NAG Library Routine Document

F08JAF (DSTEV)

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 Parameters

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

Errors or warnings detected by the routine:

$INFO < 0$: 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 Further Comments

The total number of floating point operations is proportional to

n^{2}

JOBZ ='N'

and is proportional to

n^{3}

JOBZ ='V'

9 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 DocumentF08JAF (DSTEV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08JAF (DSTEV)