nag_lapack_dsteqr (f08je) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.

Syntax

[d, e, z, info] = f08je(compz, d, e, 'n', n, 'z', z)

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

Description

nag_lapack_dsteqr (f08je) 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 may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix

A

which has been reduced to tridiagonal form

T

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

In this case, the matrix

Q

must be formed explicitly and passed to nag_lapack_dsteqr (f08je), 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_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 $vect ='V'$ .

nag_lapack_dsteqr (f08je) 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 factor

\pm 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_dpteqr (f08jg).

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

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

2: $z (ldz :)$ – double 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 orthogonal matrix

Q

from the reduction to tridiagonal form.

compz ='I'

, z need not be set.

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: $z (ldz :)$ – double 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 orthogonally 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 floating-point operations is typically about

24 n^{2}

compz ='N'

and about

7 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 complex analogue of this function is nag_lapack_zsteqr (f08js).

Example

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

T

, where

T = (\begin{array}{r} - 6.99 & - 0.44 & 0.00 & 0.00 \\ - 0.44 & 7.92 & - 2.63 & 0.00 \\ 0.00 & - 2.63 & 2.34 & - 1.18 \\ 0.00 & 0.00 & - 1.18 & 0.32 \end{array}) .

See also the examples for nag_lapack_dorgtr (f08ff), nag_lapack_dopgtr (f08gf) or nag_lapack_dsbtrd (f08he), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.

Open in the MATLAB editor: f08je_example

function f08je_example


fprintf('f08je example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
n = 4;
d = [-6.99;     7.92;     2.34;     0.32];
e = [-0.44;    -2.63;    -1.18];

% All eigenvalues and eigenvectors of A
compz = 'I';
z = zeros(n, n);
[w, ~, z, info] = f08je( ...
                         compz, d, e, 'z', z);

% 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);

f08je example results

Eigenvalues
   -7.0037   -0.4059    2.0028    8.9968

Eigenvectors
    0.9995   -0.0109   -0.0167   -0.0255
    0.0310    0.1627    0.3408    0.9254
    0.0089    0.5170    0.7696   -0.3746
    0.0014    0.8403   -0.5397    0.0509

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

Chapter Contents

Chapter Introduction

NAG Toolbox