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

NAG Library Routine Document

F07HDF (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.

 Contents

    1  Purpose
    7  Accuracy

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=UTU if UPLO='U' or A=LLT if UPLO='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 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

5  Arguments

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.
UPLO='U'
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
UPLO='L'
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     KD – INTEGERInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: KD0.
4:     ABLDAB* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least max1,N.
On entry: the n by n symmetric positive definite band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABkd+1+i-jj​ for ​max1,j-kdij;
  • if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj​ for ​jiminn,j+kd.
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: LDABKD+1.
6:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value.
If 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.
INFO>0
The leading minor of order 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).

7  Accuracy

If UPLO='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
Eck+1εUTU ,  
ck+1 is a modest linear function of k+1, and ε is the machine precision.
If UPLO='L', a similar statement holds for the computed factor L. It follows that eijck+1εaiiajj.

8  Parallelism 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.

9  Further Comments

The total number of floating-point operations is approximately n k+1 2, assuming nk.
A call to F07HDF (DPBTRF) may be followed by calls to the routines:
The complex analogue of this routine is F07HRF (ZPBTRF).

10  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 .  

10.1  Program Text

Program Text (f07hdfe.f90)

10.2  Program Data

Program Data (f07hdfe.d)

10.3  Program Results

Program Results (f07hdfe.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016