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# NAG Toolbox: nag_lapack_zhegv (f08sn)

## Purpose

nag_lapack_zhegv (f08sn) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
 Az = λBz ,   ABz = λz   or   BAz = λz , $Az=λBz , ABz=λz or BAz=λz ,$
where A$A$ and B$B$ are Hermitian and B$B$ is also positive definite.

## Syntax

[a, b, w, info] = f08sn(itype, jobz, uplo, a, b, 'n', n)
[a, b, w, info] = nag_lapack_zhegv(itype, jobz, uplo, a, b, 'n', n)

## Description

nag_lapack_zhegv (f08sn) first performs a Cholesky factorization of the matrix B$B$ as B = UHU $B={U}^{\mathrm{H}}U$, when uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or B = LLH $B=L{L}^{\mathrm{H}}$, when uplo = 'L'${\mathbf{uplo}}=\text{'L'}$. The generalized problem is then reduced to a standard symmetric eigenvalue problem
 Cx = λx , $Cx=λx ,$
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az = λBz $Az=\lambda Bz$, the eigenvectors are normalized so that the matrix of eigenvectors, z$z$, satisfies
 ZH A Z = Λ   and   ZH B Z = I , $ZH A Z = Λ and ZH B Z = I ,$
where Λ $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z $ABz=\lambda z$ we correspondingly have
 Z − 1 A Z − H = Λ   and   ZH B Z = I , $Z-1 A Z-H = Λ and ZH B Z = I ,$
and for B A z = λ z $BAz=\lambda z$ we have
 ZH A Z = Λ   and   ZH B − 1 Z = I . $ZH A Z = Λ and ZH B-1 Z = I .$

## 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:     itype – int64int32nag_int scalar
Specifies the problem type to be solved.
itype = 1${\mathbf{itype}}=1$
Az = λBz$Az=\lambda Bz$.
itype = 2${\mathbf{itype}}=2$
ABz = λz$ABz=\lambda z$.
itype = 3${\mathbf{itype}}=3$
BAz = λz$BAz=\lambda z$.
Constraint: itype = 1${\mathbf{itype}}=1$, 2$2$ or 3$3$.
2:     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'$\text{'V'}$.
3:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangles of A$A$ and B$B$ are stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangles of A$A$ and B$B$ are stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
4:     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.
5:     b(ldb, : $:$) – complex array
The first dimension of the array b 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 positive definite matrix B$B$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of b$b$ 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 b$b$ 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 arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork rwork

### 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 jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, a contains the matrix Z$Z$ of eigenvectors. The eigenvectors are normalized as follows:
• if itype = 1${\mathbf{itype}}=1$ or 2$2$, ZHBZ = I${Z}^{\mathrm{H}}BZ=I$;
• if itype = 3${\mathbf{itype}}=3$, ZHB1Z = I${Z}^{\mathrm{H}}{B}^{-1}Z=I$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, the upper triangle (if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$) or the lower triangle (if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$) of a, including the diagonal, is overwritten.
2:     b(ldb, : $:$) – complex array
The first dimension of the array b 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If 0infon$0\le {\mathbf{info}}\le {\mathbf{n}}$, the part of b containing the matrix stores the triangular factor U$U$ or L$L$ from the Cholesky factorization B = UHU$B={U}^{\mathrm{H}}U$ or B = LLH$B=L{L}^{\mathrm{H}}$.
3:     w(n) – double array
The eigenvalues in ascending order.
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: itype, 2: jobz, 3: uplo, 4: n, 5: a, 6: lda, 7: b, 8: ldb, 9: w, 10: work, 11: lwork, 12: rwork, 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 = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
If info = i${\mathbf{info}}=i$, nag_lapack_zheev (f08fn) failed to converge; i$i$ i$i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
${\mathbf{INFO}}>{\mathbf{N}}$
nag_lapack_zpotrf (f07fr) returned an error code; i.e., if info = n + i${\mathbf{info}}={\mathbf{n}}+i$, for 1in$1\le i\le {\mathbf{n}}$, then the leading minor of order i$i$ of B$B$ is not positive definite. The factorization of B$B$ could not be completed and no eigenvalues or eigenvectors were computed.

