F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08TPF (ZHPGVX)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08TPF (ZHPGVX) computes selected eigenvalues and, optionally, 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## 2  Specification

 SUBROUTINE F08TPF ( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, JFAIL, INFO)
 INTEGER ITYPE, N, IL, IU, M, LDZ, IWORK(5*N), JFAIL(*), INFO REAL (KIND=nag_wp) VL, VU, ABSTOL, W(N), RWORK(7*N) COMPLEX (KIND=nag_wp) AP(*), BP(*), Z(LDZ,*), WORK(2*N) CHARACTER(1) JOBZ, RANGE, UPLO
The routine may be called by its LAPACK name zhpgvx.

## 3  Description

F08TPF (ZHPGVX) 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 desired eigenvalues and 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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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:     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:     RANGE – CHARACTER(1)Input
On entry: if ${\mathbf{RANGE}}=\text{'A'}$, all eigenvalues will be found.
If ${\mathbf{RANGE}}=\text{'V'}$, all eigenvalues in the half-open interval $\left({\mathbf{VL}},{\mathbf{VU}}\right]$ will be found.
If ${\mathbf{RANGE}}=\text{'I'}$, the ILth to IUth eigenvalues will be found.
Constraint: ${\mathbf{RANGE}}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.
4:     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 $\text{'L'}$.
5:     N – INTEGERInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     AP($*$) – 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.
7:     BP($*$) – 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 $B$.
8:     VL – REAL (KIND=nag_wp)Input
9:     VU – REAL (KIND=nag_wp)Input
On entry: if ${\mathbf{RANGE}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{RANGE}}=\text{'A'}$ or $\text{'I'}$, VL and VU are not referenced.
Constraint: if ${\mathbf{RANGE}}=\text{'V'}$, ${\mathbf{VL}}<{\mathbf{VU}}$.
10:   IL – INTEGERInput
11:   IU – INTEGERInput
On entry: if ${\mathbf{RANGE}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If ${\mathbf{RANGE}}=\text{'A'}$ or $\text{'V'}$, IL and IU are not referenced.
Constraints:
• if ${\mathbf{RANGE}}=\text{'I'}$ and ${\mathbf{N}}=0$, ${\mathbf{IL}}=1$ and ${\mathbf{IU}}=0$;
• if ${\mathbf{RANGE}}=\text{'I'}$ and ${\mathbf{N}}>0$, $1\le {\mathbf{IL}}\le {\mathbf{IU}}\le {\mathbf{N}}$.
12:   ABSTOL – REAL (KIND=nag_wp)Input
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to
 $ABSTOL+ε maxa,b ,$
where $\epsilon$ is the machine precision. If ABSTOL is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold , not zero. If this routine returns with , indicating that some eigenvectors did not converge, try setting ABSTOL to . See Demmel and Kahan (1990).
13:   M – INTEGEROutput
On exit: the total number of eigenvalues found. $0\le {\mathbf{M}}\le {\mathbf{N}}$.
If ${\mathbf{RANGE}}=\text{'A'}$, ${\mathbf{M}}={\mathbf{N}}$.
If ${\mathbf{RANGE}}=\text{'I'}$, ${\mathbf{M}}={\mathbf{IU}}-{\mathbf{IL}}+1$.
14:   W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the first M elements contain the selected eigenvalues in ascending order.
15:   Z(LDZ,$*$) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ if ${\mathbf{JOBZ}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{JOBZ}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first M columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{W}}\left(i\right)$. 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 an eigenvector fails to converge (), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in JFAIL.
If ${\mathbf{JOBZ}}=\text{'N'}$, Z is not referenced.
Note:  you must ensure that at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ columns are supplied in the array Z; if ${\mathbf{RANGE}}=\text{'V'}$, the exact value of M is not known in advance and an upper bound of at least N must be used.
16:   LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08TPF (ZHPGVX) is called.
Constraints:
• if ${\mathbf{JOBZ}}=\text{'V'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• otherwise ${\mathbf{LDZ}}\ge 1$.
17:   WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
18:   RWORK($7×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
19:   IWORK($5×{\mathbf{N}}$) – INTEGER arrayWorkspace
20:   JFAIL($*$) – INTEGER arrayOutput
Note: the dimension of the array JFAIL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On exit: if ${\mathbf{JOBZ}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first M elements of JFAIL are zero;
• if , JFAIL contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{JOBZ}}=\text{'N'}$, JFAIL is not referenced.
21:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
If ${\mathbf{INFO}}=i$, F08GPF (ZHPEVX) failed to converge; $i$ eigenvectors failed to converge. Their indices are stored in array JFAIL.
${\mathbf{INFO}}>{\mathbf{N}}$
F07GRF (ZPPTRF) returned an error code; i.e., 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.

## 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 total number of floating point operations is proportional to ${n}^{3}$.
The real analogue of this routine is F08TBF (DSPGVX).

## 9  Example

This example finds the eigenvalues in the half-open interval $\left(-3,3\right]$, and corresponding eigenvectors, of the generalized Hermitian eigenproblem $Az=\lambda Bz$, 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
 $B = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00 .$
The example program for F08TQF (ZHPGVD) illustrates solving a generalized symmetric eigenproblem of the form $ABz=\lambda z$.

### 9.1  Program Text

Program Text (f08tpfe.f90)

### 9.2  Program Data

Program Data (f08tpfe.d)

### 9.3  Program Results

Program Results (f08tpfe.r)