f08gcf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix held in packed storage. 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 f08gcf (

job, uplo, n, ap, w, 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)	::	ap(), w(), z(ldz,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	job, uplo

C Header Interface

#include <nag.h>

void	f08gcf_ (const char job, const char uplo, const Integer n, double ap[], double w[], double z[], const Integer ldz, double work[], const Integer lwork, Integer iwork[], const Integer liwork, Integer *info, const Charlen length_job, const Charlen length_uplo)

The routine may be called by the names f08gcf, nagf_lapackeig_dspevd or its LAPACK name dspevd.

3 Description

f08gcf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix

A

(held in packed storage). In other words, it can compute the spectral factorization of

A

A = 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

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

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: $job$ – 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: $uplo$ – Character(1) Input

On entry: indicates whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: 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: $ap (*)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n \times (n + 1) / 2)

On entry: the upper or lower triangle of the

n \times n

symmetric matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

On exit: ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of

A

5: $w (*)$ – Real (Kind=nag_wp) array Output

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

\max (1, n)

On exit: the eigenvalues of the matrix

A

in ascending order.

6: $z (ldz, *)$ – 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

A

job ='N'

, z is not referenced.

7: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08gcf is called.

Constraints:

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

8: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the required minimal size of lwork.

9: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08gcf 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 $n \leq 1$ , $lwork \geq 1$ or $lwork = - 1$ ;
if $job ='N'$ and $n > 1$ , $lwork \geq 2 \times n$ or $lwork = - 1$ ;
if $job ='V'$ and $n > 1$ , $lwork \geq n^{2} + 6 \times n + 1$ or $lwork = - 1$ .

10: $iwork (\max (1, liwork))$ – Integer array Workspace

On exit: if

info = 0

iwork (1)

contains the required minimal size of liwork.

11: $liwork$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08gcf 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 $liwork = - 1$ .

12: $info$ – 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.

$info > 0$: If $info = ⟨ value ⟩$ and $job ='N'$ , the algorithm failed to converge; $⟨ value ⟩$ elements of an intermediate tridiagonal form did not converge to zero; if $info = ⟨ value ⟩$ and $job ='V'$ , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $⟨ value ⟩ / (n + 1)$ through $⟨ value ⟩ mod (n + 1)$ .

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

f08gcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08gcf 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 complex analogue of this routine is f08gqf.

10 Example

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

A

, where

A = (\begin{matrix} 1.0 & 2.0 & 3.0 & 4.0 \\ 2.0 & 2.0 & 3.0 & 4.0 \\ 3.0 & 3.0 & 3.0 & 4.0 \\ 4.0 & 4.0 & 4.0 & 4.0 \end{matrix}) .

f08gc: FL CL CPP AD

NAG FL Interfacef08gcf (dspevd)

▸▿ 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
f08gcf (dspevd)