nag_dpbtrf (f07hdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dpbtrf (f07hdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbtrf (f07hdc) computes the Cholesky factorization of a real symmetric positive definite band matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpbtrf (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, NagError *fail)

3  Description

nag_dpbtrf (f07hdc) forms the Cholesky factorization of a real symmetric positive definite band matrix A either as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     kdIntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
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.
6:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
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 function 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 nag_dgbtrf (f07bdc) or as a full symmetric matrix, by calling nag_dsytrf (f07mdc).

7  Accuracy

If uplo=Nag_Upper, 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=Nag_Lower, a similar statement holds for the computed factor L. It follows that eijck+1εaiiajj.

8  Parallelism and Performance

nag_dpbtrf (f07hdc) is not threaded by NAG in any implementation.
nag_dpbtrf (f07hdc) 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 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 nag_dpbtrf (f07hdc) may be followed by calls to the functions:
The complex analogue of this function is nag_zpbtrf (f07hrc).

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 (f07hdce.c)

10.2  Program Data

Program Data (f07hdce.d)

10.3  Program Results

Program Results (f07hdce.r)

nag_dpbtrf (f07hdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014