# NAG Library Routine Document

## 1Purpose

f07hdf (dpbtrf) computes the Cholesky factorization of a real symmetric positive definite band matrix.

## 2Specification

Fortran Interface
 Subroutine f07hdf ( uplo, n, kd, ab, ldab, info)
 Integer, Intent (In) :: n, kd, ldab Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07hdf_ ( const char *uplo, const Integer *n, const Integer *kd, double ab[], const Integer *ldab, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dpbtrf.

## 3Description

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 $A$.

## 4References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn14.pdf
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{uplo}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kd}$ – IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – 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 positive definite 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:     $\mathbf{ldab}$ – 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:     $\mathbf{info}$ – 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.
If ${\mathbf{info}}=-999$, dynamic memory allocation failed. See Section 3.7 in How to Use the NAG Library and its Documentation for further information. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The leading minor of order $〈\mathit{\text{value}}〉$ 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 symmetric 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 symmetric matrix, by calling f07mdf (dsytrf).

## 7Accuracy

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 $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

## 8Parallelism and Performance

f07hdf (dpbtrf) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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).

## 10Example

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

### 10.1Program Text

Program Text (f07hdfe.f90)

### 10.2Program Data

Program Data (f07hdfe.d)

### 10.3Program Results

Program Results (f07hdfe.r)

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