f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhegst (f08ssc)

## 1  Purpose

nag_zhegst (f08ssc) reduces a complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, where $A$ is a complex Hermitian matrix and $B$ has been factorized by nag_zpotrf (f07frc).

## 2  Specification

 #include #include
 void nag_zhegst (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Complex b[], Integer pdb, NagError *fail)

## 3  Description

To reduce the complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, nag_zhegst (f08ssc) must be preceded by a call to nag_zpotrf (f07frc) which computes the Cholesky factorization of $B$; $B$ must be positive definite.
The different problem types are specified by the argument comp_type, as indicated in the table below. The table shows how $C$ is computed by the function, and also how the eigenvectors $z$ of the original problem can be recovered from the eigenvectors of the standard form.
 ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ comp_type Problem uplo $B$ $C$ $z$ $B$ $C$ $z$ $1$ $Az=\lambda Bz$ $\mathrm{Nag_Upper}$  $\mathrm{Nag_Lower}$ ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ ${U}^{-\mathrm{H}}A{U}^{-1}$  ${L}^{-1}A{L}^{-\mathrm{H}}$ ${U}^{-1}y$  ${L}^{-\mathrm{H}}y$ $U{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}L$ ${U}^{-1}A{U}^{-\mathrm{H}}$  ${L}^{-\mathrm{H}}A{L}^{-1}$ ${U}^{-\mathrm{H}}y$  ${L}^{-1}y$ $2$ $ABz=\lambda z$ $\mathrm{Nag_Upper}$  $\mathrm{Nag_Lower}$ ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ $UA{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}AL$ ${U}^{-1}y$  ${L}^{-\mathrm{H}}y$ $U{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}L$ ${U}^{\mathrm{H}}AU$  $LA{L}^{\mathrm{H}}$ ${U}^{-\mathrm{H}}y$  ${L}^{-1}y$ $3$ $BAz=\lambda z$ $\mathrm{Nag_Upper}$  $\mathrm{Nag_Lower}$ ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ $UA{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}AL$ ${U}^{\mathrm{H}}y$  $Ly$ $U{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}L$ ${U}^{\mathrm{H}}AU$  $LA{L}^{\mathrm{H}}$ $Uy$  ${L}^{\mathrm{H}}y$

## 4  References

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:     comp_typeNag_ComputeTypeInput
On entry: indicates how the standard form is computed.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C={U}^{-\mathrm{H}}A{U}^{-1}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={U}^{-1}A{U}^{-\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{-1}A{L}^{-\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={L}^{-\mathrm{H}}A{L}^{-1}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C=UA{U}^{\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={U}^{\mathrm{H}}AU$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{\mathrm{H}}AL$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C=LA{L}^{\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Constraint: ${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$, $\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$.
3:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored and $B={U}^{\mathrm{H}}U$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $B=U{U}^{\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{H}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $B={L}^{\mathrm{H}}L$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:     nIntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     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: the upper or lower triangle of a is overwritten by the corresponding upper or lower triangle of $C$ as specified by comp_type and uplo.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     b[$\mathit{dim}$]const ComplexInput
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 Cholesky factor of $B$ as specified by uplo and returned by nag_zpotrf (f07frc).
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $B$ in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     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_INT
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.

## 7  Accuracy

Forming the reduced matrix $C$ is a stable procedure. However it involves implicit multiplication by ${B}^{-1}$ (if ${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$) or $B$ (if ${\mathbf{comp_type}}=\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$). When nag_zhegst (f08ssc) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if $B$ is ill-conditioned with respect to inversion.

## 8  Parallelism and Performance

nag_zhegst (f08ssc) is not threaded by NAG in any implementation.
nag_zhegst (f08ssc) 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 real floating-point operations is approximately $4{n}^{3}$.
The real analogue of this function is nag_dsygst (f08sec).

## 10  Example

This example computes all the eigenvalues of $Az=\lambda Bz$, where
 $A = -7.36+0.00i 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49+0.00i 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12+0.00i 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54+0.00i$
and
 $B = 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $B$ is Hermitian positive definite and must first be factorized by nag_zpotrf (f07frc). The program calls nag_zhegst (f08ssc) to reduce the problem to the standard form $Cy=\lambda y$; then nag_zhetrd (f08fsc) to reduce $C$ to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

### 10.1  Program Text

Program Text (f08ssce.c)

### 10.2  Program Data

Program Data (f08ssce.d)

### 10.3  Program Results

Program Results (f08ssce.r)