# NAG CL Interfacef08tsc (zhpgst)

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## 1Purpose

f08tsc 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 f07grc, using packed storage.

## 2Specification

 #include
 void f08tsc (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, Complex ap[], const Complex bp[], NagError *fail)
The function may be called by the names: f08tsc, nag_lapackeig_zhpgst or nag_zhpgst.

## 3Description

To reduce the complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$ using packed storage, f08tsc must be preceded by a call to f07grc 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.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface 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_ComputeType Input
On entry: indicates how the standard form is computed.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C={U}^{-\mathrm{H}}A{U}^{-1}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{-1}A{L}^{-\mathrm{H}}$.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C=UA{U}^{\mathrm{H}}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{\mathrm{H}}AL$.
Constraint: ${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$, $\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$.
3: $\mathbf{uplo}$Nag_UploType Input
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 $B={U}^{\mathrm{H}}U$.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{H}}$.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{ap}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the upper or lower triangle of the $n×n$ Hermitian matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: the upper or lower triangle of ap is overwritten by the corresponding upper or lower triangle of $C$ as specified by comp_type and uplo, using the same packed storage format as described above.
6: $\mathbf{bp}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array bp must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the Cholesky factor of $B$ as specified by uplo and returned by f07grc.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

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 f08tsc 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.

## 8Parallelism and Performance

f08tsc 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 f08tec.

## 10Example

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
 $B = ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) ,$
using packed storage. Here $B$ is Hermitian positive definite and must first be factorized by f07grc. The program calls f08tsc to reduce the problem to the standard form $Cy=\lambda y$; then f08gsc to reduce $C$ to tridiagonal form, and f08jfc to compute the eigenvalues.

### 10.1Program Text

Program Text (f08tsce.c)

### 10.2Program Data

Program Data (f08tsce.d)

### 10.3Program Results

Program Results (f08tsce.r)