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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhpevd (f08gq)

## Purpose

nag_lapack_zhpevd (f08gq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix held in packed storage. 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

[ap, w, z, info] = f08gq(job, uplo, n, ap)
[ap, w, z, info] = nag_lapack_zhpevd(job, uplo, n, ap)

## Description

nag_lapack_zhpevd (f08gq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix A$A$ (held in packed storage). 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:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
4:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The upper or lower triangle of the n$n$ by n$n$ Hermitian matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.

None.

### Input Parameters Omitted from the MATLAB Interface

ldz work lwork rwork lrwork iwork liwork

### Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A$A$.
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$A$.
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: ap, 5: w, 6: z, 7: ldz, 8: work, 9: lwork, 10: rwork, 11: lrwork, 12: iwork, 13: liwork, 14: 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_dspevd (f08gc).

## Example

```function nag_lapack_zhpevd_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
2 + 1i;
3 + 1i;
4 + 1i;
2 + 0i;
3 + 2i;
4 + 2i;
3 + 0i;
4 + 3i;
4 + 0i];
[apOut, w, z, info] = nag_lapack_zhpevd(job, uplo, n, ap)
```
```

apOut =

1.0000 + 0.0000i
-5.6569 + 0.0000i
0.4020 + 0.0781i
0.5304 + 0.0613i
9.6250 + 0.0000i
-4.8846 + 0.0000i
0.4351 - 0.7543i
-0.6898 + 0.0000i
1.4423 + 0.0000i
0.0648 + 0.0000i

w =

-4.2443
-0.6886
1.1412
13.7916

z =

-0.4836 + 0.0000i  -0.6470 + 0.0000i  -0.4456 + 0.0000i  -0.3859 + 0.0000i
-0.2912 + 0.3618i   0.4984 + 0.1130i  -0.0230 - 0.5702i  -0.4441 + 0.0156i
0.3163 + 0.3696i  -0.2949 - 0.3165i   0.5331 + 0.1317i  -0.5173 - 0.0844i
0.4447 - 0.3406i   0.2241 + 0.2878i  -0.3510 + 0.2261i  -0.5277 - 0.3168i

info =

0

```
```function f08gq_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
2 + 1i;
3 + 1i;
4 + 1i;
2 + 0i;
3 + 2i;
4 + 2i;
3 + 0i;
4 + 3i;
4 + 0i];
[apOut, w, z, info] = f08gq(job, uplo, n, ap)
```
```

apOut =

1.0000 + 0.0000i
-5.6569 + 0.0000i
0.4020 + 0.0781i
0.5304 + 0.0613i
9.6250 + 0.0000i
-4.8846 + 0.0000i
0.4351 - 0.7543i
-0.6898 + 0.0000i
1.4423 + 0.0000i
0.0648 + 0.0000i

w =

-4.2443
-0.6886
1.1412
13.7916

z =

-0.4836 + 0.0000i  -0.6470 + 0.0000i  -0.4456 + 0.0000i  -0.3859 + 0.0000i
-0.2912 + 0.3618i   0.4984 + 0.1130i  -0.0230 - 0.5702i  -0.4441 + 0.0156i
0.3163 + 0.3696i  -0.2949 - 0.3165i   0.5331 + 0.1317i  -0.5173 - 0.0844i
0.4447 - 0.3406i   0.2241 + 0.2878i  -0.3510 + 0.2261i  -0.5277 - 0.3168i

info =

0

```