Integer, Intent (In)	::	n, kd, ldab, ldz
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	ab(ldab,), z(ldz,)
Real (Kind=nag_wp), Intent (Out)	::	w(n), work(3*n-2)
Character (1), Intent (In)	::	jobz, uplo

C Header Interface

#include <nag.h>

void	f08haf_ (const char jobz, const char uplo, const Integer n, const Integer kd, double ab[], const Integer ldab, double w[], double z[], const Integer ldz, double work[], Integer *info, const Charlen length_jobz, const Charlen length_uplo)

The routine may be called by the names f08haf, nagf_lapackeig_dsbev or its LAPACK name dsbev.

3 Description

The symmetric band matrix

A

is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the

Q R

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 https://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: $jobz$ – Character(1) Input

On entry: indicates whether eigenvectors are computed.

$jobz ='N'$: Only eigenvalues are computed.
$jobz ='V'$: Eigenvalues and eigenvectors are computed.

Constraint:

jobz ='N'

'V'

2: $uplo$ – Character(1) Input

On entry: if

uplo ='U'

, the upper triangular part of

A

is stored.

uplo ='L'

, the lower triangular part of

A

is stored.

Constraint:

uplo ='U'

'L'

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $kd$ – Integer Input

On entry: if

uplo ='U'

, the number of superdiagonals,

k_{d}

, of the matrix

A

uplo ='L'

, the number of subdiagonals,

k_{d}

, of the matrix

A

Constraint:

kd \geq 0

5: $ab (ldab, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array ab must be at least

\max (1, n)

On entry: the upper or lower triangle of the

n \times n

symmetric band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

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: $ldab$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f08haf is called.

Constraint:

ldab \geq kd + 1

7: $w (n)$ – Real (Kind=nag_wp) array Output

On exit: the eigenvalues in ascending order.

8: $z (ldz, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array z must be at least

\max (1, n)

jobz ='V'

, and at least

1

otherwise.

On exit: if

jobz ='V'

, z contains the orthonormal eigenvectors of the matrix

A

, with the

i

th column of

Z

holding the eigenvector associated with

w (i)

jobz ='N'

, z is not referenced.

9: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08haf is called.

Constraints:

if $jobz ='V'$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

10: $work (3 \times n - 2)$ – Real (Kind=nag_wp) array Workspace

11: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: The algorithm failed to converge; $⟨ value ⟩$ 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

(A + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2},

and

ε

is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08haf 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.

9 Further Comments

The total number of floating-point operations is proportional to

n^{3}

jobz ='V'

and is proportional to

k_{d} n^{2}

otherwise.

The complex analogue of this routine is f08hnf.

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix

A = (\begin{matrix} 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 \end{matrix}),

together with approximate error bounds for the computed eigenvalues and eigenvectors.

f08ha: FL CL CPP AD PY MB

NAG FL Interfacef08haf (dsbev)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08haf (dsbev)