f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zpttrf (f07jrc)

## 1  Purpose

nag_zpttrf (f07jrc) computes the modified Cholesky factorization of a complex $n$ by $n$ Hermitian positive definite tridiagonal matrix $A$.

## 2  Specification

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

## 3  Description

nag_zpttrf (f07jrc) factorizes the matrix $A$ as
 $A=LDLH ,$
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 ${U}^{\mathrm{H}}DU$, where $U$ is a unit upper bidiagonal matrix.

None.

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     d[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 $LD{L}^{\mathrm{H}}$ factorization of $A$.
3:     e[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: is overwritten by the $\left(n-1\right)$ subdiagonal elements of the lower bidiagonal matrix $L$. (e can also be regarded as containing the $\left(n-1\right)$ superdiagonal elements of the upper bidiagonal matrix $U$.)
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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 ${\mathbf{d}}\left[{\mathbf{n}}-1\right]\le 0$.
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite, the factorization could not be completed.

## 7  Accuracy

The computed factorization satisfies an equation of the form
 $A+E=LDLH ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision.
Following the use of this function, nag_zpttrs (f07jsc) can be used to solve systems of equations $AX=B$, and nag_zptcon (f07juc) can be used to estimate the condition number of $A$.

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).

## 9  Example

This example factorizes the Hermitian positive definite tridiagonal matrix $A$ given by
 $A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0 .$

### 9.1  Program Text

Program Text (f07jrce.c)

### 9.2  Program Data

Program Data (f07jrce.d)

### 9.3  Program Results

Program Results (f07jrce.r)