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: $ap (:)$ – complex 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 unitary 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: $tau (n - 1)$ – complex array: Further details of the unitary 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 real floating-point operations is approximately

\frac{16}{3} n^{3}

To form the unitary matrix

Q

nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupgtr (f08gt):

[q, info] = f08gt(uplo, n, ap, tau);

To apply

Q

to an

n

p

complex matrix

C

nag_lapack_zhptrd (f08gs) may be followed by a call to nag_lapack_zupmtr (f08gu). For example,

[ap, c, info] = f08gu('Left', uplo, 'No Transpose', ap, tau, c);

forms the matrix product

Q C

The real analogue of this function is nag_lapack_dsptrd (f08ge).

Example

This example reduces the matrix

A

to tridiagonal form, where

A = (\begin{array}{r} - 2.28 + 0.00 i & 1.78 - 2.03 i & 2.26 + 0.10 i & - 0.12 + 2.53 i \\ 1.78 + 2.03 i & - 1.12 + 0.00 i & 0.01 + 0.43 i & - 1.07 + 0.86 i \\ 2.26 - 0.10 i & 0.01 - 0.43 i & - 0.37 + 0.00 i & 2.31 - 0.92 i \\ - 0.12 - 2.53 i & - 1.07 - 0.86 i & 2.31 + 0.92 i & - 0.73 + 0.00 i \end{array}),

using packed storage.

Open in the MATLAB editor: f08gs_example

function f08gs_example


fprintf('f08gs example results\n\n');

% Hermitian matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [-2.28 + 0i;   1.78 + 2.03i;   2.26 - 0.10i;  -0.12 - 2.53i;
                   -1.12 + 0i;      0.01 - 0.43i;  -1.07 - 0.86i;
                                   -0.37 + 0i;      2.31 + 0.92i;
                                   -0.73 + 0i];

% Reduce to tridiagonal form
[apf, d, e, tau, info] = f08gs( ...
                                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));

f08gs example results

Diagonal and off-diagonal elements of tridiagonal form

    i         d           e
    1    -2.28000     4.33846
    2    -0.12846     2.02259
    3    -0.16659     1.80232
    4    -1.92495

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

Chapter Contents

Chapter Introduction

NAG Toolbox