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# 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.

## Further Comments

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

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Chapter Contents
Chapter Introduction
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