nag_lapack_dspevd (f08gc) 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.

Syntax

[ap, w, z, info] = f08gc(job, uplo, n, ap)

[ap, w, z, info] = nag_lapack_dspevd(job, uplo, n, ap)

Description

nag_lapack_dspevd (f08gc) 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 .

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 http://www.netlib.org/lapack/lug

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

Parameters

Compulsory Input Parameters

1: $job$ – string (length ≥ 1)

Indicates whether eigenvectors are computed.

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

Constraint:

job ='N'

'V'

2: $uplo$ – string (length ≥ 1)

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: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $ap (:)$ – double array

The dimension of the array ap must be at least

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

The upper or lower triangle of the

n

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 $i \geq j$ .

Optional Input Parameters

None.

Output Parameters

1: $ap (:)$ – double array

The dimension of the array ap will be

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

ap stores 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

2: $w (:)$ – double array

The dimension of the array w will be

\max (1, n)

The eigenvalues of the matrix

A

in ascending order.

3: $z (ldz :)$ – double array

The first dimension,

ldz

, of the array z will be

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

The second dimension of the array z will be

\max (1, n)

job ='V'

and at least

1

job ='N'

job ='V'

, z stores the orthogonal matrix

Z

which contains the eigenvectors of

A

job ='N'

, z is not referenced.

4: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: job, 2: uplo, 3: n, 4: ap, 5: w, 6: z, 7: ldz, 8: work, 9: lwork, 10: iwork, 11: liwork, 12: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

$info > 0$: 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 $i mod (n + 1)$ .

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.

Further Comments

The complex analogue of this function is nag_lapack_zhpevd (f08gq).

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}) .

Open in the MATLAB editor: f08gc_example

function f08gc_example


fprintf('f08gc example results\n\n');

% A is symmetric matrix stored in symmetric (Lower) packed format
uplo = 'L';
n = int64(4);
ap = [1;     2;     3;     4;
             2;     3;     4;
                    3;     4;
                           4];

% Eigenvalues and vectors of A 
job = 'Vectors';
[apf, w, z, info] = f08gc( ...
                           job, uplo, n, ap);

disp('Eigenvalues');
disp(w');

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            

disp('Eigenvectors');
disp(z);

f08gc example results

Eigenvalues
   -2.0531   -0.5146   -0.2943   12.8621

Eigenvectors
    0.7003   -0.5144   -0.2767    0.4103
    0.3592    0.4851    0.6634    0.4422
   -0.1569    0.5420   -0.6504    0.5085
   -0.5965   -0.4543    0.2457    0.6144

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox