$vect ='V'$: $Q$ is returned.
$vect ='U'$: $Q$ is updated (and the array q must contain a matrix on entry).
$vect ='N'$: $Q$ is not required.

Constraint:

vect ='V'

'U'

'N'

2: $uplo$ – string (length ≥ 1)

Indicates whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored.
$uplo ='L'$: The lower triangular part of $A$ is stored.

Constraint:

uplo ='U'

'L'

3: $kd$ – int64int32nag_int scalar

uplo ='U'

, the number of superdiagonals,

k_{d}

, of the matrix

A

uplo ='L'

, the number of subdiagonals,

k_{d}

, of the matrix

A

Constraint:

kd \geq 0

4: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

\max (1, kd + 1)

The second dimension of the array ab must be at least

\max (1, n)

The upper or lower triangle of the

n

n

symmetric band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

5: $q (ldq :)$ – double array

The first dimension,

ldq

, of the array q must satisfy

if $vect ='V'$ or $'U'$ , $ldq \geq \max (1, n)$ ;
if $vect ='N'$ , $ldq \geq 1$ .

The second dimension of the array q must be at least

\max (1, n)

vect ='V'

'U'

and at least

1

vect ='N'

vect ='U'

, q must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded symmetric-definite generalized eigenproblem); otherwise q need not be set.

Optional Input Parameters

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

Constraint: $n \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – double array

The first dimension of the array ab will be

\max (1, kd + 1)

The second dimension of the array ab will be

\max (1, n)

ab stores values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix

T

are returned in ab using the same storage format as described above.

2: $d (n)$ – double array

The diagonal elements of the tridiagonal matrix

T

3: $e (n - 1)$ – double array

The off-diagonal elements of the tridiagonal matrix

T

4: $q (ldq :)$ – double array

The first dimension,

ldq

, of the array q will be

if $vect ='V'$ or $'U'$ , $ldq = \max (1, n)$ ;
if $vect ='N'$ , $ldq = 1$ .

The second dimension of the array q will be

\max (1, n)

vect ='V'

'U'

and at least

1

vect ='N'

vect ='V'

'U'

, the

n

n

matrix

Q

vect ='N'

, q is not referenced.

5: $info$ – 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: vect, 2: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: d, 8: e, 9: q, 10: ldq, 11: work, 12: 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.

Accuracy

The computed tridiagonal matrix

T

is exactly similar to a nearby matrix

(A + E)

, where

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

is a modestly increasing function of

n

, and

ε

is the machine precision.

The elements of

T

themselves may be sensitive to small perturbations in

A

or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

The computed matrix

Q

differs from an exactly orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

6 n^{2} k

vect ='N'

with

3 n^{3} (k - 1) / k

additional operations if

vect ='V'

The complex analogue of this function is nag_lapack_zhbtrd (f08hs).

Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 4.99 & 0.04 & 0.22 & 0.00 \\ 0.04 & 1.05 & - 0.79 & 1.04 \\ 0.22 & - 0.79 & - 2.31 & - 1.30 \\ 0.00 & 1.04 & - 1.30 & - 0.43 \end{array}) .

Here

A

is symmetric and is treated as a band matrix. The program first calls nag_lapack_dsbtrd (f08he) to reduce

A

to tridiagonal form

T

, and to form the orthogonal matrix

Q

; the results are then passed to nag_lapack_dsteqr (f08je) which computes the eigenvalues and eigenvectors of

A

Open in the MATLAB editor: f08he_example

function f08he_example


fprintf('f08he example results\n\n');

% Symmetric band matrix A, stored on symmetric banded format
uplo = 'L';
kd   = int64(2);
n    = int64(4);
ab   = [4.99,  1.05, -2.31, -0.43;
        0.04, -0.79, -1.30,  0;
        0.22,  1.04,  0,     0];

% Reduce A to tridiagonal form and compute Q
vect = 'V';
q = zeros(n, n);
[apf, d, e, q, info] = f08he( ...
                              vect, uplo, kd, ab, q);

% Calculate eigenvalues and eigenvectors of A
[w, ~, z, info] = f08je( ...
                         vect, d, e, 'z', q);

% 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);

f08he example results

Eigenvalues
   -2.9943   -0.7000    1.9974    4.9969

Eigenvectors
   -0.0251    0.0162    0.0113    0.9995
    0.0656   -0.5859    0.8077    0.0020
    0.9002   -0.3135   -0.3006    0.0311
    0.4298    0.7471    0.5070   -0.0071

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

Chapter Contents

Chapter Introduction

NAG Toolbox