f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhegvd (f08sqc)

## 1  Purpose

nag_zhegvd (f08sqc) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
 $Az=λBz , ABz=λz or BAz=λz ,$
where $A$ and $B$ are Hermitian and $B$ is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

## 2  Specification

 #include #include
 void nag_zhegvd (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_UploType uplo, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, double w[], NagError *fail)

## 3  Description

nag_zhegvd (f08sqc) first performs a Cholesky factorization of the matrix $B$ as $B={U}^{\mathrm{H}}U$, when ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $B=L{L}^{\mathrm{H}}$, when ${\mathbf{uplo}}=\mathrm{Nag_Lower}$. The generalized problem is then reduced to a standard symmetric eigenvalue problem
 $Cx=λx ,$
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 $Az=\lambda Bz$, the eigenvectors are normalized so that the matrix of eigenvectors, $z$, satisfies
 $ZH A Z = Λ and ZH B Z = I ,$
where $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem $ABz=\lambda z$ we correspondingly have
 $Z-1 A Z-H = Λ and ZH B Z = I ,$
and for $BAz=\lambda z$ we have
 $ZH A Z = Λ and ZH B-1 Z = I .$

## 4  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

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     itypeIntegerInput
On entry: specifies the problem type to be solved.
${\mathbf{itype}}=1$
$Az=\lambda Bz$.
${\mathbf{itype}}=2$
$ABz=\lambda z$.
${\mathbf{itype}}=3$
$BAz=\lambda z$.
Constraint: ${\mathbf{itype}}=1$, $2$ or $3$.
3:     jobNag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Only eigenvalues are computed.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$.
4:     uploNag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangles of $A$ and $B$ are stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangles of $A$ and $B$ are stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
5:     nIntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, a contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:
• if ${\mathbf{itype}}=1$ or $2$, ${Z}^{\mathrm{H}}BZ=I$;
• if ${\mathbf{itype}}=3$, ${Z}^{\mathrm{H}}{B}^{-1}Z=I$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, the upper triangle (if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$) or the lower triangle (if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$) of a, including the diagonal, is overwritten.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     b[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $B$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${B}_{ij}$ is stored in ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $B$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $B$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the triangular factor $U$ or $L$ from the Cholesky factorization $B={U}^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$.
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   w[n]doubleOutput
On exit: the eigenvalues in ascending order.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; $⟨\mathit{\text{value}}⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_INT
On entry, ${\mathbf{itype}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{itype}}=1$, $2$ or $3$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_MAT_NOT_POS_DEF
If ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}={\mathbf{n}}+⟨\mathit{\text{value}}⟩$, for $1\le ⟨\mathit{\text{value}}⟩\le {\mathbf{n}}$, then the leading minor of order $⟨\mathit{\text{value}}⟩$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

## 7  Accuracy

If $B$ 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 $B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of $B$ 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.

## 8  Parallelism and Performance

nag_zhegvd (f08sqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhegvd (f08sqc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_dsygvd (f08scc).

## 10  Example

This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem $ABz=\lambda z$, where
 $A = -7.36i+0.00 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49i+0.00 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12i+0.00 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54i+0.00$
and
 $B = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00 ,$
together with an estimate of the condition number of $B$, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for nag_zhegv (f08snc) illustrates solving a generalized Hermitian eigenproblem of the form $Az=\lambda Bz$.

### 10.1  Program Text

Program Text (f08sqce.c)

### 10.2  Program Data

Program Data (f08sqce.d)

### 10.3  Program Results

Program Results (f08sqce.r)