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_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.
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.

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)