f07 Chapter Contents
f07 Chapter Introduction
NAG 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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{d}\left[\mathit{dim}\right]$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:    $\mathbf{e}\left[\mathit{dim}\right]$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:    $\mathbf{fail}$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.
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.
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 ${\mathbf{d}}\left[{\mathbf{n}}-1\right]\le 0$.
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite, the factorization could not be completed.
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
 $A+E=LDLH ,$
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 $A$.

## 8  Parallelism and Performance

Not applicable.

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
 $A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0 .$

### 10.1  Program Text

Program Text (f07jrce.c)

### 10.2  Program Data

Program Data (f07jrce.d)

### 10.3  Program Results

Program Results (f07jrce.r)