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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhpevx (f08gp)

## Purpose

nag_lapack_zhpevx (f08gp) computes selected eigenvalues and, optionally, eigenvectors of a complex n$n$ by n$n$ Hermitian matrix A$A$ in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Syntax

[ap, m, w, z, jfail, info] = f08gp(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
[ap, m, w, z, jfail, info] = nag_lapack_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)

## Description

The Hermitian matrix A$A$ is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

## 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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
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:     range – string (length ≥ 1)
If range = 'A'${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If range = 'V'${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval (vl,vu]$\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If range = 'I'${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'${\mathbf{range}}=\text{'A'}$, 'V'$\text{'V'}$ or 'I'$\text{'I'}$.
3:     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'$\text{'L'}$.
4:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
5:     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$i\ge j$.
6:     vl – double scalar
7:     vu – double scalar
If range = 'V'${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'I'$\text{'I'}$, vl and vu are not referenced.
Constraint: if range = 'V'${\mathbf{range}}=\text{'V'}$, vl < vu${\mathbf{vl}}<{\mathbf{vu}}$.
8:     il – int64int32nag_int scalar
9:     iu – int64int32nag_int scalar
If range = 'I'${\mathbf{range}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'V'$\text{'V'}$, il and iu are not referenced.
Constraints:
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n = 0${\mathbf{n}}=0$, il = 1${\mathbf{il}}=1$ and iu = 0${\mathbf{iu}}=0$;
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n > 0${\mathbf{n}}>0$, 1 il iu n $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
10:   abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] $\left[a,b\right]$ of width less than or equal to
 abstol + ε max (|a|,|b|) , $abstol+ε max(|a|,|b|) ,$
where ε $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then ε T1 $\epsilon {‖T‖}_{1}$ will be used in its place, where T$T$ is the tridiagonal matrix obtained by reducing A$A$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am (   ) , not zero. If this function returns with ${\mathbf{INFO}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am (   ) . See Demmel and Kahan (1990).

None.

### Input Parameters Omitted from the MATLAB Interface

ldz work rwork iwork

### 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:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn$0\le {\mathbf{m}}\le {\mathbf{n}}$.
If range = 'A'${\mathbf{range}}=\text{'A'}$, m = n${\mathbf{m}}={\mathbf{n}}$.
If range = 'I'${\mathbf{range}}=\text{'I'}$, m = iuil + 1${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
3:     w(n) – double array
The selected eigenvalues in ascending order.
4:     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,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first m columns of Z$Z$ contain the orthonormal eigenvectors of the matrix A$A$ corresponding to the selected eigenvalues, with the i$i$th column of Z$Z$ holding the eigenvector associated with w(i)${\mathbf{w}}\left(i\right)$;
• if an eigenvector fails to converge (${\mathbf{INFO}}>{\mathbf{0}}$), then that column of Z$Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
5:     jfail( : $:$) – int64int32nag_int array
Note: the dimension of the array jfail must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if ${\mathbf{INFO}}>{\mathbf{0}}$, jfail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
6:     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: range, 3: uplo, 4: n, 5: ap, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: work, 16: rwork, 17: iwork, 18: jfail, 19: 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${\mathbf{info}}=i$, then i$i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.

## 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 total number of floating point operations is proportional to n3${n}^{3}$.
The real analogue of this function is nag_lapack_dspevx (f08gb).

## Example

```function nag_lapack_zhpevx_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
n = int64(4);
ap = [1;
2 - 1i;
2 + 0i;
3 - 1i;
3 - 2i;
3 + 0i;
4 - 1i;
4 - 2i;
4 - 3i;
4 + 0i];
vl = -2;
vu = 2;
il = int64(-1208245600);
iu = int64(121387641);
abstol = 0;
[apOut, m, w, z, jfail, info] = nag_lapack_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
```
```

apOut =

-0.2187 + 0.0000i
1.0422 + 0.0000i
-0.3942 + 0.0000i
0.4448 + 0.4277i
-3.4564 + 0.0000i
6.6129 + 0.0000i
0.3367 + 0.0008i
0.3567 - 0.0783i
-7.8740 + 0.0000i
4.0000 + 0.0000i

m =

2

w =

-0.6886
1.1412
0
0

z =

-0.3975 + 0.5105i  -0.3746 - 0.2414i
0.3953 - 0.3238i   0.2895 - 0.4917i
-0.4309 + 0.0383i   0.3768 + 0.3994i
0.3648 + 0.0000i  -0.4175 + 0.0000i

jfail =

0
0
0
0

info =

0

```
```function f08gp_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
n = int64(4);
ap = [1;
2 - 1i;
2 + 0i;
3 - 1i;
3 - 2i;
3 + 0i;
4 - 1i;
4 - 2i;
4 - 3i;
4 + 0i];
vl = -2;
vu = 2;
il = int64(-1208245600);
iu = int64(121387641);
abstol = 0;
[apOut, m, w, z, jfail, info] = f08gp(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
```
```

apOut =

-0.2187 + 0.0000i
1.0422 + 0.0000i
-0.3942 + 0.0000i
0.4448 + 0.4277i
-3.4564 + 0.0000i
6.6129 + 0.0000i
0.3367 + 0.0008i
0.3567 - 0.0783i
-7.8740 + 0.0000i
4.0000 + 0.0000i

m =

2

w =

-0.6886
1.1412
0
0

z =

-0.3975 + 0.5105i  -0.3746 - 0.2414i
0.3953 - 0.3238i   0.2895 - 0.4917i
-0.4309 + 0.0383i   0.3768 + 0.3994i
0.3648 + 0.0000i  -0.4175 + 0.0000i

jfail =

0
0
0
0

info =

0

```