Integer, Intent (In)	::	n, ldz
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	d(), e(), z(ldz,), work()
Character (1), Intent (In)	::	jobz

C Header Interface

#include nagmk26.h

void	f08jaf_ ( const char jobz, const Integer n, double d[], double e[], double z[], const Integer ldz, double work[], Integer info, const Charlen length_jobz)

The routine may be called by its LAPACK name dstev.

3

Description

f08jaf (dstev) computes all the eigenvalues and, optionally, all the eigenvectors of

A

using a combination of the

Q R

and

Q L

algorithms, with an implicit shift.

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: $jobz$ – 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: $n$ – IntegerInput

On entry:

n

, the order of the matrix.

Constraint:

n \geq 0

3: $d (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array d must be at least

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

A

On exit: if

info = 0

, the eigenvalues in ascending order.

4: $e (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array e must be at least

\max (1, n - 1)

On entry: the

(n - 1)

subdiagonal elements of the tridiagonal matrix

A

On exit: the contents of e are destroyed.

5: $z (ldz *)$ – Real (Kind=nag_wp) arrayOutput

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'

, then if

info = 0

, z contains the orthonormal eigenvectors of the matrix

A

, with the

i

th column of

Z

holding the eigenvector associated with

d (i)

jobz ='N'

, z is not referenced.

6: $ldz$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08jaf (dstev) is called.

Constraints:

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

7: $work (*)$ – Real (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array work must be at least

\max (1, 2 \times n - 2)

On exit: if

jobz ='N'

, work is not referenced.

8: $info$ – IntegerOutput

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.

$info > 0$: If $info = i$ , the algorithm failed to converge; $i$ off-diagonal elements of e 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

f08jaf (dstev) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jaf (dstev) 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^{2}

jobz ='N'

and is proportional to

n^{3}

jobz ='V'

10

Example

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

A = (\begin{array}{r} 1 & 1 & 0 & 0 \\ 1 & 4 & 2 & 0 \\ 0 & 2 & 9 & 3 \\ 0 & 0 & 3 & 16 \end{array}),

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

NAG Library Routine Document

f08jaf (dstev)

▸▿ 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

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