# NAG FL Interfacef08hnf (zhbev)

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## 1Purpose

f08hnf computes all the eigenvalues and, optionally, all the eigenvectors of a complex $n×n$ Hermitian band matrix $A$ of bandwidth $\left(2{k}_{d}+1\right)$.

## 2Specification

Fortran Interface
 Subroutine f08hnf ( jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info)
 Integer, Intent (In) :: n, kd, ldab, ldz Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Out) :: w(n), rwork(3*n-2) Complex (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), z(ldz,*) Complex (Kind=nag_wp), Intent (Out) :: work(n) Character (1), Intent (In) :: jobz, uplo
#include <nag.h>
 void f08hnf_ (const char *jobz, const char *uplo, const Integer *n, const Integer *kd, Complex ab[], const Integer *ldab, double w[], Complex z[], const Integer *ldz, Complex work[], double rwork[], Integer *info, const Charlen length_jobz, const Charlen length_uplo)
The routine may be called by the names f08hnf, nagf_lapackeig_zhbev or its LAPACK name zhbev.

## 3Description

The Hermitian band 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.

## 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 https://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: $\mathbf{jobz}$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: $\mathbf{uplo}$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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{kd}$Integer Input
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: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Complex (Kind=nag_wp) array Input/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 $n×n$ 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: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f08hnf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7: $\mathbf{w}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the eigenvalues in ascending order.
8: $\mathbf{z}\left({\mathbf{ldz}},*\right)$Complex (Kind=nag_wp) array Output
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.
9: $\mathbf{ldz}$Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08hnf 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: $\mathbf{work}\left({\mathbf{n}}\right)$Complex (Kind=nag_wp) array Workspace
11: $\mathbf{rwork}\left(3×{\mathbf{n}}-2\right)$Real (Kind=nag_wp) array Workspace
12: $\mathbf{info}$Integer Output
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.
${\mathbf{info}}>0$
The algorithm failed to converge; $⟨\mathit{\text{value}}⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

## 7Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08hnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hnf 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 ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$ and is proportional to ${k}_{d}{n}^{2}$ otherwise.
The real analogue of this routine is f08haf.

## 10Example

This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
 $A = ( 1 2-i 3-i 0 0 2+i 2 3-2i 4-2i 0 3+i 3+2i 3 4-3i 5-3i 0 4+2i 4+3i 4 5-4i 0 0 5+3i 5+4i 5 ) ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.

### 10.1Program Text

Program Text (f08hnfe.f90)

### 10.2Program Data

Program Data (f08hnfe.d)

### 10.3Program Results

Program Results (f08hnfe.r)