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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dstedc (f08jh)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dstedc (f08jh) computes all the eigenvalues and, optionally, all the eigenvectors of a real

n

n

symmetric tridiagonal matrix, or of a real full or banded symmetric matrix which has been reduced to tridiagonal form.

Syntax

[d, e, z, info] = f08jh(compz, d, e, z, 'n', n)

[d, e, z, info] = nag_lapack_dstedc(compz, d, e, z, 'n', n)

Description

nag_lapack_dstedc (f08jh) computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix

T

. That is, the function computes the spectral factorization of

T

given by

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues,

λ_{i}

, of

T

and

Z

is an orthogonal matrix whose columns are the eigenvectors,

z_{i}

, of

T

. Thus

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

The function may also be used to compute all the eigenvalues and vectors of a real full, or banded, symmetric matrix

A

which has been reduced to tridiagonal form

T

A = Q T Q^{T},

where

Q

is orthogonal. The spectral factorization of

A

is then given by

A = (Q Z) Λ {(Q Z)}^{T} .

In this case

Q

must be formed explicitly and passed to nag_lapack_dstedc (f08jh) in the array z, and the function called with

compz ='V'

. Functions which may be called to form

T

and

Q

are

full matrix	nag_lapack_dsytrd (f08fe) and nag_lapack_dorgtr (f08ff)
full matrix, packed storage	nag_lapack_dsptrd (f08ge) and nag_lapack_dopgtr (f08gf)
band matrix	nag_lapack_dsbtrd (f08he), with $vect ='V'$

When only eigenvalues are required then this function calls nag_lapack_dsterf (f08jf) to compute the eigenvalues of the tridiagonal matrix

T

, but when eigenvectors of

T

are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than nag_lapack_dsteqr (f08je), although more storage is required.

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: $compz$ – string (length ≥ 1)

Indicates whether the eigenvectors are to be computed.

$compz ='N'$: Only the eigenvalues are computed (and the array z is not referenced).
$compz ='V'$: The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz ='I'$: The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).

Constraint:

compz ='N'

'V'

'I'

2: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

The diagonal elements of the tridiagonal matrix.

3: $e (:)$ – double array

The dimension of the array e must be at least

\max (1, n - 1)

The subdiagonal elements of the tridiagonal matrix.

4: $z (ldz :)$ – double array

The first dimension,

ldz

, of the array z must satisfy

if $compz ='V'$ or $'I'$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

The second dimension of the array z must be at least

\max (1, n)

compz ='V'

'I'

, and at least

1

otherwise.

compz ='V'

, z must contain the orthogonal matrix

Q

used in the reduction to tridiagonal form.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array d.
$n$ , the order of the symmetric tridiagonal matrix $T$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $d (:)$ – double array

The dimension of the array d will be

\max (1, n)

info = 0

, the eigenvalues in ascending order.

2: $e (:)$ – double array

The dimension of the array e will be

\max (1, n - 1)

3: $z (ldz :)$ – double array

The first dimension,

ldz

, of the array z will be

if $compz ='V'$ or $'I'$ , $ldz = \max (1, n)$ ;
otherwise $ldz = 1$ .

The second dimension of the array z will be

\max (1, n)

compz ='V'

'I'

and

1

otherwise.

compz ='V'

, z contains the orthonormal eigenvectors of the original symmetric matrix

A

, and if

compz ='I'

, z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix

T

compz ='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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: compz, 2: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: lwork, 9: iwork, 10: liwork, 11: 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.

W $info > 0$: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns $info / (n + 1)$ through $info mod (n + 1)$ .

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.

See Section 4.7 of Anderson et al. (1999) for further details. See also nag_lapack_ddisna (f08fl).

Further Comments

If only eigenvalues are required, the total number of floating-point operations is approximately proportional to

n^{2}

. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as nag_lapack_dsteqr (f08je), but for large matrices nag_lapack_dstedc (f08jh) is usually much faster.

The complex analogue of this function is nag_lapack_zstedc (f08jv).

Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix

A = (\begin{array}{r} 4.99 & 0.04 & 0.22 & 0 \\ 0.04 & 1.05 & - 0.79 & 1.04 \\ 0.22 & - 0.79 & - 2.31 & - 1.30 \\ 0 & 1.04 & - 1.30 & - 0.43 \end{array}) .

A

is first reduced to tridiagonal form by a call to nag_lapack_dsbtrd (f08he).

Open in the MATLAB editor: f08jh_example

function f08jh_example


fprintf('f08jh example results\n\n');

% Symmetric band matrix A, stored on symmetric banded format
uplo = 'L';
kd   = int64(2);
n    = int64(4);
ab   = [4.99,  1.05, -2.31, -0.43;
        0.04, -0.79, -1.30,  0;
        0.22,  1.04,  0,     0];

% Reduce A to tridiagonal form and compute Q
vect = 'V';
q = zeros(n, n);
[apf, d, e, q, info] = f08he( ...
                              vect, uplo, kd, ab, q);

% Calculate eigenvalues and eigenvectors of A
compz = 'V';
[w, ~, z, info] = f08jh( ...
                         compz, d, e, q);

% 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('Eigenvalues');
disp(w');
disp('Eigenvectors');
disp(z);

f08jh example results

Eigenvalues
   -2.9943   -0.7000    1.9974    4.9969

Eigenvectors
   -0.0251    0.0162    0.0113    0.9995
    0.0656   -0.5859    0.8077    0.0020
    0.9002   -0.3135   -0.3006    0.0311
    0.4298    0.7471    0.5070   -0.0071

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Chapter Contents

Chapter Introduction

NAG Toolbox