NAG Library Routine Document

F08JCF (DSTEVD)

+− 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

1 Purpose

F08JCF (DSTEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal 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

Q L

Q R

algorithm.

2 Specification

SUBROUTINE F08JCF (

JOB, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

INTEGER	N, LDZ, LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO
REAL (KIND=nag_wp)	D(), E(), Z(LDZ,*), WORK(max(1,LWORK))
CHARACTER(1)	JOB

The routine may be called by its LAPACK name dstevd.

3 Description

F08JCF (DSTEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix

T

. In other words, it can compute the spectral factorization of

T

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the orthogonal matrix whose columns are the eigenvectors

z_{i}

. Thus

T z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: JOB – CHARACTER(1)Input

On entry: indicates whether eigenvectors are computed.

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

Constraint:

JOB ='N'

'V'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

T

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

T

On exit: the eigenvalues of the matrix

T

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)

On entry: the

n - 1

off-diagonal elements of the tridiagonal matrix

T

. The

n

th element of this array is used as workspace.

On exit: E is overwritten with intermediate results.

5: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (1, N)

JOB ='V'

and at least

1

JOB ='N'

On exit: if

JOB ='V'

, Z is overwritten by the orthogonal matrix

Z

which contains the eigenvectors of

T

JOB ='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 F08JCF (DSTEVD) is called.

Constraints:

if $JOB ='V'$ , $LDZ \geq \max (1, N)$ ;
if $JOB ='N'$ , $LDZ \geq 1$ .

7: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

contains the required minimal size of LWORK.

8: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08JCF (DSTEVD) is called.

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 $JOB ='N'$ or $N \leq 1$ , $LWORK \geq 1$ or $LWORK = - 1$ ;
if $JOB ='V'$ and $N > 1$ , $LWORK \geq N^{2} + 4 \times N + 1$ or $LWORK = - 1$ .

9: IWORK( $\max (1, LIWORK)$ ) – INTEGER arrayWorkspace

On exit: if

INFO = 0

IWORK (1)

contains the required minimal size of LIWORK.

10: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08JCF (DSTEVD) is called.

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 $JOB ='N'$ or $N \leq 1$ , $LIWORK \geq 1$ or $LIWORK = - 1$ ;
if $JOB ='V'$ and $N > 1$ , $LIWORK \geq 5 \times N + 3$ or $LIWORK = - 1$ .

11: 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 $INFO = i$ and $JOB ='N'$ , the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if $INFO = i$ and $JOB ='V'$ , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i / (N + 1)$ through $i mod (N + 1)$ .

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(T + E)

, where

{‖E‖}_{2} = O (ε) {‖T‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq c (n) ε {‖T‖}_{2},

where

c (n)

is a modestly increasing function of

n

z_{i}

is the corresponding exact eigenvector, and

{\tilde{z}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{z}}_{i}, z_{i})

between them is bounded as follows:

θ ({\tilde{z}}_{i}, z_{i}) \leq \frac{c (n) ε {‖T‖}_{2}}{\min_{i \neq j} |λ_{i} - λ_{j}|} .

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8 Further Comments

There is no complex analogue of this routine.

9 Example

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

T

, where

T = (\begin{array}{r} 1.0 & 1.0 & 0.0 & 0.0 \\ 1.0 & 4.0 & 2.0 & 0.0 \\ 0.0 & 2.0 & 9.0 & 3.0 \\ 0.0 & 0.0 & 3.0 & 16.0 \end{array}) .

NAG Library Routine DocumentF08JCF (DSTEVD)

+− 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

F08JCF (DSTEVD)