f08jcf 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

Fortran Interface

Subroutine f08jcf (

job, n, d, e, z, ldz, work, lwork, iwork, liwork, info)

Integer, Intent (In)	::	n, ldz, lwork, liwork
Integer, Intent (Out)	::	iwork(max(1,liwork)), info
Real (Kind=nag_wp), Intent (Inout)	::	d(), e(), z(ldz,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	job

C Header Interface

#include <nag.h>

void	f08jcf_ (const char job, const Integer n, double d[], double e[], double z[], const Integer ldz, double work[], const Integer lwork, Integer iwork[], const Integer liwork, Integer info, const Charlen length_job)

The routine may be called by the names f08jcf, nagf_lapackeig_dstevd or its LAPACK name dstevd.

3 Description

f08jcf 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 Arguments

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: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

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

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: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08jcf 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) array Workspace

On exit: if

info = 0

work (1)

contains the required minimal size of lwork.

8: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08jcf 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 array Workspace

On exit: if

info = 0

iwork (1)

contains the required minimal size of liwork.

10: $liwork$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08jcf 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: $info$ – 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 e did not converge to zero.

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 Parallelism and Performance

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

f08jcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jcf 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

There is no complex analogue of this routine.

10 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}) .

f08jc: FL CL CPP AD PY MB

NAG FL Interfacef08jcf (dstevd)

▸▿ 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
f08jcf (dstevd)