F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GNF (ZHPEV)

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

F08GNF (ZHPEV) computes all the eigenvalues and, optionally, all the eigenvectors of a complex $n$ by $n$ Hermitian matrix $A$ in packed storage.

## 2  Specification

 SUBROUTINE F08GNF ( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, RWORK, INFO)
 INTEGER N, LDZ, INFO REAL (KIND=nag_wp) W(N), RWORK(3*N-2) COMPLEX (KIND=nag_wp) AP(*), Z(LDZ,*), WORK(2*N-1) CHARACTER(1) JOBZ, UPLO
The routine may be called by its LAPACK name zhpev.

## 3  Description

The Hermitian matrix $A$ is first reduced to real tridiagonal form, using unitary similarity transformations, and then the $QR$ algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

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

## 5  Parameters

1:     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 $\text{'V'}$.
2:     UPLO – CHARACTER(1)Input
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangular part of $A$ is stored.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangular part of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     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: AP is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of $A$.
5:     W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
6:     Z(LDZ,$*$) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{JOBZ}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{JOBZ}}=\text{'V'}$, Z contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{W}}\left(i\right)$.
If ${\mathbf{JOBZ}}=\text{'N'}$, Z is not referenced.
7:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08GNF (ZHPEV) is called.
Constraints:
• if ${\mathbf{JOBZ}}=\text{'V'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• otherwise ${\mathbf{LDZ}}\ge 1$.
8:     WORK($2×{\mathbf{N}}-1$) – COMPLEX (KIND=nag_wp) arrayWorkspace
9:     RWORK($3×{\mathbf{N}}-2$) – REAL (KIND=nag_wp) arrayWorkspace
10:   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.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the algorithm failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

## 7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8  Further Comments

Each eigenvector is normalized so that the element of largest absolute value is real and positive.
The total number of floating point operations is proportional to ${n}^{3}$.
The real analogue of this routine is F08GAF (DSPEV).

## 9  Example

This example finds all the eigenvalues of the Hermitian matrix
 $A = 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 ,$
together with approximate error bounds for the computed eigenvalues.

### 9.1  Program Text

Program Text (f08gnfe.f90)

### 9.2  Program Data

Program Data (f08gnfe.d)

### 9.3  Program Results

Program Results (f08gnfe.r)