# NAG Library Routine Document

## 1Purpose

f07jrf (zpttrf) computes the modified Cholesky factorization of a complex $n$ by $n$ Hermitian positive definite tridiagonal matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f07jrf ( n, d, e, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*) Complex (Kind=nag_wp), Intent (Inout) :: e(*)
C Header Interface
#include nagmk26.h
 void f07jrf_ ( const Integer *n, double d[], Complex e[], Integer *info)
The routine may be called by its LAPACK name zpttrf.

## 3Description

f07jrf (zpttrf) 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.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension 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:     $\mathbf{e}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension 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:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}<{\mathbf{n}}$
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite, the factorization could not be completed.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}={\mathbf{n}}$
The leading minor of order $n$ is not positive definite, the factorization was completed, but ${\mathbf{d}}\left({\mathbf{n}}\right)\le 0$.

## 7Accuracy

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 routine, f07jsf (zpttrs) can be used to solve systems of equations $AX=B$, and f07juf (zptcon) can be used to estimate the condition number of $A$.

## 8Parallelism and Performance

f07jrf (zpttrf) is not threaded in any implementation.

## 9Further Comments

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).

## 10Example

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

### 10.1Program Text

Program Text (f07jrfe.f90)

### 10.2Program Data

Program Data (f07jrfe.d)

### 10.3Program Results

Program Results (f07jrfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017