NAG Library Routine Document

F07JRF (ZPTTRF)

1: $N$ – INTEGERInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $N \geq 0$ .
2: $D (*)$ – REAL (KIND=nag_wp) arrayInput/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) arrayInput/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: $INFO$ – INTEGEROutput: 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 $D (N) \leq 0$ .

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 (ZPTTRS) can be used to solve systems of equations

A X = B

, and F07JUF (ZPTCON) can be used to estimate the condition number of

A

Not applicable.

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 (DPTTRF).

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}) .