nag_lapack_dpteqr (f08jg) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix, or of a real symmetric positive definite matrix which has been reduced to tridiagonal form.

Syntax

[d, e, z, info] = f08jg(compz, d, e, z, 'n', n)

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

Description

nag_lapack_dpteqr (f08jg) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite 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 positive definite 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_dpteqr (f08jg), 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_dpteqr (f08jg) first factorizes

T

L D L^{T}

where

L

is unit lower bidiagonal and

D

is diagonal. It forms the bidiagonal matrix

B = L D^{\frac{1}{2}}

, and then calls nag_lapack_dbdsqr (f08me) to compute the singular values of

B

which are the same as the eigenvalues of

T

. The method used by the function allows high relative accuracy to be achieved in the small eigenvalues of

T

. The eigenvectors are normalized so that

{‖z_{i}‖}_{2} = 1

, but are determined only to within a factor

\pm 1

References

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

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: $z (ldz :)$ – 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.

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 descending order, unless

info > 0

, in which case d is overwritten.

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

$info = - i$: 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.

$info > 0$: If $info = i$ , the leading minor of order $i$ is not positive definite and the Cholesky factorization of $T$ could not be completed. Hence $T$ itself is not positive definite.

If $info = n + i$ , the algorithm to compute the singular values of the Cholesky factor $B$ failed to converge; $i$ off-diagonal elements did not converge to zero.

Accuracy

The eigenvalues and eigenvectors of

T

are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard

Q R

method. However, the reduction to tridiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.

To be more precise, let

H

be the tridiagonal matrix defined by

H = D T D

, where

D

is diagonal with

d_{i i} = t_{i i}^{- \frac{1}{2}}

, and

h_{i i} = 1

for all

i

. If

λ_{i}

is an exact eigenvalue of

T

and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq c (n) ε κ_{2} (H) λ_{i}

where

c (n)

is a modestly increasing function of

n

ε

is the machine precision, and

κ_{2} (H)

is the condition number of

H

with respect to inversion defined by:

κ_{2} (H) = ‖H‖ \cdot ‖H^{- 1}‖

z_{i}

is the corresponding exact eigenvector of

T

, 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 κ_{2} (H)}{{relgap}_{i}}

where

{relgap}_{i}

is the relative gap between

λ_{i}

and the other eigenvalues, defined by

{relgap}_{i} = \min_{i \neq j} \frac{|λ_{i} - λ_{j}|}{(λ_{i} + λ_{j})} .

Further Comments

The total number of floating-point operations is typically about

30 n^{2}

compz ='N'

and about

6 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_zpteqr (f08ju).

Example

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

T

, where

T = (\begin{array}{r} 4.16 & 3.17 & 0.00 & 0.00 \\ 3.17 & 5.25 & - 0.97 & 0.00 \\ 0.00 & - 0.97 & 1.09 & 0.55 \\ 0.00 & 0.00 & 0.55 & 0.62 \end{array}) .

Open in the MATLAB editor: f08jg_example

function f08jg_example


fprintf('f08jg example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
n = 4;
d = [4.16;     5.25;     1.09;     0.62];
e = [3.17;    -0.97;     0.55];

% All eigenvalues and eigenvectors of A
compz = 'I';
z = zeros(4, 4);
[w, ~, z, info] = f08jg( ...
                         compz, d, e, 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);

f08jg example results

Eigenvalues
    8.0023    1.9926    1.0014    0.1237

Eigenvectors
    0.6326    0.6245   -0.4191    0.1847
    0.7668   -0.4270    0.4176   -0.2352
   -0.1082    0.6071    0.4594   -0.6393
   -0.0081    0.2432    0.6625    0.7084

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

Chapter Contents

Chapter Introduction

NAG Toolbox