f08gcc 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

#include <nag.h>

void	f08gcc (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, double ap[], double w[], double z[], Integer pdz, NagError *fail)

The function may be called by the names: f08gcc, nag_lapackeig_dspevd or nag_dspevd.

3 Description

f08gcc 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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $job$ – Nag_JobType Input

On entry: indicates whether eigenvectors are computed.

$job = Nag_DoNothing$: Only eigenvalues are computed.
$job = Nag_EigVecs$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_DoNothing

Nag_EigVecs

3: $uplo$ – Nag_UploType Input

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

A

is stored.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored.
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $ap [\dim]$ – double Input/Output

Note: the dimension, dim, 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 rows or columns.

The storage of elements

A_{i j}

depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , 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

6: $w [\dim]$ – double Output

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

\max (1, n)

On exit: the eigenvalues of the matrix

A

in ascending order.

7: $z [\dim]$ – double Output

Note: the dimension, dim, of the array z must be at least

$\max (1, pdz \times n)$ when $job = Nag_EigVecs$ ;
$1$ when $job = Nag_DoNothing$ .

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

job = Nag_EigVecs

, z is overwritten by the orthogonal matrix

Z

which contains the eigenvectors of

A

job = Nag_DoNothing

, z is not referenced.

8: $pdz$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

if $job = Nag_EigVecs$ , $pdz \geq \max (1, n)$ ;
if $job = Nag_DoNothing$ , $pdz \geq 1$ .

9: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: If $fail . errnum = ⟨ value ⟩$ and $job = Nag_DoNothing$ , the algorithm failed to converge; $⟨ value ⟩$ elements of an intermediate tridiagonal form did not converge to zero; if $fail . errnum = ⟨ value ⟩$ and $job = Nag_EigVecs$ , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $⟨ value ⟩ / (n + 1)$ through $⟨ value ⟩ mod (n + 1)$ .
NE_ENUM_INT_2: On entry, $job = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_EigVecs$ , $pdz \geq \max (1, n)$ ;
if $job = Nag_DoNothing$ , $pdz \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

f08gcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08gcc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this function is f08gqc.

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 CL Interfacef08gcc (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 CL Interface
f08gcc (dspevd)