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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zstedc (f08jv)

▸▿ 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_zstedc (f08jv) computes all the eigenvalues and, optionally, all the eigenvectors of a real

n

n

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

Syntax

[d, e, z, info] = f08jv(compz, d, e, z, 'n', n)

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

Description

nag_lapack_zstedc (f08jv) 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 eigenvectors of a complex full, or banded, Hermitian matrix

A

which has been reduced to real tridiagonal form

T

A = Q T Q^{H},

where

Q

is unitary. The spectral factorization of

A

is then given by

A = (Q Z) Λ {(Q Z)}^{H} .

In this case

Q

must be formed explicitly and passed to nag_lapack_zstedc (f08jv) 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_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 $vect ='V'$

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_zsteqr (f08js), 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: $z (ldz :)$ – complex 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 unitary 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: $z (ldz :)$ – complex 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 Hermitian 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

$info = - i$: 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: rwork, 10: lrwork, 11: iwork, 12: liwork, 13: 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.

$info > 0$: 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_zsteqr (f08js), but for large matrices nag_lapack_zstedc (f08jv) is usually much faster.

The real analogue of this function is nag_lapack_dstedc (f08jh).

Example

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

A = (\begin{array}{r} - 3.13 & 1.94 - 2.10 i & - 3.40 + 0.25 i & 0 \\ 1.94 + 2.10 i & - 1.91 & - 0.82 - 0.89 i & - 0.67 + 0.34 i \\ - 3.40 - 0.25 i & - 0.82 + 0.89 i & - 2.87 & - 2.10 - 0.16 i \\ 0 & - 0.67 - 0.34 i & - 2.10 + 0.16 i & 0.50 \end{array}) .

A

is first reduced to tridiagonal form by a call to nag_lapack_zhbtrd (f08hs).

Open in the MATLAB editor: f08jv_example

function f08jv_example


fprintf('f08jv example results\n\n');

% A is Hermitian banded matrix
n = 4;
k = int64(2);
a = [ -3.13 + 0.00i,  1.94 - 2.10i, -3.40 + 0.25i,  0    + 0i;
       1.94 + 2.10i, -1.91 + 0.00i, -0.82 - 0.89i, -0.67 + 0.34i;
      -3.40 - 0.25i, -0.82 + 0.89i, -2.87 + 0.00i, -2.10 - 0.16i;
       0    + 0i,    -0.67 - 0.34i, -2.10 + 0.16i,  0.50 + 0i    ];

% Convert to general banded format ...
kl = int64(k); 
[~, abn, ifail] = f01zd( ...
                         'P', k, k, a, complex(zeros(k+k+1,n)));
% ... and chop to give 'Upper' Hermitian banded format
ab = abn(1:k+1,1:n);

% Reduce A to tridiagonal form A = ZTZ^H
compz = 'V';
uplo  = 'Upper';
[~, Td, Te, ZTZ, info] = f08hs( ...
                                compz, uplo, k, ab, complex(zeros(n, n)));

% Eigenvalues and vectors of A from tridiagonal form (ZTZ, Td, Te).
[w, ~, z, info] = f08jv( ...
                         compz, Td, Te, ZTZ);

% Normalize: largest elements are real
for i = 1:n
  [~,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);

f08jv example results

 Eigenvalues of A:
   -7.0042   -4.0038    0.5968    3.0012

 Corresponding eigenvectors:
   0.7293 + 0.0000i  -0.2128 + 0.1511i  -0.3354 - 0.1604i   0.4732 + 0.1947i
  -0.1654 - 0.2046i   0.7316 + 0.0000i  -0.2804 - 0.3413i   0.0891 + 0.4387i
   0.6081 + 0.0301i   0.3910 - 0.3843i  -0.0144 + 0.1532i  -0.5172 - 0.1938i
   0.1653 - 0.0303i   0.2775 - 0.1378i   0.8019 + 0.0000i   0.4824 + 0.0000i

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Chapter Contents

Chapter Introduction

NAG Toolbox