F07GDF (DPPTRF) (PDF version)
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F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07GDF (DPPTRF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07GDF (DPPTRF) computes the Cholesky factorization of a real symmetric positive definite matrix, using packed storage.

## 2  Specification

 SUBROUTINE F07GDF ( UPLO, N, AP, INFO)
 INTEGER N, INFO REAL (KIND=nag_wp) AP(*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpptrf.

## 3  Description

F07GDF (DPPTRF) forms the Cholesky factorization of a real symmetric positive definite matrix $A$ either as $A={U}^{\mathrm{T}}U$ if ${\mathbf{UPLO}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{UPLO}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is lower triangular, using packed storage.

## 4  References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{UPLO}}=\text{'U'}$
The upper triangular part of $A$ is stored and $A$ is factorized as ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: the $n$ by $n$ symmetric matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.
4:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the leading minor of order $i$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. To factorize a matrix which is not positive definite, call F07PDF (DSPTRF) instead.

## 7  Accuracy

If ${\mathbf{UPLO}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤cnεUTU ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If ${\mathbf{UPLO}}=\text{'L'}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

## 8  Further Comments

The total number of floating point operations is approximately $\frac{1}{3}{n}^{3}$.
A call to F07GDF (DPPTRF) may be followed by calls to the routines:
• F07GEF (DPPTRS) to solve $AX=B$;
• F07GGF (DPPCON) to estimate the condition number of $A$;
• F07GJF (DPPTRI) to compute the inverse of $A$.
The complex analogue of this routine is F07GRF (ZPPTRF).

## 9  Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ,$
using packed storage.

### 9.1  Program Text

Program Text (f07gdfe.f90)

### 9.2  Program Data

Program Data (f07gdfe.d)

### 9.3  Program Results

Program Results (f07gdfe.r)

F07GDF (DPPTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual