if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

Optional Input Parameters

None.

Output Parameters

1: $ap (:)$ – double array: The dimension of the array ap will be $\max (1, n \times (n + 1) / 2)$

ap stores the tridiagonal matrix $T$ and details of the orthogonal matrix $Q$ .
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: $tau (n - 1)$ – double array: Further details of the orthogonal matrix $Q$ .
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: uplo, 2: n, 3: ap, 4: d, 5: e, 6: tau, 7: info.

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.

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3}

To form the orthogonal matrix

Q

nag_lapack_dsptrd (f08ge) may be followed by a call to nag_lapack_dopgtr (f08gf):

[q, info] = f08gf(uplo, n, ap, tau);

To apply

Q

to an

n

p

real matrix

C

nag_lapack_dsptrd (f08ge) may be followed by a call to nag_lapack_dopmtr (f08gg). For example,

[ap, c, info] = f08gg('Left', uplo, 'No Transpose', ap, tau, c);

forms the matrix product

Q C

The complex analogue of this function is nag_lapack_zhptrd (f08gs).

Example

This example reduces the matrix

A

to tridiagonal form, where

A = (\begin{array}{r} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{array}),

using packed storage.

Open in the MATLAB editor: f08ge_example

function f08ge_example


fprintf('f08ge example results\n\n');

uplo = 'L';
n = int64(4);
ap = [2.07;     3.87;     4.2;     -1.15;
               -0.21;     1.87;     0.63;
                          1.15;     2.06;
                                   -1.81];

[apf, d, e, tau, info] = f08ge( ...
                                uplo, n, ap);

% Note: absolute values for e are displayed because the signs may change
%       with changes in sign of columns of Q.
fprintf('Diagonal and off-diagonal elements of tridiagonal form\n\n');
fprintf('    i         d           e\n');

for j = 1:n-1
  fprintf('%5d%12.5f%12.5f\n', j, d(j), abs(e(j)));
end
fprintf('%5d%12.5f\n', n, d(n));

f08ge example results

Diagonal and off-diagonal elements of tridiagonal form

    i         d           e
    1     2.07000     5.82575
    2     1.47409     2.62405
    3    -0.64916     0.91627
    4    -1.69493

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

Chapter Contents

Chapter Introduction

NAG Toolbox