_{d}{n}^{2}$ otherwise.
The complex analogue of this function is nag_lapack_zhbev (f08hn).

## Example

```function nag_lapack_dsbev_example
jobz = 'Vectors';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
0, 2, 3, 4, 5;
1, 2, 3, 4, 5];
[abOut, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab)
```
```

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

w =

-3.2474
-2.6633
1.7511
4.1599
14.9997

z =

0.0394    0.6238    0.5635   -0.5165    0.1582
0.5721   -0.2575   -0.3896   -0.5955    0.3161
-0.4372   -0.5900    0.4008   -0.1470    0.5277
-0.4424    0.4308   -0.5581    0.0470    0.5523
0.5332    0.1039    0.2421    0.5956    0.5400

info =

0

```
```function f08ha_example
jobz = 'Vectors';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
0, 2, 3, 4, 5;
1, 2, 3, 4, 5];
[abOut, w, z, info] = f08ha(jobz, uplo, kd, ab)
```
```

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

w =

-3.2474
-2.6633
1.7511
4.1599
14.9997

z =

0.0394    0.6238    0.5635   -0.5165    0.1582
0.5721   -0.2575   -0.3896   -0.5955    0.3161
-0.4372   -0.5900    0.4008   -0.1470    0.5277
-0.4424    0.4308   -0.5581    0.0470    0.5523
0.5332    0.1039    0.2421    0.5956    0.5400

info =

0

```