F07HDF (DPBTRF) (PDF version)
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F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HDF (DPBTRF)

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

F07HDF (DPBTRF) computes the Cholesky factorization of a real symmetric positive definite band matrix.

## 2  Specification

 SUBROUTINE F07HDF ( UPLO, N, KD, AB, LDAB, INFO)
 INTEGER N, KD, LDAB, INFO REAL (KIND=nag_wp) AB(LDAB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpbtrf.

## 3  Description

F07HDF (DPBTRF) forms the Cholesky factorization of a real symmetric positive definite band 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 $U$ (or $L$) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as $A$.

## 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:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{KD}}\ge 0$.
4:     AB(LDAB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ symmetric band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
On exit: the upper or lower triangle of $A$ is overwritten by the Cholesky factor $U$ or $L$ as specified by UPLO, using the same storage format as described above.
5:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HDF (DPBTRF) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
6:     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 $A$. There is no routine specifically designed to factorize a band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling F07BDF (DGBTRF) or as a full matrix, by calling F07MDF (DSYTRF).

## 7  Accuracy

If ${\mathbf{UPLO}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤ck+1εUTU ,$
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision.
If ${\mathbf{UPLO}}=\text{'L'}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

## 8  Further Comments

The total number of floating point operations is approximately $n{\left(k+1\right)}^{2}$, assuming $n\gg k$.
A call to F07HDF (DPBTRF) may be followed by calls to the routines:
• F07HEF (DPBTRS) to solve $AX=B$;
• F07HGF (DPBCON) to estimate the condition number of $A$.
The complex analogue of this routine is F07HRF (ZPBTRF).

## 9  Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .$

### 9.1  Program Text

Program Text (f07hdfe.f90)

### 9.2  Program Data

Program Data (f07hdfe.d)

### 9.3  Program Results

Program Results (f07hdfe.r)

F07HDF (DPBTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual