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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhpev (f08gn)

## Purpose

nag_lapack_zhpev (f08gn) computes all the eigenvalues and, optionally, all the eigenvectors of a complex n$n$ by n$n$ Hermitian matrix A$A$ in packed storage.

## Syntax

[ap, w, z, info] = f08gn(jobz, uplo, n, ap)
[ap, w, z, info] = nag_lapack_zhpev(jobz, uplo, n, ap)

## Description

The Hermitian matrix A$A$ is first reduced to real tridiagonal form, using unitary 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:     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.

ldz work rwork

### 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(n) – double array
The eigenvalues in ascending order.
3:     z(ldz, : $:$) – complex 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: ap, 5: w, 6: z, 7: ldz, 8: work, 9: rwork, 10: 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.

Each eigenvector is normalized so that the element of largest absolute value is real and positive.
The total number of floating point operations is proportional to n3${n}^{3}$.
The real analogue of this function is nag_lapack_dspev (f08ga).

## Example

```function nag_lapack_zhpev_example
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [complex(0);
0 + 0i;
0 + 0i;
3 - 1i;
3 - 2i;
3 + 0i;
4 - 1i;
4 - 2i;
4 - 3i;
4 + 0i];
[apOut, w, z, info] = nag_lapack_zhpev(jobz, uplo, n, ap)
```
```

apOut =

-0.0049 + 0.0000i
0.1559 + 0.0000i
-1.6564 + 0.0000i
0.3559 + 0.1524i
-3.9039 + 0.0000i
4.6613 + 0.0000i
0.3367 + 0.0008i
0.3567 - 0.0783i
-7.8740 + 0.0000i
4.0000 + 0.0000i

w =

-5.6248
-0.2182
0.0638
12.7793

z =

1.9763e-322 +6.9164e-310i

info =

0

```
```function f08gn_example
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [complex(0);
0 + 0i;
0 + 0i;
3 - 1i;
3 - 2i;
3 + 0i;
4 - 1i;
4 - 2i;
4 - 3i;
4 + 0i];
[apOut, w, z, info] = f08gn(jobz, uplo, n, ap)
```
```

apOut =

-0.0049 + 0.0000i
0.1559 + 0.0000i
-1.6564 + 0.0000i
0.3559 + 0.1524i
-3.9039 + 0.0000i
4.6613 + 0.0000i
0.3367 + 0.0008i
0.3567 - 0.0783i
-7.8740 + 0.0000i
4.0000 + 0.0000i

w =

-5.6248
-0.2182
0.0638
12.7793

z =

0.0000e+00 +6.9024e-310i

info =

0

```