1: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $d (*)$ – Real (Kind=nag_wp) array Input/Output: Note: the dimension of the array d must be at least $\max (1, n)$ .

On entry: must contain the $n$ diagonal elements of the matrix $A$ .

On exit: is overwritten by the $n$ diagonal elements of the diagonal matrix $D$ from the $L D L^{H}$ factorization of $A$ .
3: $e (*)$ – Complex (Kind=nag_wp) array Input/Output: Note: the dimension of the array e must be at least $\max (1, n - 1)$ .

On entry: must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .

On exit: is overwritten by the $(n - 1)$ subdiagonal elements of the lower bidiagonal matrix $L$ . (e can also be regarded as containing the $(n - 1)$ superdiagonal elements of the upper bidiagonal matrix $U$ .)
4: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0 and info < n$: The leading minor of order $⟨ value ⟩$ is not positive definite, the factorization could not be completed.

$info > 0 and info = n$: The leading minor of order $n$ is not positive definite, the factorization was completed, but $d (n) \leq 0$ .

7 Accuracy

The computed factorization satisfies an equation of the form

A + E = L D L^{H},

where

{‖ E ‖}_{\infty} = O (ε) {‖ A ‖}_{\infty}

and

ε

is the machine precision.

Following the use of this routine, f07jsf can be used to solve systems of equations

A X = B

, and f07juf can be used to estimate the condition number of

A

f07jrf is not threaded in any implementation.

The total number of floating-point operations required to factorize the matrix

A

is proportional to

n

The real analogue of this routine is f07jdf.

This example factorizes the Hermitian positive definite tridiagonal matrix

A

given by

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array}) .

f07jr: FL CL CPP AD