nag_lapack_zsteqr (f08js) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix which has been reduced to tridiagonal form.

Syntax

[d, e, z, info] = f08js(compz, d, e, z, 'n', n)

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

Description

nag_lapack_zsteqr (f08js) 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 .

The function stores the real orthogonal matrix

Z

in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix

A

which has been reduced to tridiagonal form

T

\begin{array}{l} A & = Q T Q^{H}, where ​ Q ​ is unitary \\ = (Q Z) Λ {(Q Z)}^{H} . \end{array}

In this case, the matrix

Q

must be formed explicitly and passed to nag_lapack_zsteqr (f08js), which must be called with

compz ='V'

. The functions which must be called to perform the reduction to tridiagonal form and form

Q

are:

full matrix	nag_lapack_zhetrd (f08fs) and nag_lapack_zungtr (f08ft)
full matrix, packed storage	nag_lapack_zhptrd (f08gs) and nag_lapack_zupgtr (f08gt)
band matrix	nag_lapack_zhbtrd (f08hs) with $vect ='V'$ .

nag_lapack_zsteqr (f08js) uses the implicitly shifted

Q R

algorithm, switching between the

Q R

and

Q L

variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that

{‖z_{i}‖}_{2} = 1

, but are determined only to within a complex factor of absolute value

1

If only the eigenvalues of

T

are required, it is more efficient to call nag_lapack_dsterf (f08jf) instead. If

T

is positive definite, small eigenvalues can be computed more accurately by nag_lapack_zpteqr (f08ju).

References

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

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn17.pdf

Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

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

T

3: $e (:)$ – double array

The dimension of the array e must be at least

\max (1, n - 1)

The off-diagonal elements of the tridiagonal matrix

T

4: $z (ldz :)$ – complex array

The first dimension,

ldz

, of the array z must satisfy

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

The second dimension of the array z must be at least

\max (1, n)

compz ='V'

'I'

and at least

1

compz ='N'

compz ='V'

, z must contain the unitary matrix

Q

from the reduction to tridiagonal form.

compz ='I'

, z need not be set.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array d and the second dimension of the array d. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $T$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $d (:)$ – double array

The dimension of the array d will be

\max (1, n)

The

n

eigenvalues in ascending order, unless

info > 0

(in which case see Error Indicators and Warnings).

2: $e (:)$ – double array

The dimension of the array e will be

\max (1, n - 1)

3: $z (ldz :)$ – complex array

The first dimension,

ldz

, of the array z will be

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

The second dimension of the array z will be

\max (1, n)

compz ='V'

'I'

and at least

1

compz ='N'

compz ='V'

'I'

, the

n

required orthonormal eigenvectors stored as columns of

Z

; the

i

th column corresponds to the

i

th eigenvalue, where

i = 1, 2, \dots, n

, unless

info > 0

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: 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 has failed to find all the eigenvalues after a total of $30 \times n$ iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix unitarily similar to $T$ . If $info = i$ , then $i$ off-diagonal elements have not converged to zero.

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.

Further Comments

The total number of real floating-point operations is typically about

24 n^{2}

compz ='N'

and about

14 n^{3}

compz ='V'

'I'

, but depends on how rapidly the algorithm converges. When

compz ='N'

, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when

compz ='V'

'I'

can be vectorized and on some machines may be performed much faster.

The real analogue of this function is nag_lapack_dsteqr (f08je).

Example

See Example in nag_lapack_zungtr (f08ft), nag_lapack_zupgtr (f08gt) or nag_lapack_zhbtrd (f08hs), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band Hermitian matrix.

Open in the MATLAB editor: f08js_example

function f08js_example


fprintf('f08js example results\n\n');

% Hermitian matrix A (Lower triangular part stored)
uplo = 'L';
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

% Reduce to tridiagonal form
[QT, d, e, tau, info] = f08fs( ...
                               uplo, a);

% Form Q
[Q, info] = f08ft( ...
                   uplo, QT, tau);

% Calculate eigenvalues/vectors of A from Q, d and e.
compz = 'V';
[w, ~, z, info] = f08js( ...
                         compz, d, e, Q);

% Normalize: largest elements are real
for i = 1:4
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp(' Eigenvalues of A:');
disp(w');
disp(' Corresponding eigenvectors:');
disp(z);

f08js example results

 Eigenvalues of A:
   -6.0002   -3.0030    0.5036    3.9996

 Corresponding eigenvectors:
   0.7299 + 0.0000i  -0.2120 + 0.1497i   0.1000 - 0.3570i   0.1991 + 0.4720i
  -0.1663 - 0.2061i   0.7307 + 0.0000i   0.2863 - 0.3353i  -0.2467 + 0.3751i
  -0.4165 - 0.1417i  -0.3291 + 0.0479i   0.6890 + 0.0000i   0.4468 + 0.1466i
   0.1743 + 0.4162i   0.5200 + 0.1329i   0.0662 + 0.4347i   0.5612 + 0.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox