Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
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 $QL$ or $QR$ 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 $A$. In other words, it can compute the spectral factorization of $A$ as
 $A=ZΛZH,$
where $\Lambda$ is a real diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_{i}$, and $Z$ is the (complex) unitary matrix whose columns are the eigenvectors ${z}_{i}$. Thus
 $Azi=λizi, i=1,2,…,n.$

## 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:     $\mathrm{job}$ – string (length ≥ 1)
Indicates whether eigenvectors are computed.
${\mathbf{job}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{job}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\text{'N'}$ or $\text{'V'}$.
2:     $\mathrm{uplo}$ – string (length ≥ 1)
Indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ Hermitian matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{job}}=\text{'V'}$, a stores the unitary matrix $Z$ which contains the eigenvectors of $A$.
2:     $\mathrm{w}\left(:\right)$ – double array
The dimension of the array w will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The eigenvalues of the matrix $A$ in ascending order.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ 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.
${\mathbf{info}}>0$
if ${\mathbf{info}}=i$ and ${\mathbf{job}}=\text{'N'}$, the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if ${\mathbf{info}}=i$ and ${\mathbf{job}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i/\left({\mathbf{n}}+1\right)$ through .

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ 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

This example computes all the eigenvalues and eigenvectors of the Hermitian matrix $A$, where
 $A = 1.0+0.0i 2.0-1.0i 3.0-1.0i 4.0-1.0i 2.0+1.0i 2.0+0.0i 3.0-2.0i 4.0-2.0i 3.0+1.0i 3.0+2.0i 3.0+0.0i 4.0-3.0i 4.0+1.0i 4.0+2.0i 4.0+3.0i 4.0+0.0i .$
The example program for nag_lapack_zheevd (f08fq) illustrates the computation of error bounds for the eigenvalues and eigenvectors.
```function f08fq_example

fprintf('f08fq example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'L';
n = 4;
a = [ 1 + 0i,  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];

% Calculate all the eigenvalues and eigenvectors of A
job = 'Vectors';
[z, w, info] = f08fq( ...
job, uplo, a);

% Normalize vectors, largest element is real and positive.
for i = 1:n
[~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

% Display results
fprintf('Eigenvalues:\n');
disp(w);

ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', z, 'Bracketed', 'F7.4', ...
'Eigenvectors', 'Integer', 'Integer', ...
ncols, indent);

```
```f08fq example results

Eigenvalues:
-4.2443
-0.6886
1.1412
13.7916

Eigenvectors
1                 2                 3                 4
1  (-0.3839,-0.2941) ( 0.6470, 0.0000) (-0.4326, 0.1068) ( 0.3309,-0.1986)
2  (-0.4512, 0.1102) (-0.4984,-0.1130) (-0.1590,-0.5480) ( 0.3728,-0.2419)
3  ( 0.0263, 0.4857) ( 0.2949, 0.3165) ( 0.5491, 0.0000) ( 0.4870,-0.1938)
4  ( 0.5602, 0.0000) (-0.2241,-0.2878) (-0.2865, 0.3037) ( 0.6155, 0.0000)
```