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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhbevd (f08hq)

## Purpose

nag_lapack_zhbevd (f08hq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL$QL$ or QR$QR$ algorithm.

## Syntax

[ab, w, z, info] = f08hq(job, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_zhbevd(job, uplo, kd, ab, 'n', n)

## Description

nag_lapack_zhbevd (f08hq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix A$A$. In other words, it can compute the spectral factorization of A$A$ as
 A = ZΛZH, $A=ZΛZH,$
where Λ$\Lambda$ is a real diagonal matrix whose diagonal elements are the eigenvalues λi${\lambda }_{i}$, and Z$Z$ is the (complex) unitary matrix whose columns are the eigenvectors zi${z}_{i}$. Thus
 Azi = λizi,  i = 1,2, … ,n. $Azi=λizi, i=1,2,…,n.$

## 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:     job – string (length ≥ 1)
Indicates whether eigenvectors are computed.
job = 'N'${\mathbf{job}}=\text{'N'}$
Only eigenvalues are computed.
job = 'V'${\mathbf{job}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: job = 'N'${\mathbf{job}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A$A$ is stored.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored.
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, : $:$) – complex 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$n$ Hermitian 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 first dimension of the array ab and the second dimension of the array ab. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldab ldz work lwork rwork lrwork iwork liwork

### Output Parameters

1:     ab(ldab, : $:$) – complex 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( : $:$) – 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 eigenvalues of the matrix A$A$ in ascending order.
3:     z(ldz, : $:$) – complex array
The first dimension, ldz, of the array z will be
• if job = 'V'${\mathbf{job}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if job = 'N'${\mathbf{job}}=\text{'N'}$, 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 job = 'V'${\mathbf{job}}=\text{'V'}$ and at least 1$1$ if job = 'N'${\mathbf{job}}=\text{'N'}$
If job = 'V'${\mathbf{job}}=\text{'V'}$, z stores the unitary matrix Z$Z$ which contains the eigenvectors of A$A$. The i$i$th column of Z$Z$ contains the eigenvector which corresponds to the eigenvalue w(i)${\mathbf{w}}\left(i\right)$.
If job = 'N'${\mathbf{job}}=\text{'N'}$, 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: job, 2: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: lwork, 12: rwork, 13: lrwork, 14: iwork, 15: liwork, 16: 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${\mathbf{info}}=i$ and job = 'N'${\mathbf{job}}=\text{'N'}$, the algorithm failed to converge; i$i$ elements of an intermediate tridiagonal form did not converge to zero; if info = i${\mathbf{info}}=i$ and job = 'V'${\mathbf{job}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column i / (n + 1)$i/\left({\mathbf{n}}+1\right)$ through i  mod  (n + 1).

## 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 real analogue of this function is nag_lapack_dsbevd (f08hc).

## Example

```function nag_lapack_zhbevd_example
job = 'V';
uplo = 'L';
kd = int64(2);
ab = [complex(1),  2 + 0i,  3 + 0i,  4 + 0i,  5 + 0i;
2 + 1i,  3 + 2i,  4 + 3i,  5 + 4i,  0 + 0i;
3 + 1i,  4 + 2i,  5 + 3i,  0 + 0i,  0 + 0i];
[abOut, w, z, info] = nag_lapack_zhbevd(job, uplo, kd, ab)
```
```

1.0000 + 0.0000i   5.2000 + 0.0000i   7.8051 + 0.0000i  -0.0720 + 0.0000i   1.0669 + 0.0000i
3.8730 + 0.0000i   8.5456 + 0.0000i   6.2114 + 0.0000i   1.9595 + 0.0000i   0.0000 + 0.0000i
3.0000 + 1.0000i   7.2377 + 3.8317i   5.4275 + 0.2013i   0.0000 + 0.0000i   0.0000 + 0.0000i

w =

-6.4185
-1.4094
1.4421
4.4856
16.9002

z =

0.2591 + 0.0000i  -0.6367 + 0.0000i   0.4516 + 0.0000i  -0.5503 + 0.0000i   0.1439 + 0.0000i
-0.0245 - 0.4344i   0.2578 - 0.2413i  -0.3029 - 0.4402i  -0.4785 - 0.2759i   0.3060 + 0.0411i
-0.5159 + 0.1095i   0.3039 + 0.3481i   0.3160 + 0.2978i  -0.2128 - 0.0465i   0.4681 + 0.2306i
-0.0004 + 0.5093i  -0.3450 + 0.0832i  -0.4088 - 0.3213i   0.1707 - 0.0200i   0.4098 + 0.3832i
0.4333 - 0.1353i   0.2469 - 0.2634i   0.0204 + 0.2262i  -0.0175 + 0.5611i   0.1819 + 0.5136i

info =

0

```
```function f08hq_example
job = 'V';
uplo = 'L';
kd = int64(2);
ab = [complex(1),  2 + 0i,  3 + 0i,  4 + 0i,  5 + 0i;
2 + 1i,  3 + 2i,  4 + 3i,  5 + 4i,  0 + 0i;
3 + 1i,  4 + 2i,  5 + 3i,  0 + 0i,  0 + 0i];
[abOut, w, z, info] = f08hq(job, uplo, kd, ab)
```
```

1.0000 + 0.0000i   5.2000 + 0.0000i   7.8051 + 0.0000i  -0.0720 + 0.0000i   1.0669 + 0.0000i
3.8730 + 0.0000i   8.5456 + 0.0000i   6.2114 + 0.0000i   1.9595 + 0.0000i   0.0000 + 0.0000i
3.0000 + 1.0000i   7.2377 + 3.8317i   5.4275 + 0.2013i   0.0000 + 0.0000i   0.0000 + 0.0000i

w =

-6.4185
-1.4094
1.4421
4.4856
16.9002

z =

0.2591 + 0.0000i  -0.6367 + 0.0000i   0.4516 + 0.0000i  -0.5503 + 0.0000i   0.1439 + 0.0000i
-0.0245 - 0.4344i   0.2578 - 0.2413i  -0.3029 - 0.4402i  -0.4785 - 0.2759i   0.3060 + 0.0411i
-0.5159 + 0.1095i   0.3039 + 0.3481i   0.3160 + 0.2978i  -0.2128 - 0.0465i   0.4681 + 0.2306i
-0.0004 + 0.5093i  -0.3450 + 0.0832i  -0.4088 - 0.3213i   0.1707 - 0.0200i   0.4098 + 0.3832i
0.4333 - 0.1353i   0.2469 - 0.2634i   0.0204 + 0.2262i  -0.0175 + 0.5611i   0.1819 + 0.5136i

info =

0

```