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: $d (:)$ – double array: The dimension of the array d must be at least $\max (1, n)$

The diagonal elements of the tridiagonal matrix $T$ .
2: $e (:)$ – 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: $n \geq 0$ .

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: $info$ – 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: n, 2: d, 3: e, 4: info.

W $info > 0$: The algorithm has failed to find all the eigenvalues after a total of $30 \times n$ iterations. If $info = i$ , then on exit $i$ elements of e have not converged to zero.

Accuracy

The computed eigenvalues 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

Further Comments

The total number of floating-point operations is typically about

14 n^{2}

, but depends on how rapidly the algorithm converges. The operations are all performed in scalar mode.

There is no complex analogue of this function.

Example

This example computes all the eigenvalues 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}) .

Open in the MATLAB editor: f08jf_example

function f08jf_example


fprintf('f08jf 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];

% Eigenvalues only of A
[w, ~, info] = f08jf( ...
                      d, e);

disp('Eigenvalues');
disp(w');

f08jf example results

Eigenvalues
   -7.0037   -0.4059    2.0028    8.9968

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

Chapter Contents

Chapter Introduction

NAG Toolbox