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:    $\mathbf{order}$Nag_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:    $\mathbf{comp_type}$Nag_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:    $\mathbf{uplo}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$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:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{b}\left[\mathit{dim}\right]$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:    $\mathbf{pdb}$IntegerInput
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:    $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 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.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific 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)