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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zheevd (f08fq)

## Purpose

nag_lapack_zheevd (f08fq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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

[a, w, info] = f08fq(job, uplo, a, 'n', n)
[a, w, info] = nag_lapack_zheevd(job, uplo, a, 'n', n)

## Description

nag_lapack_zheevd (f08fq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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 n$n$ by n$n$ Hermitian matrix A$A$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of a$a$ must be stored and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of a$a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

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

### Input Parameters Omitted from the MATLAB Interface

lda work lwork rwork lrwork iwork liwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If job = 'V'${\mathbf{job}}=\text{'V'}$, a stores the unitary matrix Z$Z$ which contains the eigenvectors 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:     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: a, 5: lda, 6: w, 7: work, 8: lwork, 9: rwork, 10: lrwork, 11: iwork, 12: liwork, 13: 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_dsyevd (f08fc).

## Example

```function nag_lapack_zheevd_example
job = 'V';
uplo = 'L';
a = [complex(1),  0 + 0i,  0 + 0i,  0 + 0i;
2 + 1i,  2 + 0i,  0 + 0i,  0 + 0i;
3 + 1i,  3 + 2i,  3 + 0i,  0 + 0i;
4 + 1i,  4 + 2i,  4 + 3i,  4 + 0i];
[aOut, w, info] = nag_lapack_zheevd(job, uplo, a)
```
```

aOut =

-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

w =

-4.2443
-0.6886
1.1412
13.7916

info =

0

```
```function f08fq_example
job = 'V';
uplo = 'L';
a = [complex(1),  0 + 0i,  0 + 0i,  0 + 0i;
2 + 1i,  2 + 0i,  0 + 0i,  0 + 0i;
3 + 1i,  3 + 2i,  3 + 0i,  0 + 0i;
4 + 1i,  4 + 2i,  4 + 3i,  4 + 0i];
[aOut, w, info] = f08fq(job, uplo, a)
```
```

aOut =

-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

w =

-4.2443
-0.6886
1.1412
13.7916

info =

0

```