## Accuracy

If B$B$ is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B$B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B$B$ would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.

The total number of floating point operations is proportional to n3${n}^{3}$.
The real analogue of this function is nag_lapack_dsygv (f08sa).

## Example

```function nag_lapack_zhegv_example
itype = int64(1);
jobz = 'Vectors';
uplo = 'Upper';
a = [-7.36,  0.77 - 0.43i,  -0.64 - 0.92i,  3.01 - 6.97i;
0 + 0i,  3.49 + 0i,  2.19 + 4.45i,  1.9 + 3.73i;
0 + 0i,  0 + 0i,  0.12 + 0i,  2.88 - 3.17i;
0 + 0i,  0 + 0i,  0 + 0i,  -2.54 + 0i];
b = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[aOut, bOut, w, info] = nag_lapack_zhegv(itype, jobz, uplo, a, b)
```
```

aOut =

-0.7721 + 1.5598i  -0.3504 + 0.6060i   0.2835 - 0.5806i   0.2310 - 1.2161i
0.6038 - 0.1627i  -0.0993 + 0.0631i  -0.3769 - 0.3194i  -0.4710 + 0.4814i
0.5954 - 0.6430i   0.6851 - 0.5987i  -0.3338 - 0.0134i  -0.2242 + 0.6335i
-0.6810 + 0.0000i  -0.8127 + 0.0000i   0.6663 + 0.0000i   0.8515 + 0.0000i

bOut =

1.7972 + 0.0000i   0.8402 - 1.0683i   1.0572 + 0.4674i   0.2337 + 1.3910i
0.0000 + 0.0000i   1.3164 + 0.0000i  -0.4702 - 0.3131i   0.0834 - 0.0368i
0.0000 + 0.0000i   0.0000 + 0.0000i   1.5604 + 0.0000i   0.9360 - 0.9900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.6603 + 0.0000i

w =

-5.9990
-2.9936
0.5047
3.9990

info =

0

```
```function f08sn_example
itype = int64(1);
jobz = 'Vectors';
uplo = 'Upper';
a = [-7.36,  0.77 - 0.43i,  -0.64 - 0.92i,  3.01 - 6.97i;
0 + 0i,  3.49 + 0i,  2.19 + 4.45i,  1.9 + 3.73i;
0 + 0i,  0 + 0i,  0.12 + 0i,  2.88 - 3.17i;
0 + 0i,  0 + 0i,  0 + 0i,  -2.54 + 0i];
b = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[aOut, bOut, w, info] = f08sn(itype, jobz, uplo, a, b)
```
```

aOut =

-0.7721 + 1.5598i  -0.3504 + 0.6060i   0.2835 - 0.5806i   0.2310 - 1.2161i
0.6038 - 0.1627i  -0.0993 + 0.0631i  -0.3769 - 0.3194i  -0.4710 + 0.4814i
0.5954 - 0.6430i   0.6851 - 0.5987i  -0.3338 - 0.0134i  -0.2242 + 0.6335i
-0.6810 + 0.0000i  -0.8127 + 0.0000i   0.6663 + 0.0000i   0.8515 + 0.0000i

bOut =

1.7972 + 0.0000i   0.8402 - 1.0683i   1.0572 + 0.4674i   0.2337 + 1.3910i
0.0000 + 0.0000i   1.3164 + 0.0000i  -0.4702 - 0.3131i   0.0834 - 0.0368i
0.0000 + 0.0000i   0.0000 + 0.0000i   1.5604 + 0.0000i   0.9360 - 0.9900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.6603 + 0.0000i

w =

-5.9990
-2.9936
0.5047
3.9990

info =

0

```

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