NAG Library Function Document

nag_zpttrf (f07jrc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_zpttrf (f07jrc) computes the modified Cholesky factorization of a complex

n

n

Hermitian positive definite tridiagonal matrix

A

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_zpttrf (Integer n, double d[], Complex e[], NagError *fail)

3 Description

nag_zpttrf (f07jrc) factorizes the matrix

A

A = L D L^{H},

where

L

is a unit lower bidiagonal matrix and

D

is a diagonal matrix with positive diagonal elements. The factorization may also be regarded as having the form

U^{H} D U

, where

U

is a unit upper bidiagonal matrix.

4 References

None.

5 Arguments

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

Constraint: $n \geq 0$ .
2: $d [\dim]$ – doubleInput/Output: Note: the dimension, dim, 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: $e [\dim]$ – ComplexInput/Output: Note: the dimension, dim, 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: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_MAT_NOT_POS_DEF: The leading minor of order $n$ is not positive definite, the factorization was completed, but $d [n - 1] \leq 0$ .

The leading minor of order $〈value〉$ is not positive definite, the factorization could not be completed.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

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 function, nag_zpttrs (f07jsc) can be used to solve systems of equations

A X = B

, and nag_zptcon (f07juc) can be used to estimate the condition number of

A

8 Parallelism and Performance

Not applicable.

9 Further Comments

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

A

is proportional to

n

The real analogue of this function is nag_dpttrf (f07jdc).

10 Example

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}) .

NAG Library Function Documentnag_zpttrf (f07jrc)