# NAG Library Routine Document

## f08hcf (dsbevd)

Warning. The specification of the arguments lwork and liwork changed at Mark 20 in the case where ${\mathbf{job}}=\text{'V'}$ and ${\mathbf{n}}>1$: the minimum dimension of the array work has been reduced whereas the minimum dimension of the array iwork has been increased.

## 1Purpose

f08hcf (dsbevd) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric 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.

## 2Specification

Fortran Interface
 Subroutine f08hcf ( job, uplo, n, kd, ab, ldab, w, z, ldz, work, info)
 Integer, Intent (In) :: n, kd, ldab, ldz, lwork, liwork Integer, Intent (Out) :: iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), w(*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: job, uplo
#include <nagmk26.h>
 void f08hcf_ (const char *job, const char *uplo, const Integer *n, const Integer *kd, double ab[], const Integer *ldab, double w[], double z[], const Integer *ldz, double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_job, const Charlen length_uplo)
The routine may be called by its LAPACK name dsbevd.

## 3Description

f08hcf (dsbevd) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix $A$. In other words, it can compute the spectral factorization of $A$ as
 $A=ZΛZT,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_{i}$, and $Z$ is the orthogonal matrix whose columns are the eigenvectors ${z}_{i}$. Thus
 $Azi=λizi, i=1,2,…,n.$

## 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 http://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:     $\mathbf{job}$ – 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:     $\mathbf{uplo}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{kd}$ – 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:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Real (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 $n$ symmetric 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:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08hcf (dsbevd) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7:     $\mathbf{w}\left(*\right)$ – 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:     $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (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 orthogonal 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:     $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08hcf (dsbevd) 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:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the required minimal size of lwork.
11:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08hcf (dsbevd) 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 ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge 2×{\mathbf{n}}$ or ${\mathbf{lwork}}=-1$;
• if ${\mathbf{job}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge 2×{{\mathbf{n}}}^{2}+5×{\mathbf{n}}+1$ or ${\mathbf{lwork}}=-1$.
12:   $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$ – Integer arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ contains the required minimal size of liwork.
13:   $\mathbf{liwork}$ – IntegerInput
On entry: the dimension of the array iwork as declared in the (sub)program from which f08hcf (dsbevd) 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:   $\mathbf{info}$ – IntegerOutput
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.
${\mathbf{info}}>0$
If ${\mathbf{info}}=〈\mathit{\text{value}}〉$ and ${\mathbf{job}}=\text{'N'}$, the algorithm failed to converge; $〈\mathit{\text{value}}〉$ elements of an intermediate tridiagonal form did not converge to zero; if ${\mathbf{info}}=〈\mathit{\text{value}}〉$ and ${\mathbf{job}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $〈\mathit{\text{value}}〉/\left({\mathbf{n}}+1\right)$ through .

## 7Accuracy

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.

## 8Parallelism and Performance

f08hcf (dsbevd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hcf (dsbevd) 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 complex analogue of this routine is f08hqf (zhbevd).

## 10Example

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

### 10.1Program Text

Program Text (f08hcfe.f90)

### 10.2Program Data

Program Data (f08hcfe.d)

### 10.3Program Results

Program Results (f08hcfe.r)