If $uplo ='U'$ , the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

a stores the tridiagonal matrix $T$ and details of the unitary matrix $Q$ as specified by uplo.
2: $d (:)$ – double array: The dimension of the array d will be $\max (1, n)$

The diagonal elements of the tridiagonal matrix $T$ .
3: $e (:)$ – double array: The dimension of the array e will be $\max (1, n - 1)$

The off-diagonal elements of the tridiagonal matrix $T$ .
4: $tau (:)$ – complex array: The dimension of the array tau will be $\max (1, n - 1)$

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: a, 4: lda, 5: d, 6: e, 7: tau, 8: work, 9: lwork, 10: 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.

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_zhetrd (f08fs) may be followed by a call to nag_lapack_zungtr (f08ft):

[a, info] = f08ft(uplo, a, tau);

To apply

Q

to an

n

p

complex matrix

C

nag_lapack_zhetrd (f08fs) may be followed by a call to nag_lapack_zunmtr (f08fu). For example,

[c, info] = f08fu('Left', uplo, 'No Transpose', a, tau, c);

forms the matrix product

Q C

The real analogue of this function is nag_lapack_dsytrd (f08fe).

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}) .

Open in the MATLAB editor: f08fs_example

function f08fs_example


fprintf('f08fs example results\n\n');

% A --> QTQ^H, for tridiagonal T
uplo = 'L';
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

[QT, d, e, tau, info] = f08fs(uplo, a);

disp('Main diagonal of T:');
disp (d');
disp('Off diagonal of T (absolute values):');
disp (abs(e)');

f08fs example results

Main diagonal of T:
   -2.2800   -0.1285   -0.1666   -1.9249

Off diagonal of T (absolute values):
    4.3385    2.0226    1.8023

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

Chapter Contents

Chapter Introduction

NAG Toolbox