F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GQF (ZHPEVD)

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.
Warning. The specification of the parameters LRWORK and LIWORK changed at Mark 20 in the case where ${\mathbf{JOB}}=\text{'V'}$ and ${\mathbf{N}}>1$: the minimum dimension of the array RWORK has been reduced whereas the minimum dimension of the array IWORK has been increased.

## 1  Purpose

F08GQF (ZHPEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix held in packed storage. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm.

## 2  Specification

 SUBROUTINE F08GQF ( JOB, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
 INTEGER N, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO REAL (KIND=nag_wp) W(*), RWORK(max(1,LRWORK)) COMPLEX (KIND=nag_wp) AP(*), Z(LDZ,*), WORK(max(1,LWORK)) CHARACTER(1) JOB, UPLO
The routine may be called by its LAPACK name zhpevd.

## 3  Description

F08GQF (ZHPEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix $A$ (held in packed storage). In other words, it can compute the spectral factorization of $A$ as
 $A=ZΛZH,$
where $\Lambda$ is a real diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_{i}$, and $Z$ is the (complex) unitary matrix whose columns are the eigenvectors ${z}_{i}$. Thus
 $Azi=λizi, i=1,2,…,n.$

## 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:     JOB – CHARACTER(1)Input
On entry: indicates whether eigenvectors are computed.
${\mathbf{JOB}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{JOB}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{JOB}}=\text{'N'}$ or $\text{'V'}$.
2:     UPLO – CHARACTER(1)Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{UPLO}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{UPLO}}=\text{'L'}$
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($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array W must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On exit: the eigenvalues of the matrix $A$ 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 ${\mathbf{JOB}}=\text{'V'}$ and at least $1$ if ${\mathbf{JOB}}=\text{'N'}$.
On exit: if ${\mathbf{JOB}}=\text{'V'}$, Z is overwritten by the unitary matrix $Z$ which contains the eigenvectors of $A$.
If ${\mathbf{JOB}}=\text{'N'}$, Z is not referenced.
7:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08GQF (ZHPEVD) is called.
Constraints:
• if ${\mathbf{JOB}}=\text{'V'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{JOB}}=\text{'N'}$, ${\mathbf{LDZ}}\ge 1$.
8:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the required minimal size of LWORK.
9:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08GQF (ZHPEVD) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Constraints:
• if ${\mathbf{N}}\le 1$, ${\mathbf{LWORK}}\ge 1$ or ${\mathbf{LWORK}}=-1$;
• if ${\mathbf{JOB}}=\text{'N'}$ and ${\mathbf{N}}>1$, ${\mathbf{LWORK}}\ge {\mathbf{N}}$ or ${\mathbf{LWORK}}=-1$;
• if ${\mathbf{JOB}}=\text{'V'}$ and ${\mathbf{N}}>1$, ${\mathbf{LWORK}}\ge 2×{\mathbf{N}}$ or ${\mathbf{LWORK}}=-1$.
10:   RWORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LRWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{RWORK}}\left(1\right)$ contains the required minimal size of ${\mathbf{LRWORK}}$.
11:   LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which F08GQF (ZHPEVD) is called.
If ${\mathbf{LRWORK}}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued.
Constraints:
• if ${\mathbf{N}}\le 1$, ${\mathbf{LRWORK}}\ge 1$ or ${\mathbf{LRWORK}}=-1$;
• if ${\mathbf{JOB}}=\text{'N'}$ and ${\mathbf{N}}>1$, ${\mathbf{LRWORK}}\ge {\mathbf{N}}$ or ${\mathbf{LRWORK}}=-1$;
• if ${\mathbf{JOB}}=\text{'V'}$ and ${\mathbf{N}}>1$, ${\mathbf{LRWORK}}\ge 2×{{\mathbf{N}}}^{2}+5×{\mathbf{N}}+1$ or ${\mathbf{LRWORK}}=-1$.
12:   IWORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LIWORK}}\right)$) – INTEGER arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{IWORK}}\left(1\right)$ contains the required minimal size of LIWORK.
13:   LIWORK – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which F08GQF (ZHPEVD) is called.
If ${\mathbf{LIWORK}}=-1$, a workspace query is assumed; the routine only calculates the minimum dimension of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued.
Constraints:
• if ${\mathbf{JOB}}=\text{'N'}$ or ${\mathbf{N}}\le 1$, ${\mathbf{LIWORK}}\ge 1$ or ${\mathbf{LIWORK}}=-1$;
• if ${\mathbf{JOB}}=\text{'V'}$ and ${\mathbf{N}}>1$, ${\mathbf{LIWORK}}\ge 5×{\mathbf{N}}+3$ or ${\mathbf{LIWORK}}=-1$.
14:   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 ${\mathbf{INFO}}=i$ and ${\mathbf{JOB}}=\text{'N'}$, the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if ${\mathbf{INFO}}=i$ and ${\mathbf{JOB}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i/\left({\mathbf{N}}+1\right)$ through .

## 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.

The real analogue of this routine is F08GCF (DSPEVD).

## 9  Example

This example computes all the eigenvalues and eigenvectors of the Hermitian matrix $A$, where
 $A = 1.0+0.0i 2.0-1.0i 3.0-1.0i 4.0-1.0i 2.0+1.0i 2.0+0.0i 3.0-2.0i 4.0-2.0i 3.0+1.0i 3.0+2.0i 3.0+0.0i 4.0-3.0i 4.0+1.0i 4.0+2.0i 4.0+3.0i 4.0+0.0i .$

### 9.1  Program Text

Program Text (f08gqfe.f90)

### 9.2  Program Data

Program Data (f08gqfe.d)

### 9.3  Program Results

Program Results (f08gqfe.r)