# NAG Library Routine Document

## 1Purpose

f08tqf (zhpgvd) 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, stored in packed format, and $B$ is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

## 2Specification

Fortran Interface
 Subroutine f08tqf ( jobz, uplo, n, ap, bp, w, z, ldz, work, info)
 Integer, Intent (In) :: itype, n, ldz, lwork, lrwork, liwork Integer, Intent (Out) :: iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (Out) :: w(n), rwork(max(1,lrwork)) Complex (Kind=nag_wp), Intent (Inout) :: ap(*), bp(*), z(ldz,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobz, uplo
#include nagmk26.h
 void f08tqf_ ( const Integer *itype, const char *jobz, const char *uplo, const Integer *n, Complex ap[], Complex bp[], double w[], Complex z[], const Integer *ldz, Complex work[], const Integer *lwork, double rwork[], const Integer *lrwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_uplo)
The routine may be called by its LAPACK name zhpgvd.

## 3Description

f08tqf (zhpgvd) first performs a Cholesky factorization of the matrix $B$ as $B={U}^{\mathrm{H}}U$, when ${\mathbf{uplo}}=\text{'U'}$ or $B=L{L}^{\mathrm{H}}$, when ${\mathbf{uplo}}=\text{'L'}$. 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
 $ZH A Z = Λ and ZH B-1 Z = I .$

## 4References

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

## 5Arguments

1:     $\mathbf{itype}$ – IntegerInput
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 $3$.
2:     $\mathbf{jobz}$ – Character(1)Input
On entry: indicates whether eigenvectors are computed.
${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{jobz}}=\text{'N'}$ or $\text{'V'}$.
3:     $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension 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 $n$ by $n$ Hermitian matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: the contents of ap are destroyed.
6:     $\mathbf{bp}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension 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 upper or lower triangle of the $n$ by $n$ Hermitian matrix $B$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $B$ must be stored with element ${B}_{ij}$ in ${\mathbf{bp}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $B$ must be stored with element ${B}_{ij}$ in ${\mathbf{bp}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: the triangular factor $U$ or $L$ from the Cholesky factorization $B={U}^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$, in the same storage format as $B$.
7:     $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
8:     $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobz}}=\text{'V'}$, z 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 ${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
9:     $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08tqf (zhpgvd) is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
10:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
11:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08tqf (zhpgvd) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Constraints:
• if ${\mathbf{n}}\le 1$, ${\mathbf{lwork}}\ge 1$;
• if ${\mathbf{jobz}}=\text{'N'}$ and ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge {\mathbf{n}}$;
• if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge 2×{\mathbf{n}}$.
12:   $\mathbf{rwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lrwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{rwork}}\left(1\right)$ returns the optimal lrwork.
13:   $\mathbf{lrwork}$ – IntegerInput
On entry: the dimension of the array rwork as declared in the (sub)program from which f08tqf (zhpgvd) is called.
If ${\mathbf{lrwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Constraints:
• if ${\mathbf{n}}\le 1$, ${\mathbf{lrwork}}\ge 1$;
• if ${\mathbf{jobz}}=\text{'N'}$ and ${\mathbf{n}}>1$, ${\mathbf{lrwork}}\ge {\mathbf{n}}$;
• if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{lrwork}}\ge 1+5×{\mathbf{n}}+2×{{\mathbf{n}}}^{2}$.
14:   $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$ – Integer arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ returns the optimal liwork.
15:   $\mathbf{liwork}$ – IntegerInput
On entry: the dimension of the array iwork as declared in the (sub)program from which f08tqf (zhpgvd) is called.
If ${\mathbf{liwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork, lrwork or liwork is issued.
Constraints:
• if ${\mathbf{jobz}}=\text{'N'}$ or ${\mathbf{n}}\le 1$, ${\mathbf{liwork}}\ge 1$;
• if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{liwork}}\ge 3+5×{\mathbf{n}}$.
16:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
f07grf (zpptrf) or f08gqf (zhpevd) returned an error code:
 $\le {\mathbf{n}}$ if ${\mathbf{info}}=i$, f08gqf (zhpevd) failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero; $>{\mathbf{n}}$ if ${\mathbf{info}}={\mathbf{n}}+i$, for $1\le i\le {\mathbf{n}}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

## 7Accuracy

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.

## 8Parallelism and Performance

f08tqf (zhpgvd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08tqf (zhpgvd) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this routine is f08tcf (dspgvd).

## 10Example

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 f08tnf (zhpgv) illustrates solving a generalized Hermitian eigenproblem of the form $Az=\lambda Bz$.

### 10.1Program Text

Program Text (f08tqfe.f90)

### 10.2Program Data

Program Data (f08tqfe.d)

### 10.3Program Results

Program Results (f08tqfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017