F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08HQF (ZHBEVD)

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 arguments 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

F08HQF (ZHBEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix. 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 F08HQF ( JOB, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
 INTEGER N, KD, LDAB, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO REAL (KIND=nag_wp) W(*), RWORK(max(1,LRWORK)) COMPLEX (KIND=nag_wp) AB(LDAB,*), Z(LDZ,*), WORK(max(1,LWORK)) CHARACTER(1) JOB, UPLO
The routine may be called by its LAPACK name zhbevd.

## 3  Description

F08HQF (ZHBEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix $A$. 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  Arguments

1:     $\mathrm{JOB}$ – 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:     $\mathrm{UPLO}$ – 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:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     $\mathrm{KD}$ – INTEGERInput
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the number of superdiagonals, ${k}_{d}$, of the matrix $A$.
If ${\mathbf{UPLO}}=\text{'L'}$, the number of subdiagonals, ${k}_{d}$, of the matrix $A$.
Constraint: ${\mathbf{KD}}\ge 0$.
5:     $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the upper or lower triangle of the $n$ by $n$ Hermitian band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
On exit: AB is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in AB using the same storage format as described above.
6:     $\mathrm{LDAB}$ – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F08HQF (ZHBEVD) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
7:     $\mathrm{W}\left(*\right)$ – 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.
8:     $\mathrm{Z}\left({\mathbf{LDZ}},*\right)$ – 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 ${\mathbf{JOB}}=\text{'V'}$ 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 $A$. The $i$th column of $Z$ contains the eigenvector which corresponds to the eigenvalue ${\mathbf{W}}\left(i\right)$.
If ${\mathbf{JOB}}=\text{'N'}$, Z is not referenced.
9:     $\mathrm{LDZ}$ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08HQF (ZHBEVD) is called.
Constraints:
• if ${\mathbf{JOB}}=\text{'V'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{JOB}}=\text{'N'}$, ${\mathbf{LDZ}}\ge 1$.
10:   $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the required minimal size of LWORK.
11:   $\mathrm{LWORK}$ – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 ${\mathbf{N}}>1$, ${\mathbf{LWORK}}\ge 2×{{\mathbf{N}}}^{2}$ or ${\mathbf{LWORK}}=-1$.
12:   $\mathrm{RWORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LRWORK}}\right)\right)$ – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{RWORK}}\left(1\right)$ contains the required minimal size of ${\mathbf{LRWORK}}$.
13:   $\mathrm{LRWORK}$ – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 ${\mathbf{LRWORK}}=-1$.
14:   $\mathrm{IWORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LIWORK}}\right)\right)$ – INTEGER arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{IWORK}}\left(1\right)$ contains the required minimal size of LIWORK.
15:   $\mathrm{LIWORK}$ – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which F08HQF (ZHBEVD) 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 ${\mathbf{LIWORK}}=-1$.
16:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.
${\mathbf{INFO}}>0$
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.

## 8  Parallelism and Performance

F08HQF (ZHBEVD) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08HQF (ZHBEVD) 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 real analogue of this routine is F08HCF (DSBEVD).

## 10  Example

This example computes all the eigenvalues and eigenvectors of the Hermitian band matrix $A$, where
 $A = 1+0i 2-1i 3-1i 0+0i 0+0i 2+1i 2+0i 3-2i 4-2i 0+0i 3+1i 3+2i 3+0i 4-3i 5-3i 0+0i 4+2i 4+3i 4+0i 5-4i 0+0i 0+0i 5+3i 5+4i 5+0i .$

### 10.1  Program Text

Program Text (f08hqfe.f90)

### 10.2  Program Data

Program Data (f08hqfe.d)

### 10.3  Program Results

Program Results (f08hqfe.r)